# Łojasiewicz inequalities for maps of the 2-sphere

Alex Waldron  University of Wisconsin, Madison<EMAIL_ADDRESS>
###### Abstract.

We prove a _Łojasiewicz-Simon inequality_

$\left|E(u)-4\pi n\right|\leq C\|{\mathcal{T}}(u)\|^{\alpha}$

for maps $u\in W^{2,2}\left(S^{2},S^{2}\right).$ The inequality holds with
$\alpha=1$ in general and with $\alpha>1$ unless $u$ is nearly constant on an
open set. We obtain polynomial convergence of weak solutions of harmonic map
flow $u(t):S^{2}\to S^{2}$ as $t\to\infty$ on compact domains away from the
singular set, assuming that the body map is nonconstant. The proof uses
Topping’s repulsion estimates together with polynomial lower bounds on the
energy density coming from a bubble-tree induction argument.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Pre-Łojasiewicz estimates
  4. 4 Three-annulus and multi-annulus estimates
  5. 5 Bubble-tree estimates
  6. 6 Main theorems

##  1\. Introduction

###  1.1. Background and main result

Let $(M,g)$ and $(N,h)$ be compact Riemannian manifolds. Recall that a
sufficiently regular map $v:M\to N$ is called _harmonic_ if its tension field

${\mathcal{T}}(v)={\operatorname{tr}}_{g}\nabla dv$

vanishes identically. Equivalently, $v$ is a critical point of the Dirichlet
energy functional

$E(u)=\frac{1}{2}\int_{M}|du|^{2}\,dV.$

We are chiefly concerned with the fundamental case $M=N=S^{2},$ although some
of our results are stated more generally.

A classical result due to Lemaire and Wood [14, (11.5)] states that any
harmonic map $v:S^{2}\to S^{2}$ must either be holomorphic or antiholomorphic;
hence

(1.1) $E(v)=4\pi|\deg(v)|$

and $v$ attains the minimum allowable energy within its homotopy class. The
proof relies on a famous trick: since the Hopf differential (a certain
quadratic expression in the components of $du$) of a harmonic map is
holomorphic, it necessarily vanishes, forcing the map $v$ to be holomorphic or
antiholomorphic.

It is natural to ask whether a quantitative version of this result holds true;
specifically, whether the energy of a map with small tension must be close to
a predetermined value. In his thesis [30], Topping obtained a remarkable first
result in this direction: any map $u:S^{2}\to S^{2}$ with

(1.2) $E(u)<4\pi|\deg(u)|+\delta_{0},$

must obey the estimate

(1.3) $E(u)\leq 4\pi|\deg(u)|+C\|{\mathcal{T}}(u)\|^{2}.$

For initial data satisfying (1.2), Topping used (1.3) to prove that Struwe’s
weak solution of the harmonic map flow converges exponentially away from a
finite set of singular points.

Without the assumption (1.2), one can no longer expect a map with low tension
to be nearly holomorphic; instead, one might expect such a map to be close to
a _sum_ of holomorphic and anti-holomorphic maps. This intuition was borne out
by the work of Ding-Tian [12], Qing [21], and Wang [35], establishing that a
sequence of maps $u_{i}$ with $\|{\mathcal{T}}(u_{i})\|_{L^{2}}\to 0$ must
subconverge to a _bubble tree_ of harmonic maps. The reader may see Theorem
5.4 of the current paper for a detailed statement of this theory, or recent
work of Jendrej, Lawrie, and Schlag [15] for more precise results in the
parabolic setting.

In fact, the energies $E(u_{i})$ converge to the sum of the energies of the
maps in the tree, which by (1.1), is an integer multiple of $4\pi:$

$E(u_{i})\to 4\pi n.$

Another remarkable result of Topping [32] generalizes the estimate (1.3) to
this setting: under certain assumptions on the nature of the bubble tree
(described below), we have

(1.4) $|E(u_{i})-4\pi n|\leq C^{\prime}\|{\mathcal{T}}(u_{i})\|^{2}.$

Topping was able to use (1.4), which he calls a “quantization estimate,” to
again obtain exponential convergence of harmonic map flow on compact domains
away from the singular set.

The mechanism of Topping’s second theorem is more complex than the first,
incorporating a “repulsion” effect between the body map, assumed to be
holomorphic, and any antiholomorphic bubbles—in other words, a quantification
of the Hopf-differential trick. To ensure a sufficiently strong repulsion
effect, Topping makes the following assumptions: (1) holomorphic and
antiholomorphic bubbles do not occur at the same points, and (2) the energy
density of the body map does not vanish at any of the antiholomorphic bubble
points—see [32] for precise statements. As shown later [31], these hypotheses
are necessary for an estimate of the form (1.4) to hold.

However, if one settles for a weaker estimate in place of (1.4), namely, a
_Łojasiewicz(-Simon) inequality_

(1.5) $|E(u)-4\pi n|\leq C\|{\mathcal{T}}(u)\|^{\alpha},$

then it turns out that Topping’s assumptions can be removed. We shall prove
(1.5) with $\alpha=1$ in full generality, and with $\alpha>1$ assuming only
that the body map is not identically constant. This allows us to conclude that
for a solution of harmonic map flow $u(t):S^{2}\to S^{2},$ if the weak limit
along some sequence of times tending to infinity is nonconstant, then $u(t)$
converges polynomially on compact domains away from the singular set; in
particular, weak subsequential limits are unique in this case.

###  1.2. Related work

Our main theorem can be compared with Łojasiewicz inequalities and uniqueness-
of-subsequential-limit results that have appeared in similar contexts.

Assuming that $u$ is $C^{2,\alpha}$-close to a fixed harmonic map, (1.5)
follows directly from fundamental work of Leon Simon [27]. Historically,
extensions of Simon’s theorem to singular contexts have been exceedingly rare,
Topping’s theorems [30, 32] being the best and perhaps earliest examples.
Another instance is the work of Daskalopoulos and Wentworth [8, 9]
(generalized by Sibley [25] and Sibley-Wentworth [26]) establishing
convergence modulo gauge and uniqueness of the bubbling set for Hermitian
Yang-Mills flow on holomorphic bundles over compact Kähler manifolds. In his
thesis [34, Theorem 3.7-Corollary 3.8], the author also gave an exponential
convergence result for 4D Yang-Mills flow analogous to Topping’s first
convergence theorem, although in that context the bubbling set turned out to
be empty.

More recently, Malchiodi, Rupflin, and Sharp [19] obtained a Łojasiewicz
inequality for the $H$-functional that applies near “simple bubble trees,”
consisting of a single bubble on a constant body map (assuming that the domain
is a surface of genus at least one). Rupflin [22] extended this result to the
Dirichlet energy of harmonic maps and succeeded to work with a general real-
analytic target. Using similar estimates, Rupflin [23] has also obtained
Łojasiewicz inequalities for low-energy maps into 3-manifolds with generic
metrics. These results appear to be almost completely disjoint from those of
the present paper. (For recent progress on the related question of closeness
of $u$ to a genuine holomorphic map under the hypothesis (1.2), see Bernand-
Mantel, Muratov, and Simon, [1], Topping [33], and Rupflin [24].)

A related trend initiated by Colding and Minicozzi [5] aims to develop
Łojasiewicz-type estimates suitable for proving uniqueness of blowup limits at
“generic” singularities. These estimates are designed for analyzing type-I
singularities of the mean curvature flow, and have been remarkably successful
on that front [6, 7, 2, 3, 4]. Lotay, Schulze, and Szekelyhidi [18] have also
obtained a strong uniqueness result for certain singularities of Lagrangian
mean curvature flow, which are of type II. As we shall demonstrate in future
work, the inequality (1.5) can also be used to analyze (type-II) finite-time
singularities of 2D harmonic map flow.

Finally, we mention the recent Łojasiewicz inequality of Deruelle-Ozuch [10]
for a version of Perelman’s $\lambda$-functional near a Ricci-flat ALE space,
which has applications to infinite-time convergence of Ricci-DeTurck flow
[11].

###  1.3. Detailed statements

The simplest version of our main result is as follows.

######  Theorem 1.1.

Suppose that $u\in W^{2,2}(S^{2},S^{2})$ is a map with $E(u)\leq 4\pi k,$ for
$k\in{\mathbb{N}}.$

(a) There exists an integer $n\in\\{0,\ldots,k\\}$ such that

$\left|E(u)-4\pi n\right|\leq C_{\ref{thm:loj}a}\|{\mathcal{T}}(u)\|,$

where $C_{\ref{thm:loj}a}$ depends only on $k.$

(b) There exists $L\in{\mathbb{N}},$ depending only on $k,$ as follows. Let
$1<\alpha<\frac{2L+2}{2L+1}$ and $0<\kappa<{\varepsilon}_{0},$ depending on
$N.$ Suppose that there exist open sets $\Gamma\Subset\hat{\Gamma}\subset
S^{2}$ such that

$E(u,\hat{\Gamma})<{\varepsilon}_{0}\quad\text{and}\quad
E(u,\Gamma)\geq\kappa.$

We then have

$\left|E(u)-4\pi n\right|\leq
C_{\ref{thm:loj}b}\|{\mathcal{T}}(u)\|^{\alpha}.$

Here $C_{\ref{thm:loj}b}$ depends on $k,\kappa,\alpha,$ and the geometry of
$\Gamma$ and $\hat{\Gamma}.$

We also have the following more quantitative version.

######  Theorem 1.2.

Fix $\ell\in\\{0,\ldots,k\\},$ $\beta>0,$ and $\kappa>0.$ Let
$0<\lambda_{i}\leq\lambda\leq\frac{1}{\sqrt{\ell}}$ and $x_{i}\in S^{2},$ for
$i=1,\ldots,\ell,$ with $B_{2\lambda_{i}}(x_{i})\cap
B_{2\lambda_{j}}(x_{j})=\emptyset$ for $i\neq j,$ and let

$\Omega=S^{2}\setminus\cup_{i}B_{\lambda_{i}}(x_{i}),\qquad
U_{i}=U_{\lambda_{i}}^{2\lambda_{i}}(x_{i}),\qquad\hat{U}_{i}=U_{\lambda_{i}/2}^{4\lambda_{i}}(x_{i}).$

(a) Suppose that $u\in W^{2,2}(S^{2},S^{2})$ satisfies $E(u)\leq 4\pi k,$ as
well as

(1.6) $\max_{i}E\left(u,\hat{U}_{i}\right)\wedge
E_{{\bar{\partial}}}\left(u,\Omega\right)<{\varepsilon}_{0}$

and

(1.7) $\kappa E_{{\bar{\partial}}}\left(u,U_{i}\right)\leq
E_{{\partial}}\left(u,U_{i}\right)$

for $i=1,\ldots,\ell.$ There exists $n\in\\{0,\ldots,k\\}$ such that

$\left|E(u)-4\pi n\right|\leq
C_{\ref{thm:quantitativeloj}a}\left(\|{\mathcal{T}}(u)\|^{2}+\lambda^{1-\beta}\|{\mathcal{T}}(u)\|\right).$

Here, $C_{\ref{thm:quantitativeloj}a}$ depends on $k,\kappa,$ and $\beta.$

(b) Let $1<\alpha<\frac{2L+2}{2L+1},$ where $L\in{\mathbb{N}}$ depends on $k.$
In place of (1.7), assume

(1.8) $\kappa\leq\min_{i}E_{{\partial}}\left(u,U_{i}\right).$

We then have

$\left|E(u)-4\pi n\right|\leq
C_{\ref{thm:quantitativeloj}b}\|{\mathcal{T}}(u)\|^{\alpha}.$

Here $C_{\ref{thm:quantitativeloj}b}$ depends on $k,$ $\kappa,$ and $\alpha.$

Similar results hold after reversing the roles of $E_{\partial}$ and
$E_{\bar{\partial}}.$

Theorems 1.1 and 1.2 will both follow by contradiction from Theorem 6.1 below,
which states the same estimates in the context of sequences of maps with
tension fields tending to zero in $L^{2}$ (so-called “almost-harmonic
sequences”).

We can now state our harmonic-map-flow results.

######  Theorem 1.3.

There exists $\delta_{0}>0,$ depending on $k,\kappa,$ $\alpha,$ and
$\min\lambda_{i},$ as follows.

Let $u_{0}:S^{2}\to S^{2}$ be a map as in Theorem 1.2$b,$ satisfying (1.6) and
(1.8), and suppose further that

${\mathrm{dist}}\left(E(u_{0}),4\pi{\mathbb{Z}}\right)\leq\delta<\delta_{0},$

where $n\leq k.$ Let $u(t)$ be the Struwe solution of harmonic map flow with
$u(0)=u_{0},$ and let $0<T\leq\infty$ be the maximal time such that $u(t)$
satisfies

(1.9) ${\mathrm{dist}}\left(E(u(t)),4\pi{\mathbb{Z}}\right)<\delta$

for $0<t<T.$

(a) For $0\leq t<T,$ we have

${\mathrm{dist}}\left(E(u(t)),4\pi{\mathbb{Z}}\right)\leq\left(\delta^{\frac{\alpha-2}{\alpha}}+\frac{2-\alpha}{\alpha}\min\left[t,T-t\right]\right)^{\frac{\alpha}{\alpha-2}},$
$\int_{t}^{T-t}\|{\mathcal{T}}(u(s))\|_{L^{2}(S^{2})}\,ds\leq\frac{\alpha}{\alpha-1}\left(\delta^{\frac{\alpha-2}{\alpha}}+\frac{2-\alpha}{\alpha}t\right)^{\frac{\alpha-1}{\alpha-2}},$

and

$\|u(t)-u_{0}\|_{L^{2}(S^{2})}\leq\frac{\alpha}{\alpha-1}\delta^{\frac{\alpha-1}{\alpha}}.$

(b) Suppose $T=\infty.$ Then there exists a nonconstant holomorphic map
$u_{\infty}:S^{2}\to S^{2}$ such that

(1.10)
$\|u(t)-u_{\infty}\|_{L^{2}(S^{2})}\leq\frac{\alpha}{\alpha-1}\left(\delta^{\frac{\alpha-2}{\alpha}}+\frac{2-\alpha}{\alpha}t\right)^{\frac{\alpha-1}{\alpha-2}}$

Moreover, there exists a finite set of points $z_{j}\in S^{2}$ such that given
any domain $\Gamma\Subset S^{2}\setminus\\{z_{j}\\}$ and $m\in{\mathbb{N}},$
there exists $D>0$ such that

(1.11) $\|u(t)-u_{\infty}\|_{C^{m}(\Gamma)}<Dt^{\frac{\alpha-1}{\alpha-2}}$

for $t$ sufficiently large.

######  Corollary 1.4.

Let $t_{i}$ be any sequence of times tending to infinity. Given a Struwe
solution $u(t):S^{2}\to S^{2}$ of harmonic map flow with $E(u(0))\leq 4\pi k,$
pass to any subsequence such that $u(t_{i})\to u_{\infty},$ where $u_{\infty}$
is harmonic. If $u_{\infty}$ is nonconstant, then we have $u(t)\to u_{\infty}$
polynomially in $L^{2}(S^{2}),$ in the sense of (1.10), and in $C^{\infty}$ on
compact domains away from the bubbling set, in the sense of (1.11).

###  1.4. Idea of proof

We follow a stategy of decomposing an almost-harmonic map into an alternating
sum of almost-holomorphic and almost-antiholomorphic maps. The basic estimate
of Topping’s first theorem can be applied on each map, modulo a boundary term;
the goal is to make sure that the boundary terms are taken on annuli where the
holomorphic and antiholomorphic energies are comparable and bounded below by a
power of the radius. A key observation is that in this case, the boundary term
can be controlled using Topping’s repulsion estimate, giving precisely a
Łojasiewicz inequality.

The main difficulty in exploiting this observation is to obtain the required
lower (and upper) bounds on the holomorphic and antiholomorphic energies in
the neck regions. These ultimately follow from a basic three-annulus estimate
and its generalization to a “multi-annulus estimate,” needed to pass the lower
bounds across ghost bubbles; however, a tricky inductive argument over the
bubble tree is required to put the estimates together.

###  1.5. Acknowledgement

The author was partially suppored by NSF DMS-2004661 during the preparation of
this article.

##  2\. Preliminaries

In §2.1, we recall the Kähler formalism for harmonic maps. In §2.2, we recall
the explicit formulae for the quantities defined in §2.1 in the case
$\Sigma=N=S^{2},$ viewed as the unit sphere inside ${\mathbb{R}}^{3}.$ For the
purposes of understanding our main results, the reader may feel free to skip
§2.1 and refer only to the formulae in §2.2.

###  2.1. Harmonic maps to Kähler manifolds

Let $\Sigma$ be a Riemannian surface and $N$ a compact Kähler manifold of
complex dimension $n.$ The complexified tangent bundles decompose according to
type:

(2.1) $\begin{split}T\Sigma^{\mathbb{C}}=T^{1,0}\Sigma\oplus
T^{0,1}\Sigma,\qquad TN^{\mathbb{C}}=T^{1,0}N\oplus T^{0,1}N.\end{split}$

Given a local holomorphic coordinate $z$ on $\Sigma,$ we have the local
coordinate frames

$\frac{{\partial}}{{\partial}z}=\frac{1}{2}\left(\frac{{\partial}}{{\partial}x}-i\frac{{\partial}}{{\partial}y}\right),\qquad\frac{{\partial}}{{\partial}\bar{z}}=\frac{1}{2}\left(\frac{{\partial}}{{\partial}x}+i\frac{{\partial}}{{\partial}y}\right)$

for $T^{1,0}\Sigma$ and $T^{0,1}\Sigma,$ respectively, as well as the dual
frames

$dz=dx+idy,\qquad d\bar{z}=dx-idy$

for $T^{*}{}^{1,0}\Sigma$ and $T^{*}{}^{0,1}\Sigma,$ respectively. Given local
holomorphic coordinates $w^{\alpha},$ $\alpha=1,\ldots,n,$ on $N,$ we have
similar formulae for the frames $\frac{{\partial}}{{\partial}w^{\alpha}},$
$\frac{{\partial}}{{\partial}\bar{w}^{\alpha}},$ $dw^{\alpha},$ and
$d\bar{w}^{\alpha}.$

The metric tensor $g$ on $\Sigma$ may be extended complex-linearly to
$TN^{\mathbb{C}}.$ We obtain a hermitian metric by the formula

$\left\langle v,w\right\rangle=g(v,\bar{w}),$

which agrees with the real-valued metric on $TN\subset TN^{\mathbb{C}}.$ If
$z$ is a local coordinate, we may let

$\sigma(z)=\sqrt{2g\left(\frac{{\partial}}{{\partial}z},\frac{{\partial}}{{\partial}\bar{z}}\right)}.$

The metric tensor and Kähler form on $\Sigma$ are given locally by

$g=\frac{1}{2}\sigma^{2}\left(dz\otimes d\bar{z}+d\bar{z}\otimes
dz\right),\qquad\omega_{\Sigma}=\frac{i}{2}\sigma^{2}dz\wedge d\bar{z}.$

Similarly, we may write

$h_{\alpha\bar{\beta}}=h\left(\frac{{\partial}}{{\partial}w^{\alpha}},\frac{{\partial}}{{\partial}\bar{w}^{\beta}}\right),$

which is a Hermitian matrix. The Kähler form on $N$ then has the local
expression

$\omega_{N}=ih_{\alpha\bar{\beta}}dw^{\alpha}\wedge d\bar{w}^{\beta}.$

See e.g. [28, §2.4], for a few more details.

