# Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature

Gabriel Khan  Department of Mathematics, Iowa State University, Ames, IA, USA.
<EMAIL_ADDRESS>,  Xuan Hien Nguyen  Department of Mathematics, Iowa State
University, Ames, IA, USA<EMAIL_ADDRESS>,  Malik Tuerkoen  Department
of Mathematics, University of California, Santa Barbara, CA, USA.
<EMAIL_ADDRESS>and  Guofang Wei  Department of Mathematics, University of
California, Santa Barbara, CA, USA<EMAIL_ADDRESS>
###### Abstract.

We study the log-concavity of the first Dirichlet eigenfunction of the
Laplacian for convex domains. For positively curved surfaces satisfying a
condition involving the curvature and its second derivative, we show that the
first eigenfunction is strongly log-concave. Previously, the log-concavity of
these eigenfunctions were only known for convex domains of $\mathbb{R}^{n}$
and $\mathbb{S}^{n}$. Using this estimate, we establish lower bounds on the
fundamental gap of such regions. Furthermore, we study the behavior of these
estimates under Ricci flow and other deformations of the metric.

G. Khan was supported in part by Simons Collaboration Grant 849022

X.H Nguyen was supported in part by Simons Collaboration Grant 579756

G. Wei is partially supported by NSF DMS 2104704.

##  1\. Introduction

Given a bounded smooth connected domain $\Omega\subset M^{n}$ of a Riemannian
manifold, the eigenvalue equation of the Laplacian on $\Omega$ with Dirichlet
boundary conditions is

$\Delta\phi=-\lambda\,\phi,\ \ \phi|_{\partial\Omega}=0.$

The eigenvalues satisfy
$0<\lambda_{1}<\lambda_{2}\leq\lambda_{3}\cdots\to\infty.$ As the first
eigenvalue is simple, the fundamental (or mass) gap refers to the difference
between the first two eigenvalues

$\Gamma(\Omega)=\lambda_{2}-\lambda_{1}>0$

of the Laplacian. This quantity has many applications in mathematics and
physics, and can also be defined for a general Schrödinger operator.

The log-concavity estimate of the first eigenfunction plays a key role in
fundamental gap estimates for convex domains in $\mathbb{R}^{n}$ and
$\mathbb{S}^{n}$ (see Section 1.1 for some history). In this paper, we
establish a very general result about the log-concavity of the first
eigenfunction for convex domains in an arbitrary Riemannian manifold following
the approach of [SWYY85, LW87] for $\mathbb{R}^{n},\mathbb{S}^{n}$, see
Theorem 2.1. We then use this result on surfaces of positive curvature to
obtain the following estimate for the first eigenfunction $u_{1}$.

######  Theorem 1.1.

Suppose $\Omega$ is a convex domain in a surface whose sectional curvature
$\kappa$ is positive and satisfies

(1.1)
$-\Delta\log\kappa<4\lambda_{1}(\Omega)-5\kappa-4C\left(\tfrac{C}{\kappa}+3\right)$

for some $C\in\mathbb{R}$. Then the first eigenfunction satisfies

(1.2) $\operatorname{Hess}\,(\log u_{1})\leq-C-\frac{\kappa}{2}.$

Using a result of Ling [Lin06] to bound $\lambda_{1}$ from below, we can write
(1.1) solely in terms of the curvatures, see Corollary 3.1. Taking $C=0$, the
assumption is always satisfied for $\mathbb{S}^{2}$ as
$\lambda_{1}(\Omega)\geq 2$ for all convex domains $\Omega\subset S^{2}$. In
Subsection 3.1, we show all relatively round ellipsoids also satisfy the
condition.

Theorem 1.1 has several consequences. First, the log-concavity estimate (1.2)
implies a lower bound on the fundamental gap.

######  Corollary 1.2.

Given a convex domain $\Omega$ in a surface whose curvature satisfies (1.1)
with $C\in[-\kappa_{\min{}},\infty)$, we have the following estimate on the
fundamental gap:

$\lambda_{2}-\lambda_{1}>\frac{\pi^{2}}{D^{2}}+\inf_{\Omega}\kappa+C.$

Secondly, Hamilton [Ham88] showed that compact surface with positive curvature
converges to space form under normalized Ricci flow. Using an estimate of
Hamilton for $\log\kappa$ under Ricci flow, we give an explicit estimate on
the time that will satisfy (1.1) for a positively pinched surface, see
Corollary 4.3. Therefore the condition (1.1), which is originally a fourth-
order requirement on the metric can now be satisfied by a pinching of
curvature if one is amenable to flowing the surface by Ricci flow for a fixed
amount of time.