Now, given a map $u:\Sigma\to N,$ its differential $du$ is naturally a section
of $\mathcal{E}=T^{*}M\otimes u^{*}TN.$ The complexification
$\mathcal{E}^{\mathbb{C}}=T^{*}M^{\mathbb{C}}\otimes_{\mathbb{C}}\left(u^{*}TN\right)^{\mathbb{C}}$
decomposes into four factors corresponding to the direct sum decompositions
(2.1). The components of $du$ under this splitting can be written
schematically as

$du=\left(\begin{array}[]{cc}{\partial}u&{\bar{\partial}}u\\\\[5.69054pt]
\overline{{\bar{\partial}}u}&\overline{{\partial}u}\end{array}\right),$

where in local coordinates, we have

${\partial}u=w_{z}^{\alpha}dz\otimes\frac{{\partial}}{{\partial}w^{\alpha}},\qquad{\bar{\partial}}u=w^{\alpha}_{\bar{z}}d\bar{z}\otimes\frac{{\partial}}{{\partial}w^{\alpha}}.$

Here we have written
$w^{\alpha}_{z}=\frac{{\partial}w^{\alpha}(u(z))}{{\partial}z},$ etc. The
Dirichlet energy density of $u,$

$e(u)=\frac{1}{2}|du|^{2},$

decomposes under the above splitting as

$\begin{split}e(u)&=\frac{1}{2}\left(|{\partial}u|^{2}+|{\bar{\partial}}u|^{2}+|\overline{{\partial}u}|^{2}+|\overline{{\bar{\partial}}u}|^{2}\right)\\\
&=|{\partial}u|^{2}+|{\bar{\partial}}u|^{2}\\\
&=:e_{\partial}(u)+e_{\bar{\partial}}(u).\end{split}$

Letting

$E_{\partial}(u)=\int_{\Sigma}e_{\partial}(u)\,dV,\qquad
E_{\bar{\partial}}(u)=\int_{\Sigma}e_{{\bar{\partial}}}(u)\,dV,$

the Dirichlet energy splits as

$E(u)=E_{\partial}(u)+E_{\bar{\partial}}(u).$

Meanwhile, a brief calculation shows that

$u^{*}\omega_{N}=\left(e_{\partial}(u)-e_{\bar{\partial}}(u)\right)\omega_{\Sigma}.$

Since $d\omega_{N}=0,$ the pullback is again closed and its integral is
invariant under homotopy of $u.$ We conclude that

$E_{\partial}(u)-E_{\bar{\partial}}(u)=\int_{\Sigma}u^{*}\omega_{N}=:\kappa$

is a constant depending only on the homotopy class of $u.$ Rearranging, we
obtain

(2.2) $E(u)=4\pi\kappa+2E_{\bar{\partial}}(u).$

Note that if $N$ also has complex dimension one, then $\kappa=4\pi\deg(u).$

Recall that the tension field ${\mathcal{T}}(u)$ is the negative $L^{2}$
gradient of the Dirichlet functional. We can decompose ${\mathcal{T}}(u)$ as
follows:

${\mathcal{T}}(u)=\tau(u)+\overline{\tau(u)}\in u^{*}T^{1,0}N\oplus
u^{*}T^{0,1}N.$

In view of (2.2), $\tau(u)$ is half the $L^{2}$ gradient of the functional
$E_{\bar{\partial}}(u),$ which can also be computed by integration-by-parts.
This yields the formula

$\tau(u)=-{\bar{\partial}}_{u}^{*}{\bar{\partial}}u.$

Here,
${\bar{\partial}}_{u}:\Omega^{0,0}(u^{*}TN^{1,0})\to\Omega^{0,1}(u^{*}TN^{1,0})$
is the coupled ${\bar{\partial}}$-operator using the pullback of the Levi-
Civita connection on $N.$ Similarly, we have

$\tau(u)=-{\partial}_{u}^{*}{\partial}u.$

By the Kähler identities, we also have

$\tau(u)=-\Lambda{\partial}_{u}{\bar{\partial}}u=\Lambda{\bar{\partial}}_{u}{\partial}u.$

In local coordinates, we have the formula

$\tau(u)^{\alpha}=\sigma^{-2}(z)\left(w^{\alpha}_{z\bar{z}}+{}^{N}\Gamma^{\alpha}_{\beta\gamma}w^{\beta}_{z}w^{\gamma}_{\bar{z}}\right)$

and ${\mathcal{T}}(u)=2\operatorname{Re}(\tau(u)).$

We have the _Weitzenbock formula_ , for
$\alpha\in\Omega^{0,1}(u^{*}T^{1,0}N):$

$\left\langle\alpha,{\bar{\partial}}{\bar{\partial}}^{*}\alpha\right\rangle=\left\langle\alpha,\nabla^{*}\nabla\alpha\right\rangle+K_{\Sigma}|\alpha|^{2}+q_{1}({\partial}u,\alpha)+q_{2}({\bar{\partial}}u,\alpha),$

where

$\begin{split}q_{1}({\partial}u,\alpha)\stackrel{{\scriptstyle
loc}}{{=}}\sigma^{-4}{}^{N}R_{\beta\bar{\gamma}\delta\bar{\eta}}\left(\alpha_{\bar{z}}^{\beta}\overline{\alpha_{\bar{z}}^{\gamma}}w_{z}^{\delta}\overline{w_{z}^{\eta}}\right)\end{split}$

and

$\begin{split}q_{2}({\bar{\partial}}u,\alpha)\stackrel{{\scriptstyle
loc}}{{=}}-\sigma^{-4}{}^{N}R_{\beta\bar{\gamma}\delta\bar{\eta}}\left(\alpha_{\bar{z}}^{\beta}\overline{\alpha_{\bar{z}}^{\gamma}}w_{\bar{z}}^{\delta}\overline{w_{\bar{z}}^{\eta}}\right).\end{split}$

By the Kato inequality, we have

(2.3)
$\left\langle\alpha,{\bar{\partial}}{\bar{\partial}}^{*}\alpha\right\rangle\leq-2|\nabla\alpha|^{2}+\Delta|\alpha|^{2}+K_{\Sigma}|\alpha|^{2}+q_{1}({\partial}u,\alpha)+q_{2}({\bar{\partial}}u,\alpha),$

In the case that $N$ has nonnegative holomorphic bisectional curvature, the
$q_{1}$ term is nonpositive for general $\alpha,$ and we obtain

(2.4)
$\left\langle\alpha,{\bar{\partial}}{\bar{\partial}}^{*}\alpha\right\rangle\leq-2|\nabla\alpha|^{2}+\Delta|\alpha|^{2}+K_{\Sigma}|\alpha|^{2}+C_{2}|{\bar{\partial}}u|^{2}|\alpha|^{2}.$

Similar identities hold with ${\partial}u$ in place of ${\bar{\partial}}u.$ We
shall apply these inequalities below with $\alpha={\bar{\partial}}u,$ in §3.1,
and with $\alpha=\bar{f}(z){\bar{\partial}}u$ for a holomorphic function
$f(z),$ in §4.3.

###  2.2. The 2-sphere

The round 2-sphere $S^{2}\subset{\mathbb{R}}^{3}$ carries an integrable
almost-complex structure $I:TS^{2}\to TS^{2}$ derived from the cross-product
on ${\mathbb{R}}^{3}:$

$I_{u}(v)=u\times v,\quad v\perp u.$

This agrees with the complex multiplication on tangent vectors coming from the
identification $S^{2}\cong{\mathbb{C}}{\mathbb{P}}^{1},$ given by the
stereographic coordinate charts

$\begin{split}z&\mapsto\frac{1}{1+|z|^{2}}\left(2\operatorname{Re}z,2\operatorname{Im}z,1-|z|^{2}\right)\\\
w=z^{-1}&\mapsto\frac{1}{1+|w|^{2}}\left(2\operatorname{Re}w,-2\operatorname{Im}w,|w|^{2}-1\right).\end{split}$

A similar chart centered at any point in $S^{2}$ can be obtained by post-
composing with an ${\mathrm{SO}}(3)$ rotation. We have

$\sigma(z)=\frac{2}{1+|z|^{2}}$

and

$\mathrm{Vol}_{S^{2}}=\sigma^{2}\mathrm{Vol}_{{\mathbb{R}}^{2}}.$

Letting $z=x+iy,$ and viewing $u:{\mathbb{C}}\cong{\mathbb{R}}^{2}\to S^{2}\to
S^{2}\subset{\mathbb{R}}^{3}$ as a vector-valued function on
${\mathbb{R}}^{2},$ we have

$\begin{split}e_{\partial}(u)&=\frac{1}{4\sigma^{2}}|u_{x}-u\times
u_{y}|^{2}\\\ e_{\bar{\partial}}(u)&=\frac{1}{4\sigma^{2}}|u_{x}+u\times
u_{y}|^{2}\\\
{\mathcal{T}}(u)&=\frac{1}{\sigma^{2}}\left(\Delta_{{\mathbb{R}}^{2}}u+|\nabla_{{\mathbb{R}}^{2}}u|^{2}u\right)\\\
\|{\mathcal{T}}(u)\|_{L^{2}(S^{2})}&=\|\sigma{\mathcal{T}}(u)\|_{L^{2}({\mathbb{R}}^{2})}.\end{split}$

See Topping [32] for these formulae.

##  3\. Pre-Łojasiewicz estimates

###  3.1. ${\varepsilon}$-regularity

Given $x\in S^{2}$ and $0<r<\infty,$ let $D_{r}(x)$ denote the image of the
disk of radius $r$ in the stereographic coordinate chart centered at $x,$ i.e.

$D_{r}(x)=B_{2\arctan(r)}(x).$

We shall also write

$U^{\rho_{2}}_{\rho_{1}}(x)=D_{\rho_{2}}(x)\setminus\overline{D_{\rho_{1}}(x)}.$

In contrast with [32], by default, we will take norms with respect to the
metric and volume form of $S^{2}$ rather than with respect to the Euclidean
metric in the stereographic chart. For $D_{\rho}$ with $\rho\leq C,$ these are
equivalent up to constants, so the distinction can almost always be ignored.

######  Lemma 3.1.

There exists ${\varepsilon}_{0}>0,$ depending on the geometry of $N,$ as
follows. Let $\sigma>1$ and $\rho\leq 1,$ and put
$D=D_{\rho}(x),\hat{D}=D_{\sigma\rho}(x).$

(a) Given $u\in W^{2,2}(\hat{D},N)$ satisfying
$\rho^{2}\|{\mathcal{T}}(u)\|_{L^{2}(\hat{D})}\leq 1$ and

$E(u,\hat{D})<{\varepsilon}_{0},$

we have

$\rho^{2}\|du\|^{2}_{W^{1,2}\left(D\right)}\leq
C_{\sigma}\left(E(u,\hat{D})+\rho^{2}\|{\mathcal{T}}(u)\|_{L^{2}(\hat{D)})}^{2}\right).$

(b) Supposing that $N$ is Kähler, we also have

$\rho^{2}\|{\partial}u\|^{2}_{W^{1,2}\left(D\right)}\leq
C_{\sigma}\left(E_{\partial}(u,\hat{D})+\rho^{2}\|{\mathcal{T}}(u)\|_{L^{2}(\hat{D})}^{2}\right)$

and

$\rho^{2}\|{\bar{\partial}}u\|^{2}_{W^{1,2}\left(D\right)}\leq
C_{\sigma}\left(E_{\bar{\partial}}(u,\hat{D})+\rho^{2}\|{\mathcal{T}}(u)\|_{L^{2}(\hat{D})}^{2}\right).$

The same statements hold after replacing $D$ and $\hat{D}$ by annuli
$U_{R}^{\sigma R}(x)$ and $U_{\sigma^{-1}R}^{\sigma^{2}R}(x),$ respectively.

###### Proof.

The proof of ($a$) is a minor adaptation of the proof by Ding-Tian [12, Lemma
2.1], which we omit.

To prove (b), we apply (2.3) with $\alpha={\bar{\partial}}u,$ to get

$\left\langle{\bar{\partial}}u,{\bar{\partial}}\tau(u)\right\rangle\leq-2|\nabla{\bar{\partial}}u|^{2}+\Delta|{\bar{\partial}}u|^{2}+\left(K_{\Sigma}+C|\nabla
u|^{2}\right)|{\bar{\partial}}u|^{2}.$

The result follows by a similar argument. ∎

######  Lemma 3.2.

Suppose that $N$ is Kähler. Let $\frac{4}{3}\leq q\leq 2,$ $K\geq 1,$
$0<\rho\leq 1,$ and $x_{0}\in S^{2}.$ Let $U=U_{\rho}^{2\rho}(x_{0})$ and
$\hat{U}=U_{\rho/2}^{4\rho}(x_{0})\subset S^{2}.$ Suppose that
$\rho^{2}\|{\mathcal{T}}(u)\|\leq 1,$ $E(u,\hat{U})<{\varepsilon}_{0},$ and

(3.1)
$E_{\bar{\partial}}(u,\hat{U})\wedge\rho^{2}\|{\mathcal{T}}(u)\|^{2}_{L^{2}\left(\hat{U}\right)}\leq
KE_{\bar{\partial}}(u,U).$

Let $V\subset U$ be any measurable set with

(3.2) $|V|\geq\left(1-(2CK)^{-2}\right)|U|,$

with $C$ sufficiently large. We then have

$E_{\bar{\partial}}(u,U)\leq
CK\rho^{2-\frac{4}{q}}\|e_{{\bar{\partial}}}\|_{L^{\frac{q}{2}}(V)}.$

###### Proof.

We may assume without loss of generality that $\rho=1,$ since the general
statement follows by scaling invariance. Let

$p=\frac{q^{*}}{2}=\frac{q}{2(q-1)}\in\left[1,2\right].$

By Lemma 3.1 and the Sobolev inequality, we have

(3.3) $\|e_{\partial}\|_{L^{2}(U)}=\|{\bar{\partial}}u\|^{2}_{L^{4}(U)}\leq
C\left(E_{\bar{\partial}}(u,\hat{U})+\|{\mathcal{T}}(u)\|^{2}_{L^{2}\left(\hat{U}\right)}\right)\leq
CKE_{\bar{\partial}}(u,U),$

where we have applied the assumption (3.1). Then (3.3) and Hölder’s inequality
give

(3.4) $\|e_{{\bar{\partial}}}\|_{L^{p}(U)}\leq
C\|e_{\bar{\partial}}\|_{L^{2}(U)}\leq CKE_{\bar{\partial}}(u,U).$

We have

$\begin{split}E_{\bar{\partial}}(u,U)&=E_{\bar{\partial}}(u,V^{c})+E_{\bar{\partial}}(u,V).\end{split}$

Applying Hölder’s inequality on the first term and the interpolation
inequality on the second, we obtain

$\begin{split}E_{\bar{\partial}}(u,U)&\leq|V^{c}|^{\frac{1}{2}}\|e_{{\bar{\partial}}}\|_{L^{2}(U)}+\|e_{\bar{\partial}}\|_{L^{p}(V)}^{\frac{1}{2}}\|e_{{\bar{\partial}}}\|_{L^{\frac{q}{2}}(V)}^{\frac{1}{2}}.\end{split}$

By (3.2), we have

(3.5) $|V^{c}|^{\frac{1}{2}}\leq\frac{1}{2CK}.$

Inserting (3.4-3.5), we obtain

$\begin{split}E_{\bar{\partial}}(u,U)&\leq\frac{1}{2}E_{\bar{\partial}}(u,U)+\left(CKE_{\bar{\partial}}(u,U)\right)^{\frac{1}{2}}\|e_{{\bar{\partial}}}\|_{L^{\frac{q}{2}}(V)}^{\frac{1}{2}}.\end{split}$

Rearranging and cancelling exponents, we have the desired estimate. ∎

######  Lemma 3.3.

Given $K\geq 1,$ there exist ${\varepsilon}_{2}>0$ (depending on $K,N$) and
$c_{1}>0$ (depending on $K$) as follows. Suppose that $u:\hat{U}\to N$
satisfies

$E(u,\hat{U})<{\varepsilon}_{2},$ (3.6)
$\rho^{2}\|{\mathcal{T}}(u)\|^{2}_{L^{2}(\hat{U})}\leq{\varepsilon}_{2}E_{\partial}(u,U),$

and

(3.7) $E_{\partial}(u,\hat{U})\leq KE_{\partial}(u,U).$

Then

($a$) $c_{1}\sup_{\rho\leq r\leq
2\rho}\int_{S^{1}_{r}}e_{\partial}\,d\theta\leq\inf_{\rho\leq r\leq
2\rho}\int_{S^{1}_{r}}e_{\partial}\,d\theta$

and

($b$) $\mathrm{Area}\left\\{x\in U\mid\rho^{2}e_{\partial}(u)(x)\leq
c_{1}E_{\partial}(u,U)\right\\}\leq(2CK)^{-2}\mathrm{Area}(U).$

###### Proof.

This will follow by applying a standard contradiction argument to the
$(1,0)$-form ${\partial}u.$ We may assume $\rho=1$ without loss of generality.
We view $U$ and $\hat{U}$ as subsets of the ball $B_{4}(0)\subset{\mathbb{C}}$
by stereographic projection.