Lastly, equation (1.1) combined with Corollary 1.2 shows that the inequality

$\inf_{\Omega\text{ convex }\subset M}\Gamma(\Omega)D(\Omega)^{2}\geq\pi^{2}$

is stable if the manifold $M$ is a $C^{4}$ perturbation of $\mathbb{S}^{2}$
(here, $D(\Omega)$ is the diameter of $\Omega$). For a fixed domain
$\Omega\subset M$, the fundamental gap $\Gamma(\Omega)$ varies continuously
under metric deformations, which is a consequence of the regularity of
elliptic partial differential equations. However, the property above is
required to hold for _all_ convex subsets when we perturb a metric, not simply
on each individual convex set in a manifold. In contrast, when considering the
Euclidean plane, any perturbation that introduces negative curvature gives a
manifold violating the property above [KN22].

###  1.1. An abridged history of the fundamental gap problem

The study of the fundamental gap has a long history and is important both in
mathematics and physics. In 2011, Andrews and Clutterbuck [AC11] proved the
_fundamental gap conjecture_ : for convex domains
$\Omega\subset\mathbb{R}^{n}$ with diameter $D$,

$\Gamma(\Omega)\geq 3\pi^{2}/D^{2}.$

The result is optimal, with the limiting case being rectangles that collapse
to a segment. We refer to their paper for the history and earlier works on
this important subject (see also the survey article [DSW19]). As discovered in
[SWYY85], the log-concavity of the first eigenfunction is closely related to
the fundamental gap estimate (see (2.10)). For convex domains in
$\mathbb{R}^{n}$, the log-concavity of the first eigenfunction,
$\operatorname{Hess}\,(\log u_{1})\leq 0$, was first obtained in [BL76]. The
fundamental gap conjecture was proved by sharpening this estimate to obtain an
optimal modulus of log-concavity [AC11] (see also [Ni13]). In order to do so,
Andrews-Clutterbuck applied a novel two-point maximum principle.

For convex domains in $\mathbb{S}^{n}$, Lee-Wang [LW87] proved
$\operatorname{Hess}\,(\log u_{1})\leq 0$ using the continuity method as in
[SWYY85], which implies the gap estimate $\geq\pi^{2}/D^{2}$. Later F.-Y. Wang
[Wan00] improved the log-concavity estimate, see Section 2.1. Recently, the
last author joined with Dai, He, Seto, Wang (in various subsets) [SWW19,
HWZ20, DSW21] generalized the two-point modulus of log-concavity estimate to
convex domains in $\mathbb{S}^{n}$, and established the gap estimate

$\lambda_{2}-\lambda_{1}>3\pi^{2}/D^{2},$

among other things.

On the other hand, the size of the fundamental gap behaves very differently in
spaces of negative or mixed curvature. For convex domains in $\mathbb{H}^{n}$,
the first eigenfunction may not be log-concave, or even quasiconcave [Shi89].
This is reproved in [BCN+22], where it is furthermore shown that there is _no
lower bound_ on the product of the fundamental gap and the square of the
diameter for convex domains in the hyperbolic space with arbitrary fixed
diameter. In [NSW21] it is also shown $(\lambda_{2}-\lambda_{1})D^{2}$ can be
arbitrary small even for convex domains in $\mathbb{H}^{n}$ with first
eigenfunction log-concave. Building off the results in hyperbolic space, the
first and second named authors showed that for spaces of negatively pinched
curvature, it is possible to find convex domains of arbitrary diameter whose
fundamental gap is small [KN22]. Furthermore, in manifolds whose sectional
curvature has a mixed sign, it is possible to find convex domains whose
fundamental gap is arbitrarily small (although the diameter of these domains
will need to be small).