First let $\alpha=f(z)dz$ be a ${\mathbb{C}}$-valued holomorphic 1-form on
$U^{\prime}=U^{3\rho}_{2\rho/3},$ satistfying

${\bar{\partial}}\alpha=0,\quad\int_{U}|\alpha|^{2}=1,\,\,\,\,\text{and}\,\,\,\,\int_{U^{\prime}}|\alpha|^{2}\leq
K.$

By elementary complex analysis, $\alpha$ must obey estimates of the form

(3.8) $2c_{1}\sup_{\rho\leq r\leq
2\rho}\int_{S^{1}_{r}}|\alpha|^{2}\,d\theta\leq\inf_{\rho\leq r\leq
2\rho}\int_{S^{1}_{r}}|\alpha|^{2}\,d\theta$

and

$\mathrm{Area}\left\\{x\in U\mid|\alpha(x)|^{2}\leq
2c_{1}\right\\}\leq(2CK)^{-2}.$

The same statements hold for a holomorphic 1-form valued in a flat holomorphic
bundle over $U^{\prime}.$

We can now prove ($a$) by contradiction. Assume that $u_{i}:\hat{U}\to N$ is a
sequence of nonconstant maps such that

(3.9) $E(u_{i},\hat{U})\leq\frac{1}{i},$ $E_{\partial}(u_{i},\hat{U})\leq
KE_{\partial}(u_{i},U),$

and

(3.10)
$\|{\mathcal{T}}(u_{i})\|^{2}_{L^{2}(\hat{U})}\leq\frac{K}{i}E_{\partial}(u_{i},U),$

but for which

(3.11) $c_{1}\sup_{\rho\leq r\leq
2\rho}\int_{S^{1}_{r}}|{\bar{\partial}}u_{i}|^{2}\,d\theta\geq\inf_{\rho\leq
r\leq 2\rho}\int_{S^{1}_{r}}|{\bar{\partial}}u_{i}|^{2}\,d\theta.$

Let $U\Subset U^{\prime}\Subset\hat{U}.$ By Lemma 3.1$b$ and (3.10), we have

(3.12) $\|{\partial}u_{i}\|^{2}_{W^{1,2}(U^{\prime})}\leq
C\left(1+\frac{K}{i}\right)E_{{\partial}}(u_{i},\hat{U})\leq
CE_{{\partial}}(u_{i},U).$

Let

$\alpha_{i}=\frac{{\partial}u_{i}}{\sqrt{E_{\partial}(u,U)}}.$

We then have

$\int_{U}|\alpha_{i}|^{2}=1$ $\int_{\hat{U}}|\alpha_{i}|^{2}\leq K$
$\int_{\hat{U}}|{\bar{\partial}}_{u_{i}}\alpha_{i}|^{2}=\frac{\int|{\mathcal{T}}(u_{i})|^{2}}{E_{\partial}(u,U)}\leq\frac{K}{i}\to
0,$

and, by (3.12),

$\|\alpha_{i}\|_{W^{1,2}(U^{\prime})}\leq C.$

We may pass to a weak limit in $W^{1,2}(U^{\prime}),$

$\alpha_{i}\rightharpoonup\alpha,$

which again satisfies

(3.13) $\begin{split}\int_{U}|\alpha|^{2}&=1,\\\
\int_{U^{\prime}}|\alpha|^{2}&\leq K.\end{split}$

By (3.9-3.10) and Lemma 3.1, we have $du_{i}\to 0$ strongly in
$W^{1,2}(U^{\prime}).$ Passing to a subsequence, we may assume that the images
$u_{i}\left(U^{\prime}\right)$ are contained in a fixed coordinate ball
$B\subset N.$ We then have

${\bar{\partial}}_{u_{i}}\alpha_{i}={\bar{\partial}}\alpha_{i}+{}^{N}\Gamma(du_{i},du_{i})\\#\alpha_{i}\to
0,$

where ${\bar{\partial}}$ is the trivial ${\bar{\partial}}$-operator in $B.$
Since $du_{i}\to 0$ in $W^{1,2},$ and in particular in $L^{p}$ for all $p,$
while $\alpha_{i}$ is also bounded in $L^{p}$ for all $p,$ we have

${\bar{\partial}}\alpha_{i}\to 0$

in $L^{2},$ hence ${\bar{\partial}}\alpha=0.$ Moreover, since $\alpha_{i}$
converges weakly in $W^{1,2}(U^{\prime}),$
$\sqrt{\int_{S^{1}_{r}}|\alpha_{i}|^{2}\,d\theta}$ converges weakly in
$W^{1,2}(\left[\rho,2\rho\right])$ as a function of $r,$ and in particular
strongly in $C^{0}(\left[\rho,2\rho\right]),$ so (3.11) is preserved. But
together with (3.13), this contradicts (3.8), establishing ($a$).

The proof of ($b$) is similar. ∎

###  3.2. Topping’s estimates

The following basic estimates are due to Topping [32]. Throughout this
subsection and the next, we suppose $\Sigma=N=S^{2}.$

######  Proposition 3.4 ([32], Lemma 2.5b).

For $1\leq q<2,$ assuming $E(u)\leq 4\pi k,$ we have

$\|\sqrt{e_{\partial}(u)e_{\bar{\partial}}(u)}\|_{L^{q}(S^{2})}\leq
C_{q}\sqrt{k}\|{\mathcal{T}}(u)\|_{L^{2}(S^{2})}.$

######  Proposition 3.5 ([32], Lemma 2.3d).

Let $0<\rho\leq R<\infty,$ and assume that $D_{\rho_{i}}(x_{i})\subset
D_{\rho}(x_{0})$ for each $i.$ Let

$\Omega^{\prime}=D_{\rho}(x_{0})\setminus\cup_{i}\bar{D}_{\rho_{i}}(x_{i}),\quad\hat{\Omega}^{\prime}=D_{2R}(x_{0})\setminus\cup_{i}\bar{D}_{\rho_{i}/2}(x_{i}).$

If
$E_{{\bar{\partial}}}(u,D_{R}(x_{0}))<{\varepsilon}_{1}={\varepsilon}_{1}(\beta),$
then we have

$\begin{split}&E_{\bar{\partial}}(u,\Omega^{\prime})\\\ &\leq
C_{\ell,\beta}\left(\frac{\rho}{R}\right)^{\beta}\left(R^{2}\|{\mathcal{T}}(u)\|_{L^{2}\left(\hat{\Omega}^{\prime}\right)}^{2}+E_{\bar{\partial}}\left(u,U^{2R}_{R}(x_{0})\right)+\sum_{i=1}^{\ell}\left(\frac{R}{\rho_{i}}\right)^{-\beta}E_{\bar{\partial}}(u,U^{\rho_{i}}_{\rho_{i}/2}(x_{i}))\right).\end{split}$

###### Proof.

After homothetically rescaling so that $R=1,$ the $E_{\bar{\partial}}$ terms
are unchanged and the $R^{2}\|{\mathcal{T}}(u)\|^{2}$ term decreases; hence,
we may assume without loss of generality that $R=1.$ Letting $\mu=1$ in
Topping’s notation, the proof of [32, Lemma 2.3d] on p. 486 gives

$\begin{split}E_{\bar{\partial}}(u,\Omega^{\prime})&\leq
C\rho^{\frac{2(q-1)}{q}}\|u\|_{L^{\frac{2q}{2-q}}(\hat{\Omega}^{\prime})}\\\
&\leq
C_{q}\rho^{\frac{2(q-1)}{q}}\left(\|{\mathcal{T}}(u)\|_{L^{2}\left(\hat{\Omega}^{\prime}\right)}^{2}+\|u\|_{L^{q}(U^{2}_{1})}+\sum_{i=1}^{\ell}\frac{1}{\rho_{i}}\|u\|_{L^{q}(U^{\rho_{i}}_{\rho_{i}/2}(x_{i}))}\right).\end{split}$

The proof proceeds by applying Hölder’s inequality on the remaining terms, as
in [32], and letting $\beta=\frac{4(q-1)}{q}.$ ∎

######  Proposition 3.6.

Assume $0<\rho_{i}\leq\frac{1}{\sqrt{\ell}}$ for $i=1,\ldots,\ell.$ Write

$\Omega=S^{2}\setminus\cup_{i}D_{\rho_{i}}(x_{i}),\quad\hat{\Omega}=S^{2}\setminus\cup_{i}D_{\rho_{i}/2}(x_{i}),$
$U_{i}=U^{2\rho_{i}}_{\rho_{i}}(x_{i}),\quad\hat{U}_{i}=U^{4\rho_{i}}_{\rho_{i}/2}(x_{i}).$

Supposing that $E_{\bar{\partial}}(u,\hat{\Omega})<{\varepsilon}_{0},$ we have

$E_{\bar{\partial}}(u,\Omega)\leq
C_{\ell,\beta}\left(\|{\mathcal{T}}(u)\|_{L^{2}(\hat{\Omega})}^{2}+\sum\rho_{i}^{-\beta}E_{\bar{\partial}}(u,U^{\rho_{i}}_{\rho_{i}/2}(x_{i}))\right).$

###### Proof.

Let

$R=\max\left\\{\sqrt{\ell},2(4C_{\ell,\beta})^{\frac{1}{\beta}}\right\\},$

where $C_{\ell,\beta}$ is the constant of the previous Proposition. Since
$\rho_{i}\leq\frac{1}{\sqrt{\ell}},$ there must exist two antipodal points
$x_{0},\hat{x}_{0}\in S^{2}$ such that

$D_{1/\sqrt{\ell}}(x_{0})\cap
D_{\rho_{i}}(x_{i})=\emptyset=D_{1/\sqrt{\ell}}(-x_{0})\cap
D_{\rho_{i}}(x_{i})$

for $i=1,\ldots,\ell.$ We apply the previous proposition twice, with $\rho=2$
and $R$ as above, to obtain

$\begin{split}E_{\bar{\partial}}(u,\Omega\cap
D_{2}(x_{0}))\leq\frac{1}{4}\left(R^{2}\|{\mathcal{T}}(u)\|_{L^{2}(S^{2})}^{2}+E_{\bar{\partial}}\left(u,U^{2R}_{R}(x_{0})\right)+\sum_{i=1}^{k}\left(\frac{R}{\rho_{i}}\right)^{\beta}E_{\bar{\partial}}(u,U^{\rho_{i}}_{\rho_{i}/2}(x_{i}))\right),\end{split}$
$\begin{split}E_{\bar{\partial}}(u,\Omega\cap
D_{2}(\hat{x}_{0}))\leq\frac{1}{4}\left(R^{2}\|{\mathcal{T}}(u)\|_{L^{2}(S^{2})}^{2}+E_{\bar{\partial}}\left(u,U^{2R}_{R}(\hat{x}_{0})\right)+\sum_{i=1}^{\ell}\left(\frac{R}{\rho_{i}}\right)^{\beta}E_{\bar{\partial}}(u,U^{\rho_{i}}_{\rho_{i}/2}(x_{i}))\right).\end{split}$

Since $R^{2}\geq\ell,$ we have $U^{2R}_{R}(x_{0})\subset
D_{\frac{1}{\sqrt{\ell}}}(\hat{x}_{0}),$ so
$U^{2R}_{R}(x_{0})\subset\Omega\cap D_{2}(\hat{x}_{0});$ also
$U^{2R}_{R}(\hat{x}_{0})\subset D_{\frac{1}{k}}(x_{0}),$ so
$U^{2R}_{R}(\hat{x}_{0})\subset\Omega\cap D_{2}(x_{0}).$ Therefore

$E_{\bar{\partial}}\left(u,U^{2R}_{R}(\hat{x}_{0})\right)+E_{\bar{\partial}}\left(u,U^{2R}_{R}(x_{0})\right)\leq
E_{\bar{\partial}}(u,\Omega).$

We may add the above two inequalities together and rearrange, to obtain

$\begin{split}&E_{\bar{\partial}}(u,\Omega)\leq
R^{2}\|{\mathcal{T}}(u)\|_{L^{2}(S^{2})}^{2}+\sum_{i=1}^{\ell}\left(\frac{R}{\rho_{i}}\right)^{\beta}E_{\bar{\partial}}(u,U^{\rho_{i}}_{\rho_{i}/2}(x_{i})).\end{split}$

Finally, we absorb $R$ into the constant to obtain the desired estimate. ∎

######  Lemma 3.7 ([32], Lemma 2.23).

Let $\Omega,\hat{\Omega},$ and $U_{i}$ be as above.

(a) Suppose that $\|{\mathcal{T}}(u)\|\leq 1$ and

$\sum_{i=1}^{\ell}E(u,U_{i})+E_{\bar{\partial}}(u,\Lambda)\leq{\varepsilon}<{\varepsilon}_{0}.$

Then

${\mathrm{dist}}\left(E(u,\hat{\Lambda}),{\mathbb{Z}}\right)\leq
C{\varepsilon}.$

(b) Suppose further that $\cup D_{2\rho_{i}}(x_{i})\subset
D_{\rho_{0}}(x_{0}),$ and let $\Omega^{\prime},\hat{\Omega}^{\prime}$ be as
above. If $\|{\mathcal{T}}(u)\|\leq\rho_{0}^{-1}$ and

$\sum_{i=0}^{\ell}E(u,U_{i})+E_{\bar{\partial}}(u,\Lambda^{\prime})\leq{\varepsilon}<{\varepsilon}_{0},$

then

${\mathrm{dist}}\left(E(u,\hat{\Lambda}^{\prime}),{\mathbb{Z}}\right)\leq
C{\varepsilon}.$

###  3.3. Pre-Łojasiewicz estimates

######  Theorem 3.8 (Pre-Łojasiewicz estimate).

Given $k\in{\mathbb{N}},$ $K\geq 1,$ and $0<\beta<\frac{1}{2},$ there exists
${\varepsilon}_{2}>0$ as follows.

Let $x_{i}\in S^{2}$ and $0<\rho_{i}\leq\lambda\leq\frac{\pi}{2},$ for
$i=1,\ldots,\ell,$ with $\ell\leq k,$ be such that $D_{2\rho_{i}}(x_{i})\cap
D_{2\rho_{j}}(x_{j})=\emptyset$ for $i\neq j.$ Write
$U_{i}=U^{2\rho_{i}}_{\rho_{i}}(x_{i}),$
$\hat{U}_{i}=U^{4\rho_{i}}_{\rho_{i}/2}(x_{i}),$

$\Omega=S^{2}\setminus\cup_{i}D_{\rho_{i}}(x_{i}),$

and

$\hat{\Omega}=S^{2}\setminus\cup_{i}D_{\rho_{i}/2}(x_{i}).$

($a$) Suppose that $u\in W^{2,2}(S^{2},S^{2})$ satisfies

$E(u)\leq 4\pi k,$ $\|{\mathcal{T}}(u)\|\leq 1,$ (3.14)
$\max_{i}E(u,\hat{U}_{i})\wedge
E_{\bar{\partial}}(u,\hat{\Omega})<{\varepsilon}_{2},$

and, for some $M\geq 1$ and $i=1,\ldots,\ell,$

(3.15) $E_{\partial}(u,\hat{U}_{i})\leq
KE_{{\partial}}(u,U_{i})+M\rho_{i}^{2}\|{\mathcal{T}}(u)\|_{L^{2}(\hat{U}_{i})}^{2},$
(3.16) $E_{\bar{\partial}}(u,\hat{U}_{i})\leq
KE_{{\bar{\partial}}}(u,U_{i})+M\rho_{i}^{2}\|{\mathcal{T}}(u)\|_{L^{2}(\hat{U}_{i})}^{2},$

and

(3.17) $E_{{\bar{\partial}}}(u,U_{i})\leq
M\left(E_{\partial}(u,U_{i})+\rho_{i}^{2}\|{\mathcal{T}}(u)\|_{L^{2}(\hat{U}_{i})}^{2}\right).$

Then

$\sum_{i=1}^{\ell}\rho_{i}^{-\beta}E\left(u,U_{i}\right)+{\mathrm{dist}}\left(E(u,\Omega),{\mathbb{Z}}\right)\leq
C_{\ref{thm:preloj}a}\left(\|{\mathcal{T}}(u)\|^{2}+\lambda^{1-2\beta}\|{\mathcal{T}}(u)\|\right).$

($b$) Let $m\geq 2$ and

$1<\alpha=\frac{m+\beta}{m-1+3\beta}<2.$

Suppose further that

(3.18) $\rho_{j}^{m+2\beta}\leq
M\left(E_{\partial}(u,U_{j})+\rho_{j}^{2}\|{\mathcal{T}}\|^{2}\right),$

where $\rho_{j}=\max_{i}\rho_{i}.$ We then have

$\max_{i}\rho_{i}^{m-1+3\beta}\leq C_{\ref{thm:preloj}b}\|{\mathcal{T}}\|$

and

$\sum_{i=1}^{\ell}\rho_{i}^{-\beta}E\left(u,U_{i}\right)+{\mathrm{dist}}\left(E(u,\Omega),{\mathbb{Z}}\right)\leq
C_{\ref{thm:preloj}b}\|{\mathcal{T}}(u)\|^{\alpha}.$

Here $C_{\ref{thm:preloj}a-b}$ depend on $k,K,M,$ and $\beta.$

###### Proof.

Let ${\varepsilon}_{2}$ be the minimum of the constants ${\varepsilon}_{i}$
from Lemma 3.1 and Propositions 3.5-3.6, and let $c_{1}$ be the constant of
Lemma 3.3 corresponding to the given $K.$ Let $q=\frac{2}{1+\beta},$ so that
$\frac{4}{3}\leq q<2$ and

$-\beta=1-\frac{2}{q},\qquad 1-2\beta=3-\frac{4}{q},\qquad
2-\beta=3-\frac{2}{q}.$

($a$) We will first prove

(3.19)
$\lambda^{3-\frac{4}{q}}\|{\mathcal{T}}\|+\lambda^{3-\frac{2}{q}}\|{\mathcal{T}}\|^{2}\gtrsim\sum_{i}\rho_{i}^{1-\frac{2}{q}}E_{\bar{\partial}}(u,U_{i}).$

Let

$J=\left\\{i\in\\{1,\ldots,\ell\\}\mid\rho_{i}^{2}\|{\mathcal{T}}(u)\|^{2}_{L^{2}\left(\hat{U}_{i}\right)}\leq\frac{{\varepsilon}_{2}}{M^{2}}E_{\bar{\partial}}(u,U_{i})\right\\}.$

Summing over $i\in J^{c},$ we have

(3.20) $\sum_{i\in
J^{c}}\rho_{i}^{1-\frac{2}{q}}E_{\bar{\partial}}(u,U_{i})\leq\lambda^{3-\frac{2}{q}}\sum_{i\in
J^{c}}\rho_{i}^{-2}E_{\bar{\partial}}(u,U_{i})\leq\frac{M^{2}\lambda^{3-\frac{2}{q}}}{{\varepsilon}_{2}}\|{\mathcal{T}}(u)\|^{2}.$

On the other hand, for $i\in J,$ we have

$\rho_{i}^{2}\|{\mathcal{T}}(u)\|^{2}_{L^{2}\left(\hat{U}_{i}\right)}\leq\frac{{\varepsilon}_{2}}{M^{2}}E_{\bar{\partial}}(u,U_{i}).$

Combined with (3.15-3.17), this gives the conditions (3.1) and (3.6-3.7).

Let

$V_{i}=\left\\{x\in U_{i}\mid\rho_{i}^{2}e_{\partial}(u)(x)\geq
c_{1}E_{\partial}(u,U_{i})\right\\}.$

By Lemma 3.3, we have

$|V_{i}|\geq\left(1-(CK)^{-2}\right)|U_{i}|.$

By Lemma 3.2, we have

$E_{\bar{\partial}}(u,U_{i})\leq
CK\rho_{i}^{2-\frac{4}{q}}\|e_{{\bar{\partial}}}\|_{L^{\frac{q}{2}}(V_{i})}.$

Applying Proposition 3.4, we obtain

(3.21)
$\begin{split}Ck\|{\mathcal{T}}\|^{2}&\geq\|e_{\partial}e_{\bar{\partial}}\|_{L^{\frac{q}{2}}(S^{2})}\\\
&\geq\frac{1}{\ell^{\frac{q}{2}}}\sum_{i}\|e_{\partial}e_{\bar{\partial}}\|_{L^{\frac{q}{2}}(V_{i})}\\\
&\geq\frac{c_{1}}{\ell^{\frac{q}{2}}}\sum_{i}\rho_{i}^{-2}E_{{\partial}}(u,U_{i})\|e_{\bar{\partial}}\|_{L^{\frac{q}{2}}(V_{i})}\\\
&\geq\frac{c_{1}}{MK\ell^{\frac{q}{2}}}\sum_{i}\rho_{i}^{\frac{4}{q}-4}E_{{\bar{\partial}}}(u,U_{i})^{2}\\\
&\geq\frac{c_{1}}{MK\ell^{\frac{q}{2}+1}}\left(\sum_{i}\rho_{i}^{\frac{2}{q}-2}E_{{\bar{\partial}}}(u,U_{i})\right)^{2}.\end{split}$

Rearranging and canceling squares, we get

$\begin{split}\sqrt{\frac{CMKk\ell^{\frac{q}{2}+1}}{c_{1}}}\lambda^{3-\frac{4}{q}}\|{\mathcal{T}}\|&\geq\sum_{i}\rho_{i}^{1-\frac{2}{q}}E_{{\bar{\partial}}}(u,U_{i}).\end{split}$

Combining (3.20) and (3.3) now yields the claim (3.19).

Applying Proposition 3.6, we obtain

$E_{\bar{\partial}}(u,\Omega)\leq
C\left(\|{\mathcal{T}}\|^{2}+\lambda^{1-2\beta}\|{\mathcal{T}}\|\right).$

The desired estimate now follows from Lemma 3.7$a.$

($b$) We may let $\lambda=\rho_{j}=\max\rho_{i}.$ By our assumption (3.18) and
($a$), we have

$\begin{split}\lambda^{m+\beta}\leq
M\left(\lambda^{-\beta}E(u,U_{j})+\lambda^{2-\beta}\|{\mathcal{T}}\|^{2}_{L^{2}(\hat{U}_{j})}\right)&\leq
C\left(\|{\mathcal{T}}\|+\lambda^{1-2\beta}\right)\|{\mathcal{T}}\|\end{split}$

and

$\begin{split}0&\leq\|{\mathcal{T}}\|^{2}+\lambda^{1-2\beta}\|{\mathcal{T}}\|-\frac{\lambda^{m+\beta}}{C}.\end{split}$

Adding
$\left(\lambda^{1-2\beta}-\frac{\lambda^{m-1+3\beta}}{2C}\right)\|{\mathcal{T}}\|\geq
0$ to the RHS and factoring, we have

$\begin{split}0&\leq\left(\|{\mathcal{T}}\|-\frac{\lambda^{m-1+3\beta}}{2C}\right)\left(\|{\mathcal{T}}\|+2\lambda^{1-2\beta}\right),\end{split}$

which gives the first estimate of ($b$). We therefore have

$\begin{split}\lambda^{1-2\beta}\|{\mathcal{T}}\|&\leq
C\|{\mathcal{T}}\|^{\frac{1-2\beta}{m+3\beta-1}+1}\\\
&=C\|{\mathcal{T}}\|^{\frac{m+\beta}{m+3\beta-1}}.\end{split}$

The second estimate of ($b$) now follows from ($a$). ∎

######  Theorem 3.9 (Pre-Łojasiewicz estimate, local version).

Let $x_{i}\in S^{2}$ and $0<\rho_{i}\leq\lambda\leq\frac{\pi}{2},$ for
$i=0,\ldots,\ell,$ be such that

$D_{2\rho_{i}}(x_{i})\cap D_{2\rho_{j}}(x_{j})=\emptyset$

for $i\neq j,$ and

$\bigcup_{i=1}^{\ell}D_{2\rho_{i}}(x_{i})\subset D_{\rho_{0}}(x_{0}).$

Write $U_{i}=U^{2\rho_{i}}_{\rho_{i}}(x_{i}),$
$\hat{U}_{i}=U^{4\rho_{i}}_{\rho_{i}/2}(x_{i}),$ and

$\begin{split}\Omega^{\prime}&=D_{\rho_{0}}(x_{0})\setminus\cup
D_{\rho_{i}}(x_{i})\\\
\hat{\Omega}^{\prime}&=D_{2\rho_{0}}(x_{0})\setminus\cup
D_{\rho_{i}/2}(x_{i}).\end{split}$

Suppose that
$\|{\mathcal{T}}(u)\|_{L^{2}(\hat{\Omega}^{\prime})}\leq\rho_{0}^{-1},$ and
make the assumptions (3.14-3.17) for $i=0,\ldots,\ell.$ We then have

$\sum_{i=0}^{\ell}\rho_{i}^{-\beta}E\left(u,U_{i}\right)+{\mathrm{dist}}\left(E\left(u,\Omega^{\prime}\right),{\mathbb{Z}}\right)\leq
C_{\ref{thm:localloj}}\left(\lambda^{2-\beta}\|{\mathcal{T}}\|^{2}+\lambda^{1-2\beta}\|{\mathcal{T}}(u)\|\right).$

Here $C_{\ref{thm:localloj}}$ depends on $k,K,M,$ and $\beta.$

###### Proof.

Applying Proposition 3.5 in place of Proposition 3.6 in the previous proof,
together with (3.19), we get

$E_{\bar{\partial}}(u,\Omega^{\prime})\leq
C\left(\lambda^{2}\|{\mathcal{T}}\|^{2}+\lambda^{2-\beta}\|{\mathcal{T}}\|^{2}+\lambda^{1-2\beta}\|{\mathcal{T}}\|\right).$

The desired estimate follows from Lemma 3.7$b$. ∎

##  4\. Three-annulus and multi-annulus estimates

###  4.1. Three-annulus estimates

######  Lemma 4.1.

Suppose that $N$ is Kähler. Let $n\in{\mathbb{Z}}$ with $|n|\leq L,$
$\sigma>1,$ and $0<\beta\leq\frac{1}{2}$ be such that

(4.1) $\frac{2\sigma^{2}}{\sigma^{2}+1}<\sigma^{2\beta}.$

There exists $\xi_{0}>0,$ depending on $N,L,\sigma,$ and $\beta,$ as follows.

Given any $x_{0}\in S^{2}$ and $0<\rho\leq 1,$ let
$\hat{U}=U^{\sigma^{2}\rho}_{\sigma^{-1}\rho}(x_{0}).$ Suppose that
$u:\hat{U}\to N$ is a nonconstant $W^{2,2}$ map with

(4.2) $E(u,\hat{U})<\xi_{0}$

and

(4.3)
$\rho^{2}\|{\mathcal{T}}(u)\|^{2}_{L^{2}(\hat{U})}\leq\xi_{0}E_{\partial}(u,\hat{U}).$

For $i=1,2,3,$ let

$S(i)=\sup_{\sigma^{i-2}\rho\leq
r\leq\sigma^{i-1}\rho}r^{-n}\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{0})}e_{\partial}\,d\theta}.$

The following implications hold:

($a$) $S(1)\leq\sigma^{1-\beta}S(2)\Rightarrow S(2)<\sigma^{\beta}S(3)$ ($b$)
$S(3)\leq\sigma^{1-\beta}S(2)\Rightarrow S(2)<\sigma^{\beta}S(1).$

###### Proof.

We first check the implications when ${\partial}u=g(z)dz$ is a nonzero
holomorphic 1-form on a flat annulus
$V=U^{\sigma^{2}}_{\sigma^{-1}}(0)\subset{\mathbb{C}}.$ More specifically, we
prove:

($a^{\prime}$)
$S(1)\leq\sqrt{\sigma\left(1+\frac{(\sigma-1)^{2}}{2\sigma}\right)}S(2)\Rightarrow
S(2)\leq\sqrt{\frac{\sigma}{1+\frac{(\sigma-1)^{2}}{2\sigma}}}S(3)$
($b^{\prime}$)
$S(3)\leq\sqrt{\sigma\left(1+\frac{(\sigma-1)^{2}}{2\sigma}\right)}S(2)\Rightarrow
S(2)\leq\sqrt{\frac{\sigma}{1+\frac{(\sigma-1)^{2}}{2\sigma}}}S(1)$

Under the assumption (4.1), these implications are strictly stronger than
($a$) and ($b$).

Dividing $g$ by $S(2)z^{n},$ we may reduce to the case $n=0$ and $S(2)=1.$ To
prove ($a^{\prime}$) in this case, we use the Laurent expansion of $g,$ which
reads

$g(z)=\sum_{n=-\infty}^{\infty}a_{n}z^{n}.$

This gives

(4.4)
$F(r)^{2}:=\frac{1}{2\pi}\int_{S^{1}_{r}}|v|^{2}\,d\theta=\sum|a_{n}|^{2}r^{2n}.$

Assume that the supremum $S(2)=1$ is attained at
$r_{0}\in\left[1,\sigma\right],$ so

(4.5) $F(r_{0})^{2}=1=\sum|a_{n}|^{2}r_{0}^{2n}.$

We have the identity

$1=\frac{\sigma+\sigma^{-1}}{2+\frac{(\sigma-1)^{2}}{\sigma}},$

and, for $n\in{\mathbb{Z}},$ the inequality

(4.6)
$1\leq\frac{\sigma^{-2n+1}+\sigma^{2n-1}}{2+\frac{(\sigma-1)^{2}}{\sigma}}.$

Inserting (4.6) into (4.5) for each $n\in{\mathbb{Z}},$ we have

$\begin{split}1=\sum|a_{n}|^{2}r_{0}^{2n}&\leq\frac{1}{2+\frac{(\sigma-1)^{2}}{\sigma}}\sum|a_{n}|^{2}r_{0}^{2n}\left(\sigma^{-2n+1}+\sigma^{2n-1}\right)\\\
&=\frac{1}{2+\frac{(\sigma-1)^{2}}{\sigma}}\left(\sigma^{-1}F\left(\sigma^{-1}r_{0}\right)+\sigma
F\left(\sigma r_{0}\right)\right)\\\
&\leq\frac{1}{2+\frac{(\sigma-1)^{2}}{\sigma}}\left(\sigma^{-1}S(1)^{2}+\sigma
S(3)^{2}\right).\end{split}$

We insert the assumption
$S(1)^{2}\leq\sigma\left(1+\frac{(\sigma-1)^{2}}{2\sigma}\right),$ and
rearrange, to obtain

$1\leq\frac{2\sigma}{2+\frac{(\sigma-1)^{2}}{\sigma}}S(3)^{2},$

establishing ($a^{\prime}$). The proof of ($b^{\prime}$) is identical.

The proof of ($a$) and ($b$) now follows from a standard compactness-and-
contradiction argument (similar to the proof of Lemma 3.3), which we omit. ∎

######  Lemma 4.2.

Let $0<\sigma^{-1}\leq\gamma\leq 1.$ Let
$\hat{U}=U_{\rho/2}^{2\sigma\rho}(x_{0})$ and $u:\hat{U}\to N$ as above,
satisfying (4.2-4.3), where $\xi_{0}$ depends on $N,L,\sigma,$ and $\gamma.$
There exists a constant $C_{m}>0$ such that if

$C_{m}\frac{\sqrt{E_{\partial}\left(u,U_{\rho/2}^{\rho}\right)}}{\rho}\leq\gamma\sup_{\rho\leq
r\leq\sigma\rho}\left(\frac{r}{\rho}\right)^{m+1}\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{0})}e_{\partial}\,d\theta}$

then

$\sup_{\rho\leq
r\leq\sigma\rho}\left(\frac{r}{\rho}\right)^{m}\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{0})}e_{\partial}\,d\theta}<\sigma\gamma\sup_{\sigma\rho\leq
r\leq
2\sigma\rho}\left(\frac{r}{\rho}\right)^{m}\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{0})}e_{\partial}\,d\theta}.$

###### Proof.

Let

$F(r)=\left(\frac{r}{\rho}\right)^{m}\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{0})}e_{\partial}\,d\theta}.$

We prove the contrapositive for ${\partial}u=g(z)dz$ a holomorphic 1-form, as
above; assuming

(4.7) $\sup_{\rho\leq r\leq\sigma\rho}F(r)\geq\sigma\gamma\sup_{\sigma\rho\leq
r\leq 2\sigma\rho}F(r),$

we must show

(4.8)
$C_{m}\frac{\sqrt{E_{\partial}\left(u,U_{\rho/2}^{\rho}\right)}}{\rho}>\gamma\sup_{\rho\leq
r\leq\sigma\rho}\left(\frac{r}{\rho}\right)F(r)$

for an appropriate constant $C_{m}.$

Let

(4.9) $\mu:=\frac{\sqrt{E_{\partial}\left(u,U_{\rho/2}^{\rho}\right)}}{\rho}.$

First note that by (4.9) and epsilon-regularity, we have

(4.10) $F\left(\frac{\rho}{\sqrt{2}}\right)\leq C_{m}^{\prime}\mu$

for an appropriate $C_{m}^{\prime}.$ By Hadamard’s Theorem, $\log F(r)$ is a
convex function of $\log(r).$ In view of (4.7) and (4.10), we must have

$\sup_{\rho\leq r\leq\sigma\rho}F(r)\leq C_{m}^{\prime}\mu.$

But then (4.7) implies

$F(\sigma\rho)\leq\sup_{\sigma\rho\leq r\leq
2\sigma\rho}F(r)\leq\frac{C_{m}^{\prime}\mu}{\sigma\gamma}.$

By convexity, we have

$\log F(r)\leq\left(1-\frac{\log\left(\frac{\sqrt{2}r}{\rho}\right)}{\log
2\sigma}\right)\log
C^{\prime}_{m}\mu+\frac{\log\left(\frac{\sqrt{2}r}{\rho}\right)}{\log
2\sigma}\log\left(\frac{C_{m}^{\prime}\mu}{\sigma\gamma}\right).$

for $\frac{\rho}{\sqrt{2}}\leq r\leq\sigma\rho.$ This simplifies to

$\begin{split}\log F(r)&\leq\log
C_{m}^{\prime}\mu-\frac{\log\sigma\gamma}{\log
2\sigma}\log\left(\frac{\sqrt{2}r}{\rho}\right)\\\ &\leq\log
C_{m}^{\prime}\mu-\frac{\log\sigma+\log\gamma}{\log\sigma+\log
2}\log\left(\frac{r}{\rho}\right)\\\ &\leq\log\mu+\log
C_{m}^{\prime}-\log\left(\frac{r}{\rho}\right)-\frac{\log\gamma-\log
2}{\log\sigma+\log 2}\log\left(\frac{r}{\rho}\right),\end{split}$

where we have used the fact that $\log\sigma\gamma\geq 0.$ Since
$\log\gamma-\log 2<0$ and $r/\rho\leq\sigma,$ this implies

$\begin{split}\log F(r)&\leq\log\mu+\log
C_{m}^{\prime}-\log\left(\frac{r}{\rho}\right)-\frac{\log\gamma-\log
2}{\log\sigma+\log 2}\log\sigma\\\ &\leq\log\mu+\log
C_{m}^{\prime}-\log\left(\frac{r}{\rho}\right)+\log 2-\log\gamma.\end{split}$

Exponentiating yields the desired result (4.8).

The general result can again be derived by a compactness-and-contradiction
argument. ∎

######  Proposition 4.3.

Given nonnegative integers $m,n\in\\{0,\ldots,L\\}$ and $\sigma>1,$ there
exists $\xi_{0}>0$ as follows.

Let $0<\sigma\rho\leq R\leq 1,0<\xi\leq\xi_{0},$ and $\beta$ satisfying

(4.11) $C_{L}\leq\sigma^{\beta}.$

Given $u:U^{2R}_{\rho/2}(x_{0})\to N,$ let
$\delta=\|{\mathcal{T}}(u)\|_{L^{2}\left(U^{2R}_{\rho/2}\right)}.$ Suppose
that

$\sup_{\rho/2\leq r\leq R}E(u,U_{r}^{2r})<\xi_{0}.$

Let $\mu,\nu$ be such that

$\mu\geq\frac{\sqrt{E_{\partial}\left(u,U_{\rho/2}^{\rho}\right)}}{\rho}\wedge\xi^{-1}\delta$

and

$\nu\geq\frac{\sqrt{E_{\partial}\left(u,U_{R}^{2R}\right)}}{R}\wedge\xi^{-1}\delta.$

Write

$f(r)=\max\left\\{\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{0})}e_{\partial}(u)\,d\theta},\xi^{-1}\delta\right\\}$

and, for any $\tau>1,$

$F_{\tau}(r)=\sup_{r\leq s\leq\tau r}\left(\frac{s}{\rho}\right)^{m}f(s),$
$G_{\tau}(r)=\sup_{\tau^{-1}r\leq s\leq r}\left(\frac{R}{s}\right)^{n}f(s).$

The following implications hold.

(a) If

(4.12) $C_{m}\mu\leq\sup_{\rho\leq
r\leq\sigma\rho}\left(\frac{r}{\rho}\right)^{m+1}f(r)$

then

$C^{-1}_{m}\sigma^{-1}\left(\frac{\rho}{r}\right)^{\beta}\mu\leq
F_{\sigma}(r)\leq C_{m}\left(\frac{R}{\sigma
r}\right)^{\beta}\left(\frac{R}{\rho}\right)^{m}\nu$

for each $\sigma^{2}\rho\leq r\leq R/\sigma.$

(b) If

(4.13) $C_{n}\nu\leq\sup_{\sigma^{-1}R\leq r\leq
R}\left(\frac{R}{r}\right)^{n+1}f(r)$

then

$C^{-1}_{n}\sigma^{-1}\left(\frac{r}{R}\right)^{\beta}\nu\leq
G_{\sigma}(r)\leq
C_{n}\left(\frac{r}{\sigma\rho}\right)^{\beta}\left(\frac{R}{\rho}\right)^{n}\mu.$

for each $\sigma\rho\leq r\leq R/\sigma^{2}.$

(c) If both (4.12) and (4.13) hold, then

$\begin{split}f(r)\geq C_{L}^{-1}\left(\sup_{r/2\leq s\leq
2r}f(s)\wedge\sigma^{-2}\mu\left(\frac{\rho}{r}\right)^{m+\beta}\wedge\sigma^{-2}\nu\left(\frac{r}{R}\right)^{n+\beta}\right)\end{split}$

for each $\sigma^{2}\rho\leq r\leq R/\sigma^{2}.$

###### Proof.

We first claim that given any $\tau>1$ and $\beta>0$ as in (4.11), for
$\xi_{0}>0$ sufficiently small, we have the implication

(4.14) $F_{\tau}(\tau^{-1}r)\leq\tau^{1-\beta}F_{\tau}(r)\quad\Rightarrow\quad
F_{\tau}(r)\leq\tau^{\beta}F_{\tau}(\tau r).$

This follows from Lemma 4.1$a.$ For, if

$\sup_{r\leq s\leq\tau r}\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{0})}e_{\partial}(u)\,d\theta}\leq\xi^{-1}\delta,$

then $F_{\tau}(r)=\left(\dfrac{{\tau}r}{\rho}\right)^{m}\xi^{-1}\delta$ and

$F_{\tau}({\tau}r)\geq\left(\frac{\tau^{2}r}{\rho}\right)^{m}\xi^{-1}\delta=\tau^{m}F_{\tau}(r),$

so the implication holds (since $m\geq 0$). On the other hand, if

$\sup_{r\leq s\leq{\tau}r}\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{0})}e_{\partial}(u)\,d\theta}>\xi^{-1}\delta,$

then

$C_{m,{\tau}}\xi^{2}E_{\partial}(U_{r}^{{\tau}r})\geq r^{2}\delta^{2},$

which gives the assumption (4.3), so Lemma 4.1$a$ gives (4.14).

To prove ($a$), note that by the assumption (4.12) and Lemma 4.2, after
letting $\gamma=(C_{m}\sigma^{\beta})^{-1},$ we have

$F_{\sigma}(\sigma\rho)\leq\sigma^{1-\beta}F_{\sigma}(\sigma^{2}\rho).$

We may apply Lemma 4.1 iteratively to obtain the conclusion of ($a$) for
$r\in\rho\sigma^{\mathbb{N}}.$ We can then apply (4.14) with $\tau=\sigma/2,$
to obtain ($a$) for all $r$ in the stated range. The proof of ($b$) is
identical.

To obtain the first estimate in ($c$), namely $f(r)\geq
C_{m,n}^{-1}\sup_{r/2\leq s\leq 2r}f(s),$ one can apply (4.14) with $\tau=2$
as well as the corresponding statement for $G_{\tau}.$ The last two estimates
in ($c$) then follow from ($a$) and ($b$). ∎

###  4.2. Three-annulus estimates near nonconstant maps

For technical reasons, we also need the following versions of the above
estimates.