Even though there is a vast amount of literature studying the convexity of
solutions in $\mathbb{R}^{n}$ and $\mathbb{S}^{n}$, there are relatively few
results when the curvature is non-constant and positive. In fact, it is still
open whether the first eigenfunction of convex domains in $\mathbb{CP}^{n}$ is
log-concave. On the other hand there are fundamental gap lower bound estimates
for domains satisfying an interior rolling ball condition for general
manifolds satisfying certain curvature bounds only (see [OSW99, RORWW]).

###  1.2. Overview of paper

The paper is structured as follows. In Section 2, we show a general result
about the log-concavity of the first eigenfunction for convex domains in an
arbitrary Riemannian manifold, reducing the problem of establishing concavity
estimates to finding a barrier function which satisfies a particular PDE
inequality and does not grow too quickly at the boundary, see Theorem 2.1.
Using this approach, we quickly reprove several earlier results on the
fundamental gaps for convex domains of $\mathbb{S}^{n}$ [LW87, Wan00] by
choosing appropriate barriers.

In Section 3, we find a particular barrier that works for surfaces of positive
curvature. Doing so, we prove Theorem 1.1. In subsection 3.1, we show all
relatively round ellipsoids satisfy the necessary condition.

Finally, in Section 4, we discuss the relationship between fundamental gap
estimates and metric deformations. We also study the relationship between
Ricci flow and fundamental gap estimates, and show that metrics which are
sufficiently pinched satisfies Condition (1.1) after a short period of time
under normalized Ricci flow (see Corollary 4.3).

Acknowledgements: The last author would like to thank Gunhee Cho, Xianzhe Dai,
and Shoo Seto for earlier discussions about the Lee-Wang paper [LW87].

##  2\. Log-concavity via the barrier PDE approach

In this section, we adapt the approach of [SWYY85, LW87] to a general setting
to show that establishing log-concavity can be reduced to constructing a
barrier which satisfies a particular PDE inequality. Doing so recovers all the
earlier work that could be obtained with one-point estimates, and will be
useful for proving new results as well.

First we fix some notations. Let

$R(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z$

be the $(1,3)$ curvature tensor, and $R_{X}$ the $(1,1)$-tensor given by

$R_{X}(Y)=R(Y,X)X.$

Given a strictly convex domain $\Omega\subset M^{n}$, assume there is a family
of smooth strictly convex domains $\Omega(t)\subset\Omega$ such that
$\Omega(0)$ is a sufficiently small ball in $\Omega$ and that
$\Omega(1)=\Omega$. This can be constructed explicitly when
$M^{n}=\mathbb{R}^{n}$ using support functions. Within any symmetric space of
positive curvature, mean curvature flow [Hui86] will preserve the convexity of
domains and on surfaces of positive curvature, curve shortening flow will do
the same [Gag90, Prop.1.4]. This process gives an almost round domain near
time of singularity, one can then deform this small domain to a ball. In
particular, one can construct such a deformation for any convex domains in
$\mathbb{S}^{n},\mathbb{CP}^{n}$ or convex domains in a positively curved
surface.

######  Theorem 2.1.

Let $\Omega\subset M^{n}$ be a compact strictly convex domain and $\Omega(t)$
a deformation as above. Let $v(x,t)=\log u_{1}(x,t)$, where $u_{1}$ is a
positive first eigenfunction of the Laplacian on $\Omega(t)$ with Dirichlet
boundary condition. Suppose there is a function $b:\Omega\to\mathbb{R}$ which
is uniformly bounded, solves the PDE inequality

(2.1) $\Delta b>2b^{2}-2\langle\nabla b,\nabla
v\rangle+2\operatorname{Tr}[(\operatorname{Hess}v+\nabla v\otimes\nabla
v)\circ R_{X}]+2b\operatorname{Ric}(X,X)\\\ +(\nabla_{\nabla
v}\operatorname{Ric})(X,X)-2(\nabla_{X}\operatorname{Ric})(X,\nabla v)$

for all $t\in[0,1]$ whenever $X\in T_{p}\Omega$ a unit vector satisfying
$\operatorname{Hess}\,v(X_{p},X_{p})+b(p)=0$ and
$\operatorname{Hess}\,v(X_{p},X_{p})\geq\operatorname{Hess}\,v(Y_{p},Y_{p})$
for all unit vectors $Y_{p}\in T_{p}\Omega$.