######  Lemma 4.4.

Suppose that $N$ is Kähler and let $\phi:B_{2}(0)\subset{\mathbb{C}}\to N$ be
a smooth map. Let $n\in{\mathbb{Z}}$ with $|n|\leq L,$ $\sigma>1,$ and
$0<\beta\leq\frac{1}{2},$ and suppose that $\sigma$ satisfies

(4.15) $C_{n,\phi}\leq\sigma^{\beta}.$

There exists $\xi_{1}>0,$ depending on $N,L,\phi,\sigma,$ and $\beta,$ as
follows.

Given any $x_{0}\in S^{2}$ and $0<\rho\leq\sigma^{-2},$ let
$\hat{U}=U^{\sigma^{2}\rho}_{\sigma^{-1}\rho}(x_{0}).$ Suppose that
$u:\hat{U}\to N$ is a nonconstant $W^{2,2}$ map with

(4.16)
$\left\|u\left(x_{0}+\frac{y}{\rho\sigma^{2}}\right)-\phi\left(y\right)\right\|_{W^{2,2}(\hat{U})}<\xi_{1}$

and

(4.17)
$\rho^{2}\|{\mathcal{T}}(u)\|^{2}_{L^{2}(\hat{U})}\leq\xi_{1}E_{\partial}(u,\hat{U}).$

Then the implications of Lemma 4.1 still hold.

###### Proof.

We may rescale without loss of generality so that $\rho\sigma^{2}=1.$

We first check the implications assuming that ${\partial}u=\alpha$ is a
holomorphic 1-form valued in $\phi^{*}TN.$ Since $\phi^{*}TN$ is a smooth
bundle on $B_{2}(0)\subset{\mathbb{C}},$ by the Korn-Lichtenstein theorem (see
e.g. Donaldson-Kronheimer [13, Theorem (2.1.53)]), there is a holomorphic
frame $e_{1},\ldots,e_{\dim_{\mathbb{C}}(N)}$ for the holomorphic structure
induced by ${\bar{\partial}}_{\phi}$ over $B_{1}(0).$ The norms of these
sections are bounded above and away from zero by constants. Since a general
holomorphic 1-form valued in $\phi^{*}T^{(1,0)}N$ is a holomorphic linear
combination of $\\{e_{i}\\},$ the same calculation as in Lemma 4.1 gives

($a^{\prime\prime}$)
$C_{\phi}S(1)\leq\sqrt{\sigma\left(1+\frac{(\sigma-1)^{2}}{2\sigma}\right)}S(2)\Rightarrow
S(2)\leq C_{\phi}\sqrt{\frac{\sigma}{1+\frac{(\sigma-1)^{2}}{2\sigma}}}S(3)$
($b^{\prime\prime}$)
$C_{\phi}S(3)\leq\sqrt{\sigma\left(1+\frac{(\sigma-1)^{2}}{2\sigma}\right)}S(2)\Rightarrow
S(2)\leq C_{\phi}\sqrt{\frac{\sigma}{1+\frac{(\sigma-1)^{2}}{2\sigma}}}S(1).$

The claimed implications follow by letting $\sigma^{\beta}$ absorb the
constants.

The implications for a general 1-form ${\partial}u$ can be now obtained by a
contradiction argument similar to the proof of Lemma 3.3. ∎

######  Lemma 4.5.

Let $\hat{U}=U_{\rho/2}^{2\sigma\rho}(x_{0})$ and $u:\hat{U}\to N$ as above,
satisfying (4.16-4.17). There exists $C_{m,\phi}>0$ such that if

$C_{m,\phi}\frac{\sqrt{E_{\partial}\left(u,U_{\rho/2}^{\rho}\right)}}{\rho}\leq\gamma\sup_{\rho\leq
r\leq\sigma\rho}\left(\frac{r}{\rho}\right)^{m+1}\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{0})}e_{\partial}\,d\theta}$

then

$\sup_{\rho\leq
r\leq\sigma\rho}\left(\frac{r}{\rho}\right)^{m}\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{0})}e_{\partial}\,d\theta}<\gamma\sigma\sup_{\sigma\rho\leq
r\leq
2\sigma^{\prime}\rho}\left(\frac{r}{\rho}\right)^{m}\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{0})}e_{\partial}\,d\theta}.$

###### Proof.

This can be proved as in the previous Lemma, by reference to Lemma 4.2. ∎

######  Proposition 4.6.

Given nonnegative integers $m,n\in\\{0,\ldots,L\\},$ $0<\zeta^{\prime}\leq 1,$
and smooth maps $\phi_{1},\phi_{2}:B_{2}(0)\subset{\mathbb{C}}\to N,$ as well
as $1<\sigma\leq(\zeta^{\prime})^{-1}$ and $\beta>0$ satisfying

(4.18) $C_{m,\phi_{1}}\wedge C_{n,\phi_{2}}\leq\sigma^{\beta},$

where $C_{m,\phi}$ are the constants of (4.15), there exists $\xi_{1}>0$ as
follows.

Let $\rho$ and $R\leq 1$ be such that $0<\rho\leq(\zeta^{\prime})^{2}R\leq 1.$
Given $u:U^{2R}_{\rho/2}(x_{0})\to N,$ let
$\delta=\|{\mathcal{T}}(u)\|_{L^{2}\left(U^{2R}_{\rho/2}\right)}.$ Suppose
that

(4.19) $\sup_{\rho/2\zeta^{\prime}\leq
r\leq\zeta^{\prime}R}E(u,U_{r}^{2r})<\xi_{0},$

where $\xi_{0}$ is the constant of Proposition 4.3, as well as

(4.20) $\left\|u\left(x_{0}+\rho
y\right)-\phi_{1}\left(y^{-1}\right)\right\|_{W^{2,2}\left(U_{1/2}^{(\zeta^{\prime})^{-1}}\right)}<\xi_{1},$
(4.21)
$\left\|u\left(x_{0}+Ry\right)-\phi_{2}\left(y\right)\right\|_{W^{2,2}\left(U^{2}_{\zeta^{\prime}}\right)}<\xi_{1},$

and

(4.22)
$\rho^{2}\|{\mathcal{T}}(u)\|^{2}_{L^{2}(\hat{U})}\leq\xi_{1}E_{\partial}(u,\hat{U}).$

Then the implications of Proposition 4.3 still hold, after replacing $C_{m}$
by $C_{m,\phi_{1}}$ in (4.12) and $C_{n}$ by $C_{n,\phi_{2}}$ in (4.13).

###### Proof.

This follows by applying the proof of Proposition 4.3 using Lemmas 4.4-4.5 in
place of 4.1-4.2 over the inner and outer intervals
$U_{\rho/2}^{\rho/\zeta^{\prime}}$ and $U^{2R}_{\zeta^{\prime}R}.$ ∎

###  4.3. Multi-annulus estimates

Next, we will prove a weaker version of the previous estimates which applies
across bubble maps.

Let $0<\zeta_{i}\leq\zeta_{0}\leq\frac{1}{2},$ $\sigma_{i}>1,$ and $x_{i}\in
S^{2},$ for $i=1,\ldots,\ell,$ with $\ell\leq k.$ Assume that

$D_{4\zeta_{0}}(x_{i})\cap D_{4\zeta_{0}}(x_{j})=\emptyset,$

for $i\neq j,$ and let

(4.23) $\begin{split}\Lambda&=S^{2}\setminus\cup_{i}D_{\zeta_{i}}(x_{i}),\\\
\hat{\Lambda}&=S^{2}\setminus\cup_{i}D_{\zeta_{i}/\sigma_{i}}(x_{i}).\end{split}$

We need the following split ${\varepsilon}$-regularity results, which assume
only that $E_{{\bar{\partial}}}(u)$ is small, at the price of a curvature
assumption on $N.$

######  Lemma 4.7.

Suppose that $N$ is compact Kähler with nonnegative holomorphic bisectional
curvature, and let $2\leq p<\infty.$ There exists ${\varepsilon}_{0}>0,$
depending on the geometry of $N,$ such that if $u\in
W^{2,2}\left(\hat{\Lambda},N\right)$ satisfies
$E_{\bar{\partial}}\left(u,\hat{\Lambda}\right)<{\varepsilon}_{0},$ then

$\|{\bar{\partial}}u\|^{2}_{L^{p}\left(\Lambda\right)}\leq
C_{\ref{lemma:splitepsreg}}\left(E_{\bar{\partial}}(u,\hat{\Lambda}\setminus\Lambda)+\|{\mathcal{T}}(u)\|_{L^{2}(\hat{\Lambda})}^{2}\right).$

Here $C_{\ref{lemma:splitepsreg}}$ depends on $p,k,\zeta_{i},$ and
$\sigma_{i}.$

###### Proof.

This is a minor adaptation of the proofs by Struwe [29], Topping [32], or Liu-
Yang [17]. ∎

######  Lemma 4.8.

There exists $\xi_{1}>0,$ depending on $N,k,\zeta_{i},$ and $\sigma_{i},$ as
follows. Suppose that $u\in W^{2,2}(\hat{\Lambda},N)$ satisfies

(4.24)
$E_{\bar{\partial}}(u,\hat{\Lambda})\wedge\|{\mathcal{T}}(u)\|_{L^{2}(\hat{\Lambda})}^{2}<\xi_{1},$

and let $\alpha\in\Omega^{0,1}(u^{*}T^{(1,0)}N)$ on $\hat{\Lambda}.$ We have
the estimate

$\|\alpha\|^{2}_{L^{2}(\Lambda)}\leq
C\left(\|{\bar{\partial}}_{u}^{*}\alpha\|^{2}_{L^{2}(\hat{\Lambda})}+\sum_{i=1}^{\ell}\frac{1}{\log\sigma_{i}}\sup_{\sigma_{i}^{-1}\zeta_{i}\leq
r\leq\zeta_{i}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{i})}|\alpha|^{2}\right).$

###### Proof.

We may choose $\xi_{1}$ small enough that the previous Lemma makes
$\|{\bar{\partial}}u\|^{2}_{L^{p}\left(\Lambda\right)}$ arbitrarily small, for
a fixed $p>2.$ Since $N$ has nonnegative holomorphic bisectional curvature and
the domain is the round $S^{2},$ (2.4) gives the inequality

(4.25) $\int|\nabla(\varphi\alpha)|^{2}+\int\varphi^{2}|\alpha|^{2}\leq
C\left(\int\varphi^{2}|{\bar{\partial}}_{u}^{*}\alpha|^{2}+\int\varphi^{2}e_{\bar{\partial}}(u)|\alpha|^{2}+\int|\nabla\varphi|^{2}|\alpha|^{2}\right).$

The estimate follows by letting $\varphi=\prod\varphi_{i}$ be a product of
logarithmic cutoffs, with $r|\nabla\varphi_{i}|\leq C/\log\sigma_{i},$ and
using the Sobolev and Hölder’s inequalities together with the result of Lemma
4.7. ∎

######  Proposition 4.9.

Let $0<\gamma\leq 1.$ Given $u$ and $\alpha\in\Omega^{0,1}(u^{*}T^{(1,0)}N)$
as in the previous lemma, suppose that

$\|{\bar{\partial}}_{u}^{*}\alpha\|^{2}_{L^{2}(\hat{\Lambda})}\leq\frac{\|\alpha\|^{2}_{L^{2}(\Lambda)}}{2C},$

where $C$ is the univeral constant of the previous Lemma. Assume that

(4.26) $\frac{4C}{\gamma}\leq\log\sigma_{i}$

for $i=1,\ldots,\ell.$ Fixing $j\in\\{1,\ldots,\ell\\},$ the following
implication holds:

$\begin{split}\gamma\sum_{i\neq j}\sup_{\sigma_{i}^{-1}\zeta_{i}\leq
r\leq\zeta_{i}}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{i})}|\alpha|^{2}\leq\|\alpha\|^{2}_{L^{2}(\Lambda)}&\\\
\Rightarrow\quad&\\\
\|\alpha\|^{2}_{L^{2}(\Lambda)}&<\gamma\sup_{\sigma_{j}^{-1}\zeta_{j}\leq
r\leq\zeta_{j}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{j})}|\alpha|^{2}.\end{split}$

###### Proof.

This follows by rearranging the estimate of the previous Lemma. ∎

######  Theorem 4.10.

Suppose that $N$ has nonnegative holomorphic bisectional curvature.

Given $n_{i}\in{\mathbb{Z}},$ $i=1,\ldots,\ell,$ $\ell\leq k,$ with
$|n_{i}|\leq L,$ and

$\sum n_{i}=0,$

as well as $0<\zeta\leq\zeta_{0}$ and $0<\gamma\leq 1,$ there exists
$\sigma_{0}>1$ (depending on $k,L,\zeta,$ and $\gamma$) such that given
$\sigma\geq\sigma_{0},$ there exists $\xi_{1}>0$ (depending on $k,L,\zeta,$
and $\sigma$) as follows.

Let $\zeta_{i}=\zeta$ and $\sigma_{i}=\sigma,$ for $i=1,\ldots,\ell,$ in
(4.23). Suppose that $u:\hat{\Lambda}\to N$ is a nonconstant $W^{2,2}$ map
with

$E_{\bar{\partial}}(u,\hat{\Lambda})<\xi_{1}$

and

$\|{\mathcal{T}}(u)\|^{2}_{L^{2}(\hat{\Lambda})}\leq\xi_{1}E_{\bar{\partial}}(u,\hat{\Lambda}).$

Let

$S_{i}=\sup_{\sigma^{-1}\zeta\leq
r\leq\zeta}\left(\frac{\zeta}{r}\right)^{n_{i}}\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{d(x,x_{i})=r}e_{\bar{\partial}}\left(u(x)\right)\,d\theta},$

for $i=1,\ldots,\ell.$ Fixing $j\in\\{1,\ldots,\ell\\},$ the following
implication holds:

(4.27) $\begin{split}\gamma\sum_{i\neq
j}S^{2}_{i}\leq&\,\,E_{\bar{\partial}}(u,\Lambda)\,\,\forall\,\,i\neq j\\\
&\Rightarrow\\\ &E_{\bar{\partial}}(u,\Lambda)<\gamma S_{j}.\end{split}$

###### Proof.

First notice that the case $n_{i}=0$ for all $i$ follows directly from
Proposition 4.9, where $\alpha={\bar{\partial}}u.$

To obtain the general case, we let $f(x)$ be a meromorphic function on $S^{2}$
with order $-n_{i}$ at $x_{i},$ which we can choose such that
$\int_{S^{2}\setminus\cup_{i}D_{\zeta_{0}}(x_{i})}|f|^{2}\,dV=1$—this
determines $f(x)$ uniquely up to a unit complex number. We have

$K_{1}^{-1}\leq\sup_{\sigma^{-1}\zeta\leq{\mathrm{dist}}(x,x_{i})\leq\zeta}|f(x)||x-x_{i}|^{n_{i}}\leq
K_{1}.$

We now let

$\alpha=\bar{f}\cdot{\bar{\partial}}u.$

Then

${\bar{\partial}}^{*}\alpha=\Lambda{\partial}\alpha=\bar{f}{\bar{\partial}}^{*}{\bar{\partial}}u=\bar{f}{\mathcal{T}}(u),$

since $f$ is holomorphic on $\hat{\Lambda}.$ Choosing $\sigma_{0}$ to account
for $K_{1},$ and $\xi_{1}$ to account for the size of $|f|$ on $\Lambda,$ we
again obtain the desired implication from Proposition 4.9. ∎

######  Theorem 4.11.

Suppose that $N$ has nonnegative holomorphic bisectional curvature.

Given $L\in{\mathbb{N}},$ $n_{i}\in{\mathbb{Z}},$ with $|n_{i}|\leq L,$ as
well as $0<\zeta\leq\frac{1}{2}$ and $0<\gamma\leq 1,$ there exists
$\sigma_{0}>1$ (depending on $k,L,\gamma,$ and $\zeta$), as well as
$\sigma_{1}>0$ (depending only on $k,L$ and $\gamma$), such that given
$\sigma\geq\sigma_{0}$ and $\sigma^{\prime}\geq\sigma_{1},$ there exists
$\xi_{1}>0$ (depending on $k,L,\gamma,\zeta,\sigma,$ and $\sigma^{\prime}$) as
follows.

Let $0<\lambda\leq\frac{1}{\sigma^{\prime}},$ $p\in S^{2},$ and
$x_{1},\ldots,x_{\ell}\in D_{\lambda/2}(p)\subset S^{2},$ $\ell\leq L,$ with

(4.28) $D_{2\zeta_{0}\lambda}(x_{i})\cap
D_{2\zeta_{0}\lambda}(x_{j})=\emptyset$

for each $i\neq j.$ Write

$\Lambda^{\prime}=D_{\lambda}(p)\setminus\cup_{i}D_{\zeta\lambda}(x_{i}),$
$\hat{\Lambda}^{\prime}=D_{\sigma^{\prime}\lambda}(p)\setminus\cup_{i}D_{\sigma^{-1}\zeta\lambda}(x_{i}).$

Suppose that $u:\hat{\Lambda}^{\prime}\to N$ is a nonconstant $W^{2,2}$ map
with

$E_{\bar{\partial}}(u,\hat{\Lambda}^{\prime})<\xi_{1}$

and

(4.29)
$\lambda^{2}\|{\mathcal{T}}(u)\|^{2}_{L^{2}(\hat{\Lambda})}\leq\xi_{1}E_{\bar{\partial}}(u,\hat{\Lambda}).$

Put

$m=\sum n_{i},$

and let

$S_{0}=\sup_{\lambda\leq
r\leq\sigma\lambda}\left(\frac{r}{\lambda}\right)^{m}\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(p)}e_{\bar{\partial}}(u(x))\,d\theta},$
$S^{\prime}_{0}=\sup_{\lambda\leq
r\leq\sigma\lambda}\left(\frac{r}{\lambda}\right)^{m+1}\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(p)}e_{\bar{\partial}}(u(x))\,d\theta},$
$S_{i}=\sup_{\sigma^{-1}\zeta\lambda\leq
r\leq\zeta\lambda}\left(\frac{\zeta\lambda}{r}\right)^{n_{i}}\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{i})}e_{\bar{\partial}}(u(x))\,d\theta},$

and

$S^{\prime}_{i}=\sup_{\sigma^{-1}\zeta\lambda\leq
r\leq\zeta\lambda}\left(\frac{\zeta\lambda}{r}\right)^{n_{i}+1}\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{i})}e_{\bar{\partial}}(u(x))\,d\theta},$

for $i=1,\ldots,\ell.$ We then have the following implications:

($a$)
$\begin{split}\gamma\sum_{i=1}^{\ell}S^{2}_{i}\leq\frac{E_{\bar{\partial}}(u,\Lambda^{\prime})}{\lambda^{2}}&\\\
\Rightarrow\qquad&\\\
\frac{E_{\bar{\partial}}(u,\Lambda^{\prime})}{\lambda^{2}}&<\gamma\left(S^{\prime}_{0}\right)^{2}\end{split}$

and

($b$) $\begin{split}\gamma\sum_{\stackrel{{\scriptstyle i=0}}{{i\neq
j}}}^{\ell}S^{2}_{i}\leq\frac{E_{\bar{\partial}}(u,\Lambda^{\prime})}{\lambda^{2}}&\\\
\Rightarrow\qquad&\\\
\frac{E_{\bar{\partial}}(u,\Lambda^{\prime})}{\lambda^{2}}&<\gamma\left(S^{\prime}_{j}\right)^{2}.\end{split}$

###### Proof.

Since $\hat{\Lambda}$ is contained in the hemisphere $D_{1}(p),$ where the
conformal factor is bounded by 4, we may replace $D_{\sigma\lambda}(p)$ by the
flat ball $B_{\sigma}(0)\subset{\mathbb{C}},$ and assume $\lambda=1.$

We will prove the case $m=0=n_{i}$ of ($a$) by contradiction. Let

$\alpha=\frac{{\bar{\partial}}u}{\sqrt{E_{{\bar{\partial}}}(u,\Lambda^{\prime})}},$

and suppose that both

(4.30) $\left(S^{\prime}_{0}\right)^{2}\leq\frac{1}{\gamma}$

and

(4.31) $\sum_{i=1}^{\ell}S_{i}^{2}\leq\frac{1}{\gamma}.$

For $1\leq r\leq\sigma^{\prime},$ from (4.30), we have

$\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(p)}|\alpha|^{2}}\leq\frac{S^{\prime}_{0}}{\gamma
r}.$

It follows that the Lorentz norm of $\alpha$ is bounded as follows:

(4.32)
$\begin{split}\|\alpha\|_{L^{2,\infty}(\hat{\Lambda}^{\prime})}^{2}&\leq\|\alpha\|^{2}_{L^{2}\left(\hat{\Lambda}^{\prime}\cap
B_{1}(0)\right)}+\frac{1}{\gamma^{2}}\left(S_{0}^{\prime}\right)^{2}\\\
&\leq\frac{2}{\gamma^{2}}\|\alpha\|^{2}_{L^{2}\left(\hat{\Lambda}^{\prime}\cap
B_{1}(0)\right)}.\end{split}$

We choose $\varphi=\varphi_{0}\cdot\prod_{i=1}^{\ell}\varphi_{i}$ to be a
product of logarithmic cutoffs, with
${\mathrm{supp}\,}\varphi\subset\hat{\Lambda}$ and $\varphi\equiv 1$ on
$\Lambda.$ In view of (4.31) and the fact that
$\|\alpha\|_{L^{2}(\Lambda^{\prime})}=1,$ we must have

$\|\varphi\alpha\|^{2}_{L^{2}(B_{1}(0))}\leq 1+\frac{1}{\gamma}.$

Hence (4.32) reduces to

(4.33)
$\begin{split}\|\varphi\alpha\|_{L^{2,\infty}({\mathbb{R}}^{2})}^{2}&\leq\frac{2\left(1+\frac{1}{\gamma}\right)}{\gamma^{2}}.\end{split}$

By [20], Ladyzhenskaya’s inequality allows a Lorentz space on the RHS, giving:

$\|\varphi\alpha\|_{L^{4}({\mathbb{R}}^{2})}^{2}\leq
C\|\varphi\alpha\|_{L^{2,\infty}}\|\nabla(\varphi\alpha)\|_{L^{2}}.$

Inserting (4.33), we have

(4.34)
$\|\varphi\alpha\|_{L^{4}({\mathbb{R}}^{2})}^{2}\leq\frac{C}{\gamma}\|\nabla(\varphi\alpha)\|_{L^{2}}.$

Integrating the Weitzenbock formula (2.4) against $\varphi,$ instead of
(4.25), we have

(4.35) $\int|\nabla(\varphi\alpha)|^{2}\leq
C\left(\int\varphi^{2}|{\bar{\partial}}_{u}^{*}\alpha|^{2}+\int\varphi^{2}e_{\bar{\partial}}(u)|\alpha|^{2}+\int|\nabla\varphi|^{2}|\alpha|^{2}\right).$

Applying Hölder’s inequality on the RHS, we get

(4.36) $\int|\nabla(\varphi\alpha)|^{2}\leq
C\left(\int\varphi^{2}|{\bar{\partial}}_{u}^{*}\alpha|^{2}+\left(\int
e_{\bar{\partial}}(u)^{2}\right)^{\frac{1}{2}}\left(\int\left(\varphi|\alpha|\right)^{4}\right)^{\frac{1}{2}}+\int|\nabla\varphi|^{2}|\alpha|^{2}\right).$

Assuming from Lemma 4.7 that $\left(\int
e_{\bar{\partial}}(u)^{2}\right)^{\frac{1}{2}}<\delta,$ we can apply (4.34)
and rearrange (4.36) to obtain

(4.37)
$\|\nabla(\varphi\alpha)\|_{L^{2}}\left(\|\nabla(\varphi\alpha)\|_{L^{2}}-C\delta\right)\leq
C\left(\int\varphi^{2}|{\bar{\partial}}_{u}^{*}\alpha|^{2}+\int|\nabla\varphi|^{2}|\alpha|^{2}\right).$

By Hölder, we have

$1=\|\alpha\|_{L^{2}(\Lambda^{\prime})}\leq
C\|\varphi\alpha\|_{L^{4}}\leq\frac{C}{\gamma}\|\nabla(\varphi\alpha)\|_{L^{2}},$

so (4.37) gives

$\begin{split}\frac{\gamma}{C}&\leq
C\left(\int\varphi^{2}|{\bar{\partial}}_{u}^{*}\alpha|^{2}+\int|\nabla\varphi|^{2}|\alpha|^{2}\right)\\\
&\leq C\xi_{1}+C\left(\frac{1}{\log\sigma^{\prime}}\sup_{1\leq
r\leq\sigma^{\prime}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(p)}|\alpha|^{2}+\sum_{i}\frac{1}{\log\sigma}\sup_{\sigma^{-1}\zeta\leq
r\leq\zeta}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x_{i})}|\alpha|^{2}\right).\end{split}$

Choosing $\xi_{1}$ small enough and $\sigma,\sigma^{\prime}$ large enough
gives a contradiction.

To obtain the general case of ($a$), let

$f(z)=\Pi(z-z_{i})^{n_{i}}.$

On $D_{\sigma^{\prime}}(0)\setminus D_{1}(0),$ we have

$K_{1}^{-1}\leq\left|\frac{f(w)}{w^{m}}\right|\leq K_{1},$

where $K_{1}$ may depend on $\\{n_{i}\\}$ but not on $\zeta$ or the points
$x_{i}\in B_{1/2}(0).$ We can now obtain the conclusions by applying
Proposition 4.9 to the form $\alpha=\bar{f}\cdot{\bar{\partial}}u.$ The proof
of ($b$) follows similarly by choosing $f(z)=(z-z_{j})^{n_{j}+1}\Pi_{i\neq
j}(z-z_{i})^{n_{i}}.$ ∎

##  5\. Bubble-tree estimates

######  Definition 5.1.

A sequence $u_{i}:\Sigma\to N$ of $W^{2,2}$ maps with $E(u_{i})\leq 4\pi L$
and $\|{\mathcal{T}}(u_{i})\|_{L^{2}}\to 0$ as $i\to\infty$ will be called an
_almost-harmonic sequence_.

######  Definition 5.2.

A collection

$\mathcal{B}=\\{J,\zeta_{0},\\{z_{j}\\},\\{y_{j_{1},\ldots,j_{q}}\\},\phi_{j_{1},\ldots,j_{q}},u_{\infty}\\}$

will be referred to as _bubble-tree data_. Here,
$J=\\{(j_{1},\ldots,j_{q})\\}_{q=1}^{q_{max}}$ is a finite indexing set of
nonnegative integers; $0<\zeta_{0}\leq\frac{1}{2}$ is a positive real number;
$\\{z_{j}\\}\subset\Sigma,$ for $(j)\in J,$ and $y_{j_{1},\ldots,j_{q}}\in
B_{1}(0)\subset{\mathbb{R}}^{2},$ for $(j_{1},\ldots,j_{q})\in J$ ($q\geq 2$),
are finite sets of points; and

$u_{\infty}=\phi_{0}:\Sigma\to N,$
$\phi_{j_{1},\ldots,j_{q}}:{\mathbb{R}}^{2}\to N,$

for $(j_{1},\ldots,j_{q})\in J,$ are finite-energy harmonic maps. These are
required to satisfy:

$|J|\leq 4\pi L/{\varepsilon}_{0},$ $D_{2\zeta_{0}}(z_{j})\cap
D_{2\zeta_{0}}(z_{k})=\emptyset,$
$D_{2\zeta_{0}}\left(y_{j_{1},\ldots,j_{q},j}\right)\cap
D_{2\zeta_{0}}\left(y_{j_{1},\ldots,j_{q},k}\right)=\emptyset$

for $j\neq k,$ and, if $(j_{1},\ldots,j_{q})\in J$ is a terminal index (i.e.
$(j_{1},\ldots,j_{q},k)\not\in J$ for all $k$), then
$\phi_{j_{1},\ldots,j_{q}}$ is nonconstant.

######  Definition 5.3 ([28], Definition 4.2).

Given a $W^{1,2}$ map $u:B_{R}(x_{0})\to N,$ the _outer energy scale_
$\lambda_{{\varepsilon},R,x_{0}}(u)$ is the smallest number $\lambda\geq 0$
such that

(5.1)
$\sup_{\lambda<\rho<R}E\left(u,U^{\rho}_{\rho/2}(x_{0})\right)<{\varepsilon}.$

Note that $\lambda=R$ satisfies (5.1) vacuously, so
$0\leq\lambda_{{\varepsilon},R,x_{0}}(u)\leq R$ by definition.

######  Theorem 5.4 (Bubble-tree compactness for almost-harmonic sequences).

Given an almost-harmonic sequence $u_{i}:\Sigma\to N,$ we may pass to a
subsequence (again called $u_{i}$) which “converges in the bubble-tree sense,”
i.e., for which there exists a set of bubble-tree data $\mathcal{B}$ as in
Definition 5.2, as follows.

Given $0<{\varepsilon}\leq{\varepsilon}_{0},$ there exists
$0<\zeta\leq\zeta_{0}$ (with $\zeta=\zeta_{0}$ if
${\varepsilon}={\varepsilon}_{0}$), and for all $i$ sufficiently large, points
$x^{i}_{j_{1},\ldots,j_{q}}\in\Sigma$ and positive numbers
$\lambda^{i}_{j_{1},\ldots,j_{q}}\to 0,$ such that:

  * •

$x^{i}_{j_{1},\ldots,j_{q}}\to z_{j_{1}}$ as $i\to\infty,$

  * •

$\lambda^{i}_{j_{1}}=\lambda_{\zeta,{\varepsilon},x^{i}_{j_{1}}}(u_{i}),$

  * •

$\lambda^{i}_{j_{1},\ldots,j_{q}}=\lambda_{\zeta\lambda^{i}_{j_{1},\ldots,j_{q-1}},{\varepsilon},x^{i}_{j_{1},\ldots,j_{q}}}(u_{i}(t_{i})),$

  * •

$x^{i}_{j_{1},\ldots,j_{q}}=x^{i}_{j_{1},\ldots,j_{q-1}}+\lambda^{i}_{j_{1},\ldots,j_{q-1}}y_{j_{1},\ldots,j_{q}},$

  * •

$\dfrac{\lambda^{i}_{j_{1},\ldots,j_{q}}}{\lambda^{i}_{j_{1},\ldots,j_{q-1}}}\to
0$ as $i\to\infty,$

  * •

$u_{i}\left(x^{i}_{j_{1},\ldots,j_{q}}+\lambda^{i}_{j_{1},\ldots,j_{q}}y\right)\to\phi_{j_{1},\ldots,j_{q}}(y)$
in
$W^{2,2}_{loc}\left({\mathbb{R}}^{2}\setminus\cup_{k}y_{j_{1},\ldots,j_{q-1},k}\right)$
as $i\to\infty,$

  * •

Given any ${\varepsilon}^{\prime}\leq{\varepsilon},$ there exists
$\zeta^{\prime}>0$ such that for all $i$ sufficiently large, we have
$E\left(u_{i},U_{(\zeta^{\prime})^{-1}\lambda_{j_{1},\ldots,j_{q}}}^{\zeta^{\prime}\lambda_{j_{1},\ldots,j_{q-1}}}(x_{j_{1},\ldots,j_{q}})\right)<{\varepsilon}^{\prime}.$

###### Proof.

The proof is by now standard (see Song-Waldron [28] for a recent sketch), and
we omit it. ∎

Suppose now that $u_{i}$ converges in the bubble-tree sense and that
$\phi_{j_{1},\ldots,j_{q}}$ and $u_{\infty}$ are each either holomorphic,
antiholomorphic, or constant, as is the case for $\Sigma=N=S^{2}.$ Denote
these disjoint subsets of $J$ by

$J=J_{\partial}\amalg J_{\bar{\partial}}\amalg J_{0},$

respectively. We also write

$J_{T}\subset J_{\partial}\cup J_{\bar{\partial}}$

for the set of “terminal” bubbles in the tree, which are all nonconstant by
construction.

Next, given any index $\vec{\jmath}=(j_{1},\ldots,j_{q}),$ write

$\reflectbox{$\vec{\reflectbox{$\jmath$}}$}_{{\partial}}(\vec{\jmath})=(j_{1},\ldots,j_{p})\in
J_{\partial}\cup\\{0\\},$

with $p\leq q,$ for the nearest holomorphic index preceding $\vec{\jmath}$ in
the bubble tree, or
$\reflectbox{$\vec{\reflectbox{$\jmath$}}$}_{{\partial}}(\vec{\jmath})=(0)$ if
there are only ghost (i.e. constant) bubbles between $\vec{\jmath}$ and the
body map. Write

$\vec{J}_{{\partial}}(\vec{\jmath})=\\{(j_{1},\ldots,j_{q},\ldots,j_{r})\\}\subset
J_{\partial}$

for the set of holomorphic indices succeeding $\vec{\jmath}$ in the bubble
tree, i.e.,
$(j_{1},\ldots,j_{q},\ldots,j_{r})\in\vec{J}_{\partial}(\vec{\jmath})$ iff
$(j_{1},\ldots,j_{q},\ldots,j_{r})\in J_{\partial}$ but
$(j_{1},\ldots,j_{p},\ldots,j_{s})\not\in J_{\partial}$ for any $p\leq s<r.$
We define
$\reflectbox{$\vec{\reflectbox{$\jmath$}}$}_{{\bar{\partial}}}(\vec{\jmath})$
and $\vec{J}_{{\bar{\partial}}}(\vec{\jmath})$ similarly.

######  Definition 5.5.

For each $\vec{\jmath}=(j_{1},\ldots,j_{q})\in J,$ define a nonnegative
integer $M^{{\partial}}_{j_{1},\ldots,j_{q}}$ as follows. For $\vec{\jmath}\in
J_{\partial},$ suppose first that $(j_{1},\ldots,j_{q})\in J_{\partial},$ and
let

$M^{{\partial}}_{j_{1},\ldots,j_{q}}=2+\left(\text{order-of-vanishing of
${\partial}\phi_{j_{1},\ldots,j_{q}}$ at $\infty$}\right).$

For a terminal, antiholomorphic index $(j_{1},\ldots,j_{q})\in J_{T}\cap
J_{{\bar{\partial}}},$ let

$M^{\partial}_{j_{1},\ldots,j_{q}}=0.$

For an antiholomorphic or ghost index $\vec{\jmath}\in J_{\partial}^{c},$
define

$M^{\partial}_{j_{1},\ldots,j_{q}}=\sum
M^{\partial}_{j_{1},\ldots,j_{q},\ell_{q+1},\ldots,\ell_{n}},$

where the sum runs over all succeeding holomorphic indices
$(j_{1},\ldots,j_{q},\ell_{q+1},\ldots,\ell_{n})\in\vec{J}_{\partial}(\vec{\jmath}).$

######  Definition 5.6.

Define nonnegative integers $N^{{\partial}}_{j_{1},\ldots,j_{q}}$ as follows.
Supposing first that

$(j_{1},\ldots,j_{q-1})\in J_{\partial},$

let

$N^{{\partial}}_{j_{1},\ldots,j_{q}}=\text{order-of-vanishing of
${\partial}\phi_{j_{1},\ldots,j_{q-1}}$ at $y_{j_{1},\ldots,j_{q}}$}.$

Suppose next that $(j_{1},\ldots,j_{q-1})\in J_{\partial}^{c}$ is an
antiholomorphic or ghost index. Assuming $q=1$ (i.e. the body map $\phi_{0}$
is antiholomorphic or constant), define

$N^{\partial}_{j_{1}}=\sum_{j\neq j_{1}}M^{\partial}_{j}.$

Assuming $q>1,$ write
$\reflectbox{$\vec{\reflectbox{$\jmath$}}$}_{\partial}(\vec{\jmath})=(j_{1},\ldots,j_{p}),$
and define

$N^{\partial}_{j_{1},\ldots,j_{q}}=N^{\partial}_{j_{1},\ldots,j_{p}}+\sum_{n=p+1}^{q}\sum_{j\neq
j_{n}}M_{j_{1},\ldots,j_{p},\ldots,j_{n-1},j}.$

Define $M^{\bar{\partial}}_{*}$ and $N^{\bar{\partial}}_{*}$ similarly,
replacing holomorphic by anti-holomorphic and vice-versa.

######  Remark 5.7.

These integers can be bounded above by an integer $L=L(k)$ depending only on
$k$ and ${\varepsilon}_{0}$ (which can be replaced by $1$ when $N=S^{2}$). In
simple cases, $L$ can be made explicit: for instance in the situation of
Topping’s second theorem [32], the holomorphic exponents $M_{*}^{\partial}$
and $N^{\partial}_{*}$ are all zero, so one may take $L=0$ in Theorems 1.1$b$
and 1.2$b.$

Now, given $0<{\varepsilon}\leq{\varepsilon}_{0},$ let

$x^{i}_{j_{1},\ldots,j_{q}}\in S^{2},\qquad
0<\lambda^{i}_{j_{1},\ldots,j_{q}}\leq\zeta,$

for $i=1,\ldots,\infty,$ be the sequences of points and scales guaranteed by
Theorem 5.4. For $q\geq 1,$ also let

$\Lambda^{i}_{j_{1},\ldots,j_{q}}=D_{\lambda^{i}_{j_{1},\ldots,j_{q}}}(x^{i}_{j_{1},\ldots,j_{q}})\setminus\cup_{k}\bar{D}_{\zeta\lambda^{i}_{j_{1},\ldots,j_{q},k}}(x^{i}_{j_{1},\ldots,j_{q},k}),$
$V^{i}_{j_{1},\ldots,j_{q}}=U^{2\zeta\lambda^{i}_{j_{1},\ldots,j_{q-1}}}_{\zeta\lambda^{i}_{j_{1},\ldots,j_{q-1}}}(x^{i}_{j_{1},\ldots,j_{q}}),$

and

$U^{i}_{j_{1},\ldots,j_{q}}=U^{\lambda^{i}_{j_{1},\ldots,j_{q}}}_{\lambda^{i}_{j_{1},\ldots,j_{q}}/2}(x^{i}_{j_{1},\ldots,j_{q}}).$

In addition, let

$\Lambda^{i}_{0}=S^{2}\setminus\cup_{j}\bar{D}_{\zeta}\left(x^{i}_{j}\right).$

Given $\xi_{1}>0,$ for $i$ large enough, we may assume that for each
$(j_{1},\ldots,j_{q})\in J_{\partial},$ there holds

$E_{\bar{\partial}}(u_{i},\Lambda^{i}_{j_{1},\ldots,j_{q}})<\xi_{1},$

while for $(j_{1},\ldots,j_{q})\in J_{\bar{\partial}},$

(5.2) $E_{\partial}(u_{i},\Lambda^{i}_{j_{1},\ldots,j_{q}})<\xi_{1},$

and for $(j_{1},\ldots,j_{q})\in J_{0},$

$E(u_{i},\Lambda^{i}_{j_{1},\ldots,j_{q}})<\xi_{1}.$

We are finally ready to prove our main bubble-tree estimate.