Then the function $v(1,x)$ satisfies the concavity estimate

(2.2) $\operatorname{Hess}\,v+b\,g\leq 0\quad\textup{on }\Omega.$

Note that the derivatives of curvature terms only involve Ricci tensors, so
this equation simplifies considerably for Einstein manifolds, not just
symmetric spaces.

The proof of this follows along the same lines as [SWYY85] for
$\mathbb{R}^{n}$ and [LW87] for $\mathbb{S}^{n}$. We provide a complete proof
for a general manifold, filling some details in [LW87]. One of the key
computations in the proof is commuting derivatives. For convenience, we state
these formulas now.

For a Riemannian manifold $M^{n}$, let $f\in C^{4}(M),$ and $\\{e_{i}\\}$ a
basis of $T_{p}M$. Denote

$\displaystyle f_{ij}=\textup{Hess}\,f(e_{i},e_{j}),\ \ \
f_{ij,k}=[\nabla_{e_{k}}\textup{Hess}\,f](e_{i},e_{j}),\ \ \
f_{ij,kl}=[\nabla_{e_{l}}\nabla_{e_{k}}\textup{Hess}\,f](e_{i},e_{j}).$

We will use the following formulas to change order of derivatives, see e.g.
[CM20, Page 26, Lemma 2.1].

(2.3) $\displaystyle f_{ij,k}$ $\displaystyle=f_{ik,j}+\sum_{t}f_{t}R_{tijk},$
(2.4) $\displaystyle\sum_{i}f_{jj,ii}$
$\displaystyle=\sum_{i}\left[f_{ii,jj}+2f_{ij}\operatorname{Ric}_{ij}+2f_{i}\operatorname{Ric}_{ij,j}-f_{i}\operatorname{Ric}_{jj,i}-\sum_{s}2f_{si}R_{jsji}\right],$

where $R_{ijkl}=\langle R(e_{j},e_{i})e_{k},e_{l}\rangle$.

###### Proof.

First we note (2.2) is satisfied with a strict inequality near the boundary of
$\Omega(t)$ for each $t\in[0,1]$ since $\Omega(t)$ are strictly convex, and
$b$ is bounded (see, e.g., Lemma 3.4 in [SWW19]).

Furthermore, when $\Omega(0)$ is a sufficiently small ball, then (2.2) also
holds with a strict inequality on $\Omega(0)$. Indeed, for a small ball with
radius $\rho$, if we scale the metric by $\rho^{-2}$ so it becomes a unit
ball, then (2.2) becomes

$\displaystyle\operatorname{Hess}\,v+\rho^{2}b\bar{g}\leq 0,$

where $\bar{g}=\rho^{-2}$ is the new metric, since $\operatorname{Hess}\,v$ is
scale invariant. Since $\bar{g}$ can be made arbitrarily close to the
Euclidean metric by making $\rho$ sufficiently small, and the first
eigenfunction of a unit ball in Euclidean space is strongly log-concave, the
same result holds in our setting as well but in our case the radius of
$\Omega(0)$ depends on $b$.

Now we argue by contradiction. Suppose that
$\operatorname{Hess}\,v(X,X)_{p}+b(p)>0$ for some unit vector $X\in
T_{p}\Omega,$ $p\in\Omega.$ Using the preceding argument and the continuity in
$t$, there must be a time $t_{0}\in(0,1)$ such that the maximum of the mapping

$\displaystyle U\Omega(t)\rightarrow\mathbb{R},\quad
X_{q}\mapsto\operatorname{Hess}\,v(X_{q},X_{q})+b(q)$

is equal to zero, where $U\Omega(t)$ is the unit tangent bundle of
$\Omega(t)$. That is, there exists a unit vector $X_{p}\in T_{p}\Omega(t_{0})$
such that

$\displaystyle 0=\operatorname{Hess}\,v(X_{p},X_{p})+b(p)=\max_{Y_{q}\in
U\Omega(t_{0})}\left(\operatorname{Hess}\,v(Y_{q},Y_{q})+b(q)\right).$

By the estimates at the boundary for $\Omega(t)$, the maximum must be achieved
at a point $p$ in the interior of $\Omega(t_{0})$.