######  Theorem 5.8.

Let $0<\beta<\frac{1}{2}.$ Suppose that $u_{i}:S^{2}\to N,$ with nonnegative
holomorphic bisectional curvature, is a bubble-tree convergent almost-harmonic
sequence, with bubble-tree data $\mathcal{B},$ for which the body map and all
bubbles are either holomorphic or antiholomorphic. Write

$\delta_{i}:=\|{\mathcal{T}}(u_{i})\|\to 0.$

Given $0<{\varepsilon}\leq{\varepsilon}_{0},$ there exist $\kappa,\xi>0,$
depending on ${\varepsilon},$ $\beta,$ and $\mathcal{B},$ as follows.

Let $\zeta,$ $\Lambda^{i}_{\vec{\jmath}},$ $\hat{\Lambda}^{i}_{\vec{\jmath}},$
etc., be as in Theorem 5.4, and put

$\mu^{i}_{\vec{\jmath}}=\max\left\\{\frac{\sqrt{\min\left\\{E_{\partial}\left(u_{i},\hat{\Lambda}^{i}_{\vec{\jmath}}\right),{\varepsilon}\right\\}}}{\lambda^{i}_{\vec{\jmath}}},\xi^{-1}\delta_{i}\right\\}$

and

$f^{i}_{\vec{\jmath}}(r)=\max\left\\{\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x^{i}_{\vec{\jmath}})}e_{\partial}(u_{i})\,d\theta},\xi^{-1}\delta_{i}\right\\}$

for $\lambda^{i}_{j_{1},\ldots,j_{q}}\leq
r\leq\zeta\lambda^{i}_{j_{1},\ldots,j_{q-1}}$ (where
$\vec{\jmath}=(j_{1},\ldots,j_{q})$). For $i$ sufficiently large, we have

($a$)
$f^{i}_{j_{1},\ldots,j_{q}}(r)\geq\kappa\left(\mu^{i}_{j_{1},\ldots,j_{q}}\left(\frac{\lambda^{i}_{j_{1},\ldots,j_{q}}}{r}\right)^{M^{\partial}_{j_{1},\ldots,j_{q}}+\beta}+\mu^{i}_{j_{1},\ldots,j_{q-1}}\left(\frac{r}{\lambda^{i}_{j_{1},\ldots,j_{q-1}}}\right)^{N^{\partial}_{j_{1},\ldots,j_{q}}+\beta}\right)$

for each $\vec{\jmath}\in J$ and $\lambda^{i}_{j_{1},\ldots,j_{q}}\leq
r\leq\zeta\lambda^{i}_{j_{1},\ldots,j_{q-1}},$ as well as

($b$) $f^{i}_{j_{1},\ldots,j_{q}}(r)\geq c_{L}\sup_{r/2\leq s\leq
2r}f^{i}_{j_{1},\ldots,j_{q}}(s)$

for $\kappa^{-1}\lambda^{i}_{j_{1},\ldots,j_{q}}\leq
r\leq\kappa\zeta\lambda^{i}_{j_{1},\ldots,j_{q-1}}.$

###### Proof.

We will let $\sigma$ increase and $\xi$ decrease finitely many times during
the proof.

Claim $1.$ Given $0<\gamma\leq 1,$ we can choose $\sigma$ large enough such
that for each $\vec{\jmath}\in J$ and $i$ sufficiently large, there holds

(5.3)
$\mu^{i}_{j_{1},\ldots,j_{q}}\leq\gamma\sup_{\lambda^{i}_{\vec{\jmath}}\leq
r\leq\sigma\lambda^{i}_{\vec{\jmath}}}\left(\frac{r}{\lambda^{i}_{\vec{\jmath}}}\right)^{M^{\partial}_{\vec{\jmath}}+1}f^{i}_{\vec{\jmath}}(r).$

Proof of claim. First assume that $\vec{\jmath}\in J_{\partial}$ is a
holomorphic index. It follows from the definition of
$M^{\partial}_{\vec{\jmath}}$ on a holomorphic index that

(5.4)
$\mu^{i}_{j_{1},\ldots,j_{q}}\leq\kappa^{-1}\sup_{\frac{1}{2}\sigma\lambda^{i}_{\vec{\jmath}}\leq
r\leq\sigma\lambda^{i}_{\vec{\jmath}}}\left(\frac{r}{\lambda^{i}_{\vec{\jmath}}}\right)^{M^{\partial}_{\vec{\jmath}}}f^{i}_{\vec{\jmath}}(r).$

for any $\sigma$ and $i$ sufficiently large, which implies the claim after
letting $\sigma\geq 2\gamma^{-1}\kappa^{-1}.$

Next, assume that $\vec{\jmath}\in J_{T}\cap J_{\partial}^{c}$ is a terminal
but non-holomorphic index, in which case $M^{\partial}_{\vec{\jmath}}=0.$ Then
(5.2) holds for $i$ sufficiently large, and we can apply Theorem 4.11$a$ with
$m=0$ and $\\{n_{i}\\}=\emptyset,$ to obtain the claim.

We can now prove the claim for the remaining indices
$J_{\partial}^{c}=J_{\bar{\partial}}\cup J_{0}$ by “inward” induction on the
bubble tree, the base case being that of the holomorphic/terminal bubbles. Let
$(j_{1},\ldots,j_{q})\in J_{{\partial}}^{c}.$ Note from Definition 5.5 that

(5.5)
$M^{\partial}_{j_{1},\ldots,j_{q}}=\sum_{k}M^{\partial}_{j_{1},\ldots,j_{q},k}.$

Assume that the claim has been proven for all indices succeeding a given index
$\vec{\jmath}=(j_{1},\ldots,j_{q});$ in particular

$\mu^{i}_{j_{1},\ldots,j_{q},k}\leq\gamma\sup_{\lambda^{i}_{j_{1},\ldots,j_{q},k}\leq
r\leq\sigma\lambda^{i}_{j_{1},\ldots,j_{q},k}}\left(\frac{r}{\lambda^{i}_{j_{1},\ldots,j_{q},k}}\right)^{M^{\partial}_{j_{1},\ldots,j_{q},k}+1}f^{i}_{j_{1},\ldots,j_{q},k}(r)$

for each $k$ such that $(j_{1},\ldots,j_{q},k)\in J.$ Let
${\varepsilon}^{\prime}=\xi_{0},$ the constant of Proposition 4.3
corresponding to $\sigma.$ By the last item in Theorem 5.4, we may choose
$\zeta^{\prime}>0$ such that the hypothesis (4.19) is satisfied, with
$\rho=\lambda^{i}_{j_{1},\ldots,j_{q},k}$ and
$R=\zeta\lambda^{i}_{j_{1},\ldots,j_{q}}.$ By choosing $i$ sufficiently large,
we may assume that (4.20-4.21) are satisfied for the relevant bubble maps.
Corollary 4.6$a$ then gives us

(5.6) $\begin{split}\sup_{\sigma^{-1}\zeta\lambda^{i}_{j_{1},\ldots,j_{q}}\leq
r\leq\zeta\lambda^{i}_{j_{1},\ldots,j_{q}}}\left(\frac{r}{\zeta\lambda^{i}_{j_{1},\ldots,j_{q}}}\right)^{M^{\partial}_{j_{1},\ldots,j_{q},k}}f^{i}_{j_{1},\ldots,j_{q},k}(r)&\leq\gamma\frac{\sqrt{E_{\partial}\left(u,V_{j_{1},\ldots,j_{q},k}\right)}}{\lambda^{i}_{j_{1},\ldots,j_{q}}}\\\
&\leq\gamma\mu^{i}_{j_{1},\ldots,j_{q},k}.\end{split}$

We may now apply Theorem 4.11$a,$ with $m=M^{\partial}_{j_{1},\ldots,j_{q}}$
and $n_{k}=M^{\partial}_{j_{1},\ldots,j_{q},k}$ (in view of (5.5)), to
complete the induction step. (To account for the case
$\mu^{i}_{\vec{\jmath}}=\xi^{-1}\delta_{i},$ we can simply decrease $\xi.$)
$\diamond$

Claim $2.$ For each $\vec{\jmath}=(j_{1},\ldots,j_{q})\in J$ and $i$
sufficiently large, we have

(5.7)
$\mu^{i}_{j_{1},\ldots,j_{q}}\leq\gamma\sup_{\sigma^{-1}\zeta\lambda^{i}_{j_{1},\ldots,j_{q-1}}\leq
r\leq\zeta\lambda^{i}_{j_{1},\ldots,j_{q-1}}}\left(\frac{\zeta\lambda^{i}_{j_{1},\ldots,j_{q-1}}}{r}\right)^{N^{\partial}_{\vec{\jmath}}+1}f^{i}_{\vec{\jmath}}(r).$

Proof of claim. First assume that $\vec{\jmath}\in J_{\partial}$ is a
holomorphic index. Then, taking $\sigma$ larger if necessary, the claim
follows as above from the definition of $N^{\partial}_{\vec{\jmath}}$ on a
holomorphic index.

Next, assume that the body map is either antiholomorphic or constant; then
(5.2) holds for $i$ sufficiently large. Fix an index $j_{1},$ let
$n_{j_{1}}=N^{\partial}_{j_{1}},$ and $n_{k}=-M^{\partial}_{k}$ for $k\neq
j_{1}.$ In view of (5.6), we may apply Theorem 4.10 to obtain the claim, with
$\vec{\jmath}=(j_{1}).$

We can now prove the claim for the remaining indices by “outward” induction on
the bubble tree. Assume that the claim has been proven for all indices up to
and including a given index $\vec{\jmath}=(j_{1},\ldots,j_{q});$ we shall
prove it for $(j_{1},\ldots,j_{q+1})\in
J_{\partial}^{c}=J_{\bar{\partial}}\cup J_{0}.$ Note that from Definition 5.6,
we have

$N^{\partial}_{j_{1},\ldots,j_{q},j_{q+1}}=N^{\partial}_{j_{1},\ldots,j_{q}}+\sum_{k\neq
j_{q+1}}M^{\partial}_{j_{1},\ldots,j_{q},k}.$

Setting $m=N^{\partial}_{j_{1},\ldots,j_{q}},$
$n_{j_{q+1}}=N^{\partial}_{j_{1},\ldots,j_{q+1}},$ and
$n_{k}=-M^{\partial}_{j_{1},\ldots,j_{q},k}$ for $k\neq j_{q+1},$ in view of
the induction hypothesis (5.7) and the established inequality (5.6), Theorem
4.11$b$ implies the claim for $(j_{1},\ldots,j_{q+1}),$ completing the
induction. $\diamond$

Claims 1 and 2 imply the hypotheses (4.12) and (4.13) of Corollary 4.6$c,$
which gives the desired bounds ($a$) and ($b$). In the statement, we absorb
all dependence on $\sigma$ into the constant $\kappa.$ ∎

######  Corollary 5.9.

Let $\vec{\jmath}=(j_{1},\ldots,j_{q})\in J,$ and put

$\reflectbox{$\vec{\reflectbox{$\jmath$}}$}_{\partial}(\vec{\jmath})=(j_{1},\ldots,j_{p}).$

(a) Supposing that
$\reflectbox{$\vec{\reflectbox{$\jmath$}}$}_{\partial}(\vec{\jmath})$ is
holomorphic (and not constant), we have

$f^{i}_{\vec{\jmath}}(r)\geq\kappa{\varepsilon}\left(\frac{r}{\lambda^{i}_{\reflectbox{$\vec{\reflectbox{$\jmath$}}$}_{\partial}(\vec{\jmath})}}\right)^{N_{\vec{\jmath}}+\beta}.$

(b) Supposing that
$\vec{k}=(j_{1},\ldots,j_{q},k_{m+1},\ldots,k_{n})\in\vec{J}_{\partial}(\vec{\jmath})$
is holomorphic (and not constant), we have

$f^{i}_{\vec{\jmath}}(r)\geq\kappa{\varepsilon}\left(\frac{\lambda^{i}_{\vec{k}}}{r}\right)^{M_{\vec{\jmath}}+\beta}.$

(c) Let $p\leq m\leq q$ and consider three indices of the form

$\vec{\ell}=(j_{1},\ldots,j_{m}),$
$\vec{\jmath}=(j_{1},\ldots,j_{m},j_{m+1},\ldots,j_{q}),$
$\vec{k}=(j_{1},\ldots,j_{m},k_{m+1},\ldots,k_{n}).$

Suppose that $\vec{k}\in\vec{J}_{\partial}(\vec{\ell})$ is a holomorphic index
succeeding $\vec{\ell},$ and that
$\vec{J}_{\partial}(\vec{\jmath})\subset\vec{J}_{\partial}(\vec{\ell}),$ i.e.,
there are no holomorphic indices between $\vec{\ell}$ and $\vec{k}$ or between
$\vec{\ell}$ and $\vec{\jmath}.$ For $i$ sufficiently large, we then have

$f^{i}_{j_{1},\ldots,j_{q}}(r)\geq\kappa{\varepsilon}\frac{\left(\lambda^{i}_{\vec{k}}\right)^{M^{{\partial}}_{\vec{\ell}}+\beta}r^{N^{{\partial}}_{\vec{\jmath}}+\beta}}{\left(\lambda^{i}_{\vec{\ell}}\right)^{M^{{\partial}}_{\vec{\ell}}+N^{{\partial}}_{\vec{\jmath}}+2\beta}}.$

###### Proof.

According to the hypotheses, the relevant indices are connected only by
antiholomorphic or flat maps. The estimates then follow by running Theorem
5.8$a$ across the connecting indices. ∎

We can encapsulate the results of this section in the following statement.

######  Corollary 5.10.

Given $k\in{\mathbb{N}},$ $\beta>0,$ and ${\varepsilon}>0,$ there exists
$\delta>0$ as follows. Given any $u\in W^{2,2}\left(S^{2},S^{2}\right)$ with
$E(u)\leq 4\pi k$ and

$\|{\mathcal{T}}(u)\|<\delta,$

there exists at least one bubble-tree decomposition of $u$ such the estimates
of Theorem 5.4 and Corollary 5.9$a$-$c$ hold, with $\kappa=\xi=\delta.$

###### Proof.

This follows by contradiction from the last two results. ∎

##  6\. Main theorems

The following is our main technical result.

######  Theorem 6.1.

Let $u_{i}:S^{2}\to S^{2}$ be a bubble-tree convergent almost-harmonic
sequence with $E(u_{i})\leq 4\pi k.$ There exist a constant $K\geq 1,$
depending only on $k,$ as well as constants $C_{\ref{thm:sequenceloj}b-d}$ and
$M\geq 1,$ depending on the bubble-tree data $\mathcal{B}$ and $\beta>0,$ such
that for all $i$ sufficiently large, $u_{i}$ satisfies each of the following
statements.

(a) There exist $0<\rho^{i}_{j}\leq\zeta_{0},$ $j=1,\ldots,\ell,$ as follows:
letting

$\Omega_{i}=S^{2}\setminus\cup_{j}\bar{D}_{\rho^{i}_{j}}(z_{j}),\qquad\hat{\Omega}_{i}=S^{2}\setminus\cup_{j}\bar{D}_{\rho^{i}_{j}/2}(z_{j}),\qquad
U^{i}_{j}=U^{\rho^{i}_{j}}_{\rho^{i}_{j}/2}(z_{j}),$

we have

(6.1) $E_{\partial}(u_{i},\hat{\Omega}_{i})\vee
E_{\bar{\partial}}(u_{i},\hat{\Omega}_{i})<{\varepsilon}_{2},$

and, for $j=1,\ldots,\ell,$

(6.2) $E(u,\hat{U}^{i}_{j})<{\varepsilon}_{2},$ (6.3)
$E_{\partial}(u,\hat{U}^{i}_{j})\leq
KE_{{\partial}}(u,U^{i}_{j})+M\left(\rho^{i}_{j}\right)^{2}\|{\mathcal{T}}(u)\|_{L^{2}(\hat{U}^{i}_{j})}^{2},$

and

(6.4) $E_{\bar{\partial}}(u,\hat{U}_{i})\leq
KE_{{\bar{\partial}}}(u,U^{i}_{j})+M\left(\rho^{i}_{j}\right)^{2}\|{\mathcal{T}}(u)\|_{L^{2}(\hat{U}^{i}_{j})}^{2}.$

Moreover, we either have

(6.5)
$E_{\bar{\partial}}(u_{i},\hat{\Omega}_{i})<{\varepsilon}_{2}\quad\text{and}\quad
E_{\bar{\partial}}(u_{i},U^{i}_{j})\leq
M\left(E_{{\partial}}(u_{i},U^{i}_{j})+(\rho^{i}_{j})^{2}\|{\mathcal{T}}(u_{i})\|_{L^{2}(\hat{U}_{i})}^{2}\right)$

for $i=1,\ldots,\ell,$ or

(6.6)
$E_{\partial}(u_{i},\hat{\Omega}_{i})<{\varepsilon}_{2}\quad\text{and}\quad
E_{\partial}(u_{i},U^{i}_{j})\leq
M\left(E_{{\bar{\partial}}}(u_{i},U^{i}_{j})+(\rho^{i}_{j})^{2}\|{\mathcal{T}}(u_{i})\|_{L^{2}(\hat{U}_{i})}^{2}\right).$

Here ${\varepsilon}_{2}$ is the constant of Theorems 3.8-3.9.

(b) Given any $\rho^{i}_{j}$ such that (6.1-6.6) hold for some $K,M\geq 1$,
let $\lambda_{i}=\max_{j}\rho^{i}_{j}.$ We then have

${\mathrm{dist}}\left(E(u_{i},S^{2}),4\pi{\mathbb{Z}}\right)\leq
C_{\ref{thm:sequenceloj}b}\left(\|{\mathcal{T}}(u_{i})\|^{2}+\lambda_{i}^{1-\beta}\|{\mathcal{T}}(u_{i})\|\right).$

Here $C_{\ref{thm:sequenceloj}b}$ may depend additionally on $K$ and $M.$

(c) Let $L\in{\mathbb{N}}$ be the integer described in Remark 5.7, and choose
$1<\alpha<\frac{2L+2}{2L+1}.$ Suppose that the body may $u_{\infty}$ is
nonconstant and holomorphic, or that the outermost nonconstant bubbles on each
branch are holomorphic. Then

${\mathrm{dist}}\left(E(u_{i}),4\pi{\mathbb{Z}}\right)\leq
C_{\ref{thm:sequenceloj}c}\|{\mathcal{T}}(u_{i})\|^{\alpha}.$

Here $C_{\ref{thm:sequenceloj}c}$ may depend additionally on $\alpha.$

(d) Under the same assumption as (c), given any antiholomorphic bubble index
$\vec{\jmath}\in J_{\bar{\partial}},$ we have

$\left(\lambda^{i}_{\vec{\jmath}}\right)^{2L+1+\beta}\leq
C_{\ref{thm:sequenceloj}d}\|{\mathcal{T}}(u_{i})\|.$

###### Proof.

We assume that $i$ is sufficiently large that Theorem 5.8 applies to both the
holomorphic and antiholomorphic energies. We then suppress the label $i,$
writing $u=u_{i},$ $\Lambda_{\vec{\jmath}}=\Lambda^{i}_{\vec{\jmath}},$
$\delta=\delta_{i}=\|{\mathcal{T}}(u_{i})\|,$ etc.

We now decompose the map $u$ into almost-holomorphic and almost-
antiholomorphic parts, as follows. Define $\hat{J}_{\partial}\subset J$ to be
the set of all $\vec{\jmath}\in J$ for which

(6.7) $E_{\partial}\left(u,\hat{\Lambda}_{\vec{\jmath}}\right)\geq
E_{{\bar{\partial}}}\left(u,\hat{\Lambda}_{\vec{\jmath}}\right).$

Let $\hat{J}_{\bar{\partial}}=\hat{J}_{\partial}^{c},$ with
$\vec{\jmath}\in\hat{J}_{{\bar{\partial}}}$ satisfying

(6.8)
$E_{\partial}\left(u,\hat{\Lambda}_{\vec{\jmath}}\right)<E_{{\bar{\partial}}}\left(u,\hat{\Lambda}_{\vec{\jmath}}\right).$

We then have

$J=\hat{J}_{\partial}\amalg\hat{J}_{\bar{\partial}}.$

Moreover, suppose that $(j_{1},\ldots,j_{p})\in\hat{J}_{{\partial}}$ but
$(j_{1},\ldots,j_{p+1})\in\hat{J}_{\bar{\partial}},$ or vice-versa; then the
holomorphic and antiholomorphic energies must “cross” in the neck.
Specifically, let

$\begin{split}f(r)&=f^{i}_{j_{1},\ldots,j_{q}}(r)=\max\left[\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x^{i}_{j_{1},\ldots,j_{q}})}e_{\partial}(u_{i})\,d\theta},\xi^{-1}\delta_{i}\right],\\\
g(r)&=\max\left[\sqrt{\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{1}_{r}(x^{i}_{j_{1},\ldots,j_{q}})}e_{{\bar{\partial}}}(u_{i})\,d\theta},\xi^{-1}\delta_{i}\right].\end{split}$

By Theorem 5.8 and (6.7), either
$f(\lambda^{i}_{j_{1},\ldots,j_{q}})=\xi^{-1}\delta=g(\lambda^{i}_{j_{1},\ldots,j_{q}}),$
or else we have

$\begin{split}f(\lambda^{i}_{j_{1},\ldots,j_{q}})\geq\kappa
E_{\partial}(u,\Lambda_{j_{1},\ldots,j_{q}})&\geq\kappa
E_{{\bar{\partial}}}(u,\Lambda_{j_{1},\ldots,j_{q}})\\\
&\geq\kappa^{2}g(\lambda^{i}_{j_{1},\ldots,j_{q}}).\end{split}$

By (6.8), either
$f(\lambda^{i}_{j_{1},\ldots,j_{q+1}})=\xi^{-1}\delta=g(\lambda^{i}_{j_{1},\ldots,j_{q+1}}),$
or we have

$\begin{split}g(\lambda^{i}_{j_{1},\ldots,j_{q+1}})\geq\kappa
E_{\bar{\partial}}(u,\Lambda_{j_{1},\ldots,j_{q+1}})&\geq\kappa
E_{{\partial}}(u,\Lambda_{j_{1},\ldots,j_{q+1}})\\\
&\geq\kappa^{2}f(\lambda^{i}_{j_{1},\ldots,j_{q+1}}).\end{split}$

Since both $f$ and $g$ are continuous, by the intermediate value theorem and
Theorem 5.4$a,$ there exists
$\rho\in\left[\kappa^{-1}\lambda_{j_{1},\ldots,j_{q+1}},\kappa\lambda_{j_{1},\ldots,j_{q}}\right]$
such that

(6.9) $M^{-1}g(\rho)\leq f(\rho)\leq Mg(\rho),$

where $M=\kappa^{-2-L}.$ Theorem 5.4$b$ also gives (6.3-6.4), with
$K=(c_{L})^{-1}.$

To obtain ($a$), we can apply the preceding discussion and (6.9) to the body
map $\phi_{0},$ giving (6.5) or (6.6) depending on whether
$0\in\hat{J}_{\partial}$ or $\hat{J}_{{\bar{\partial}}},$ respectively.

Also note that given any $\rho^{i}_{j}$ as in ($a$), these may be decreased so
that (6.9) (rather than either (6.5) or (6.6)) is satisfied.

Now let $I_{\partial}$ and $I_{\bar{\partial}}$ be the set of connected
components of $\hat{J}_{\partial}$ and $\hat{J}_{\bar{\partial}},$
respectively, and let

$I=I_{\partial}\cup I_{\bar{\partial}}.$

We re-index $I$ by the same ordering as $J;$ accordingly, an even-length index
$(i_{1},\ldots,i_{2n})$ belongs to $I_{\partial}$ and an odd-length index
$(i_{1},\ldots,i_{2n+1})$ belongs to $I_{\bar{\partial}}.$

Given $\vec{\imath}\neq 0\in I,$ let $\vec{\jmath}$ be the “root” of the sub-
tree corresponding to $\vec{\imath},$ i.e. the smallest index in the
corresponding connected component, and let

$q_{\vec{\imath}}=x_{\vec{\jmath}}.$

Also define a radius $\rho_{\vec{\imath}}$ as follows: since, by definition,
$(j_{1},\ldots,j_{q-1})\in J_{\bar{\partial}},$ there exists a radius
$\lambda_{j_{1},\ldots,j_{p}}\leq\rho=\rho_{\vec{\imath}}\leq\zeta\lambda_{j_{1},\ldots,j_{p-1}}$
for which (6.9) is satisfied: in other words, letting

$W_{\vec{\imath}}=U_{\rho_{i_{1},\ldots,i_{q+1}}}^{2\rho_{i_{1},\ldots,i_{q+1}}}(q_{i_{1},\ldots,i_{q+1}}),$
$\hat{W}_{\vec{\imath}}=U_{\frac{1}{2}\rho_{i_{1},\ldots,i_{q+1}}}^{4\rho_{i_{1},\ldots,i_{q+1}}}(q_{i_{1},\ldots,i_{q+1}}),$

we obtain

$E\left(u,\hat{W}_{\vec{\imath}}\right)\leq
M\left(\left(E_{\bar{\partial}}\left(u,W_{\vec{\imath}}\right)\wedge
E_{\partial}\left(u,W_{\vec{\imath}}\right)\right)+\rho^{2}\delta^{2}\right).$

We further define the domains

$\Omega_{i_{1},\ldots,i_{q}}=D_{\rho_{i_{1},\ldots,i_{q}}}(q_{i_{1},\ldots,i_{q}})\setminus\cup_{k}\bar{D}_{\rho_{i_{1},\ldots,i_{q},k}}(q_{i_{1},\ldots,i_{q},k}),$
$\hat{\Omega}_{i_{1},\ldots,i_{q}}=D_{2\rho_{i_{1},\ldots,i_{q}}}(q_{i_{1},\ldots,i_{q}})\setminus\cup_{k}\bar{D}_{\frac{1}{2}\rho_{i_{1},\ldots,i_{q},k}}(q_{i_{1},\ldots,i_{q},k}),$

where the unions run over $k$ such that $(i_{1},\ldots,i_{q},k)\in
I_{\bar{\partial}}$ or $I_{\partial},$ respectively. The
$\Omega_{\vec{\imath}}$ are disjoint, with

$\cup_{\vec{\imath}}\overline{\Omega_{\vec{\imath}}}=S^{2},$ (6.10)
$\cup_{i_{2},\ldots,i_{q}}\overline{\Omega_{i_{1},i_{2},\ldots,i_{q}}}=\overline{D_{\rho_{i_{1}}}(q_{i_{1}})}$

for each $(i_{1})\in I,$ and

$\Omega_{0}=S^{2}\setminus\cup_{i_{1}}\bar{D}_{\rho_{i_{1}}}(q_{i_{1}}),$

while

$\hat{\Omega}_{i_{1},\ldots,i_{q}}\cap\hat{\Omega}_{i_{1},\ldots,i_{q+1}}=U_{i_{1},\ldots,i_{q+1}}.$

Supposing that $u=u_{i}$ is sufficiently far along the sequence, we may assume
that

$E_{\bar{\partial}}(u,\hat{\Omega}_{i_{1},\ldots,i_{2n}})<{\varepsilon}_{2}$

and

$E_{\partial}(u,\hat{\Omega}_{i_{1},\ldots,i_{2n+1}})<{\varepsilon}_{2},$

so that the hypothesis (3.14) is satisfied.

We can now assemble the desired estimates. Letting
$\lambda=\max\rho_{\vec{\imath}},$ the local pre-Łojasiewicz inequality,
Theorem 3.9, gives

${\mathrm{dist}}\left(E(u,\Omega_{\vec{\imath}}),4\pi{\mathbb{Z}}\right)\leq
C\lambda^{1-\beta}\|{\mathcal{T}}(u)\|$

for each $\vec{\imath}\neq 0.$ By the triangle inequality, in view of (6.10),
we have

${\mathrm{dist}}\left(E(u,\cup_{i}D_{\rho_{i}}),4\pi{\mathbb{Z}}\right)={\mathrm{dist}}\left(E(u,\cup_{\vec{\imath}\neq
0}\Omega_{\vec{\imath}}),4\pi{\mathbb{Z}}\right)\leq
C\lambda^{1-\beta}\|{\mathcal{T}}(u)\|.$

Now, applying the global pre-Łojasiewicz inequality, Theorem 3.8$a,$ and the
triangle inequality again, we obtain

$\begin{split}{\mathrm{dist}}\left(E(u,S^{2}),4\pi{\mathbb{Z}}\right)&\leq{\mathrm{dist}}\left(E(u,\Omega_{0}),4\pi{\mathbb{Z}}\right)+\sum_{\vec{\imath}\neq
0}{\mathrm{dist}}\left(E(u,\Omega_{\vec{\imath}}),4\pi{\mathbb{Z}}\right)\\\
&\leq
C\left(\|{\mathcal{T}}(u)\|^{2}+\lambda^{1-\beta}\|{\mathcal{T}}(u)\|\right).\end{split}$

This gives ($b$).

To obtain ($c$) and ($d$), simply notice that in case the body map is
holomorphic (or the leading bubbles are all holomorphic), the lower bound
(3.18) with $m=2L+2$ follows from Corollary 5.9$a,$ so the results follow by
using Theorem 3.8$b$ in place of 3.8$a.$ ∎

###  6.1. Proof of Theorems 1.1-1.2

Suppose for contradiction that the estimate of Theorem 1.1$a$ fails. We then
have a sequence of maps $u_{i}$ satisfying the assumptions, but for which

(6.11) ${\mathrm{dist}}(E(u),4\pi{\mathbb{Z}})\geq i\|{\mathcal{T}}(u_{i})\|,$

which implies

$\|{\mathcal{T}}(u_{i})\|\leq\frac{4\pi}{i}.$

In particular, $u_{i}$ form an almost-harmonic sequence. But we may then pass
to a bubble-tree convergent subsequence and apply Theorem 6.1$a$-$b,$ to
obtain the estimate

${\mathrm{dist}}(E(u_{i}),4\pi{\mathbb{Z}})\leq C\|{\mathcal{T}}(u_{i})\|$

for some constant $C$ and $i$ sufficiently large. This contradicts (6.11),
establishing the result.

Theorem 1.1$b$ follows similarly by contradiction from Theorem 6.1$d,$ since
${\varepsilon}$-regularity (Lemma 3.1$a$) ensures that the body map is not
constant.

Theorem 1.2$a$ follows by contradiction from Theorem 6.1$b,$ where
$\lambda_{i}\leq\lambda$ is guaranteed by the assumptions (1.6) and (1.7).
Meanwhile, Theorem 1.2$b$ follows by contradiction from Theorem 6.1$d,$ whose
hypotheses are guaranteed by the assumptions (1.6) and (1.8). ∎

###  6.2. Proof of Theorem 1.3 and Corollary 1.4

Let

$\delta(t)=\|{\mathcal{T}}(u(t))\|$

and

$\Delta(t)=E(u(t))-4\pi n,$

where $n$ is the unique integer such that

$E(u(t))\in\left(4\pi\left(n-\frac{1}{2}\right),4\pi\left(n+\frac{1}{2}\right)\right].$

To avoid confusion with $\delta(t),$ we use $\delta_{0}$ during the proof in
place of the constant $\delta$ in the statement.

By the energy identity for 2D harmonic map flow [12, 16], we know that
$E(u(t))$ can only jump by an integer multiple of $4\pi$ at each singular
time. Since $|\Delta(t)|<\delta_{0}$ for $0\leq t<T,$ the function $\Delta(t)$
is in fact continuous and decreasing, and the global energy identity

(6.12) $\Delta(t_{2})+\int_{t_{1}}^{t_{2}}\delta(t)^{2}=\Delta(t_{1})$

is valid. Let $0<T_{*}\leq T$ be the largest time such that the following hold
for $0\leq t<T_{*}$:

(6.13) $E_{{\bar{\partial}}}\left(u(t),\Omega^{\prime}\right)\leq
2{\varepsilon}_{0},$ (6.14)
$\max_{i}E\left(u(t),\hat{U}_{i}^{\prime}\right)\leq 2{\varepsilon}_{0},$
(6.15)
$\frac{\kappa}{2}\leq\min_{i}E_{{\partial}}\left(u(t),U_{i}^{\prime}\right),$

and

(6.16)
$-2\left(\delta_{0}^{\frac{\alpha-2}{\alpha}}+\frac{2-\alpha}{\alpha}(T_{*}-t)\right)^{\frac{\alpha}{\alpha-2}}\leq\Delta(t)\leq
2\left(\delta_{0}^{\frac{\alpha-2}{\alpha}}+\frac{2-\alpha}{\alpha}t\right)^{\frac{\alpha}{\alpha-2}}.$

Assuming that $\delta_{0}$ is sufficiently small, we will show that each of
these estimates holds with strict inequality on $\left[0,T_{*}\right].$ This
implies that $T_{*}$ cannot be maximal unless $T_{*}=T,$ which will give us
the desired estimates.

The following local energy identity is standard (see e.g. [28] or [32]):

(6.17)
$\left|\int\left(e_{\bar{\partial}}(u(t_{2}))-e_{\bar{\partial}}(u(t_{1}))\right)\varphi\,dV\right|\leq
C\sup|\nabla\varphi|\sqrt{k}\int_{t_{1}}^{t_{2}}\delta(t)\,dt,$

where $\varphi$ is compactly supported. The same identity holds with
$e_{{\partial}}$ or $e$ in place of $e_{\bar{\partial}}.$ Let

$U_{i}^{\prime}=U^{\frac{5}{2}\lambda_{i}}_{\frac{4}{5}\lambda_{i}}(x_{i}),\quad\hat{U}_{i}^{\prime}=U^{3\lambda_{i}}_{\frac{2}{3}\lambda_{i}}(x_{i}),\quad\Omega^{\prime}=S^{2}\setminus\cup\bar{D}_{\frac{4}{5}\lambda_{i}}(x_{i}).$

We can apply (6.17) several times, to obtain:

(6.18)
$\begin{split}E_{\bar{\partial}}(u(t),\Omega^{\prime})&<{\varepsilon}_{0}+\frac{C\sqrt{k}}{\min\lambda_{i}^{2}}\int_{0}^{t}\delta(t)\,dt,\end{split}$
(6.19)
$\begin{split}E(u(t),\hat{U}_{i}^{\prime})&<{\varepsilon}_{0}+\frac{C\sqrt{k}}{\min\lambda_{i}^{2}}\int_{0}^{t}\delta(t)\,dt,\end{split}$

and

(6.20)
$\begin{split}E_{\partial}(u(t),U^{\prime}_{i})&\geq\kappa-\frac{C\sqrt{k}}{\min\lambda_{i}^{2}}\int_{0}^{t}\delta(t)\,dt.\end{split}$

We first prove (6.16) with coefficient $1$ in place of $2$ on both sides.
Since the hypotheses of Theorem 1.2$b$ are satisfied, we have

(6.21) $\Delta(t)\leq\delta(t)^{\alpha},$

where we may ignore the constant by taking $\delta_{0}$ sufficiently small.
This gives us

$\frac{d}{dt}\Delta(t)\leq-\Delta(t)^{\frac{2}{\alpha}}$

as long as $\Delta(t)\geq 0,$ and

$\frac{d}{dt}\Delta(t)\leq-(-\Delta(t))^{\frac{2}{\alpha}}$

thereafter. The inequalities follow from Gronwall’s Theorem.

Next, we prove strict inequality in (6.13-6.15). By (6.12), we have

$\frac{d}{dt}\Delta(t)^{\frac{\alpha-1}{\alpha}}=-\frac{\alpha-1}{\alpha}\Delta(t)^{-\frac{1}{\alpha}}\delta^{2}(t).$

Together with (6.21), this gives

(6.22)
$\begin{split}\int_{t_{1}}^{t_{2}}\delta(t)\,dt=\int_{t_{1}}^{t_{2}}\delta(t)^{2}\delta(t)^{-1}\,dt&\leq\int_{t_{1}}^{t_{2}}\delta(t)^{2}\Delta(t)^{-\frac{1}{\alpha}}\,dt\\\
&=-\frac{\alpha}{\alpha-1}\int_{t_{1}}^{t_{2}}\frac{d}{dt}\Delta(t)^{\frac{\alpha-1}{\alpha}}\,dt\\\
&\leq\frac{\alpha}{\alpha-1}\left(\Delta(t_{1})^{\frac{\alpha-1}{\alpha}}-\Delta(t_{2})^{\frac{\alpha-1}{\alpha}}\right).\end{split}$

Apply (6.22) with $t_{1}=0$ and $t_{2}=T_{*},$ to obtain

(6.23)
$\int_{0}^{T_{*}}\delta(t)\,dt\leq\frac{2\alpha}{\alpha-1}\delta_{0}^{\frac{\alpha-1}{\alpha}}.$

Provided that $\delta_{0}$ is sufficiently small, we may conclude from
(6.18-6.20) that strict inequality holds in (6.13-6.15). This implies that in
fact $T_{*}=T,$ so the first estimate of Theorem 1.3$a$ now follows from
(6.16). The second estimate of Theorem 1.3$a$ follows from (6.22) by
integrating the harmonic map flow equation in time and applying the $L^{2}$
norm, then using (6.23).

To obtain Theorem 1.3$b,$ let $u_{\infty}$ be the body map along any
subsequence of times; by (6.13-6.15), this must be nonconstant and
holomorphic. The stated $L^{2}$ estimate follows by letting $t_{2}\to\infty$
in (6.22) and inserting (6.16). The $C^{k}$ improvements follow by combining
(6.22) with standard derivative estimates valid away from the bubbling set
$\\{z_{j}\\}$ (see e.g. Song-Waldron [28, §3.3]).

To prove Corollary 1.4, suppose that $u_{\infty}$ is nonconstant, and assume
without loss of generality that it is holomorphic. Let
$\\{x_{i}\\}=\\{z_{i}\\}$ be the bubbling set along the given sequence
$t_{i}\to\infty,$ put $\lambda_{j}=\zeta_{0}$ for each $j,$ and take $\Omega$
as in Theorem 1.2$b$-1.3. Since $u(t_{i})\to u_{\infty}$ strongly on $\Omega,$
there exists $\kappa>0$ such that they hypotheses (1.6) and (1.8) hold, with
$u_{0}=u(t_{i})$ for $t_{i}$ sufficiently large. Theorem 1.3 gives the desired
conclusions.

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