Let us denote $e_{1}=X_{p}$, and extend this vector to an orthonormal basis
$\\{e_{j}\\}$ of $T_{p}\Omega(t_{0})$. Now $e_{1}$ is the maximal direction of
$\textup{Hess}\,v$ at $p$, which is symmetric, so $e_{1}$ is an eigenvector of
$\textup{Hess}\,v$. Since $\\{e_{j}\\}$ is an orthonormal basis, we have at
$p$,

$\displaystyle\textup{Hess}\,v_{p}(e_{1},e_{j})=\langle\textup{Hess}\,v_{p}(e_{1}),\
e_{j}\rangle=0\quad\textup{ for }j=2,\dots,n.$

For any $q$ in a neighborhood of $p$, if we connect $q$ to $p$ by the unique
minimal geodesic and parallel translate $e_{1}$ along the minimal geodesic to
$q$, we obtain a smooth function $v_{11}=\operatorname{Hess}v(e_{1},e_{1})$
defined in a neighborhood of $p$, and the function $v_{11}+b$ achieves its
maximum at $p$. Hence by the maximum principle

(2.5) $\displaystyle e_{i}(v_{11})(p)+e_{i}(b)(p)$
$\displaystyle=0\quad\textup{for all }i=1,\dots,n$
$\displaystyle\Delta(v_{11})(p)+\Delta b(p)$ $\displaystyle\leq 0.$

To compute $\Delta(v_{11})(p)$, we extend the orthonormal basis $\\{e_{j}\\}$
of $T_{p}\Omega(t_{0})$ to the geodesic frame in a neighborhood of $p$, i.e.
$\\{e_{j}\\}$ is an orthonormal frame and $\nabla_{e_{i}}e_{j}(p)=0$ for all
$i,j$. Then at $p$

$\displaystyle\Delta(v_{11})=\sum_{i}e_{i}e_{i}v_{11}=\sum_{i}e_{i}(e_{i}v_{11}).$

In a neighborhood of $p$,

$\displaystyle e_{i}(v_{11})$
$\displaystyle=v_{11,i}+2\sum_{j}\Gamma^{j}_{i1}v_{j1}.$

Therefore at $p,$ since $\Gamma_{ij}^{k}=0$, and $v_{1j}=0$ for $j\neq 2$, we
have

$\displaystyle e_{i}e_{i}(v_{11})$
$\displaystyle=v_{11,ii}+2\sum_{j}e_{i}(\Gamma^{j}_{i1})v_{j1}+2\sum_{j}\Gamma^{j}_{i1}e_{i}(v_{j1})$
$\displaystyle=v_{11,ii}+2e_{i}(\Gamma^{1}_{i1})v_{11}.$

Since $\\{e_{j}\\}$ is an orthonormal frame, $\Gamma^{1}_{i1}=0$ in the
neighborhood of $p.$ In summary, we have

$\displaystyle\Delta(v_{11})(p)=\sum_{i}v_{11,ii}(p).$

To compute the right hand side, recall that $v$ satisfies

(2.6) $\displaystyle\|\nabla v\|^{2}=-\lambda-\Delta v,$

so we commute the derivatives. Using (2.4), we have

$\displaystyle\sum_{i}v_{11,ii}=\sum_{i}v_{ii,11}+2v_{11}\operatorname{Ric}_{11}-2\sum_{ij}v_{ji}R_{1j1i}+\sum_{i}2v_{i}\operatorname{Ric}_{i1,1}-\sum_{i}v_{i}\operatorname{Ric}_{11,i}.$

Taking the derivative of both sides of (2.6), one has that

$\displaystyle\sum_{i}v_{ii,11}=-2\sum_{i}(v_{i1}^{2}+v_{i1,1}v_{i}).$

Using (2.3) and (2.5), we have

$\displaystyle
v_{i1,1}=v_{11,i}+\sum_{j}v_{j}R_{j1i1}=-b_{i}+\sum_{j}v_{j}R_{j1i1}.$

Hence,

$\displaystyle\sum_{i}v_{ii,11}$
$\displaystyle=-2\sum_{i}\left(v_{i1}^{2}-v_{i}b_{i}+\sum_{j}v_{i}v_{j}R_{j1i1}\right)$
$\displaystyle=-2b^{2}+2\langle\nabla b,\nabla
v\rangle-2\sum_{i,j}v_{i}v_{j}R_{j1i1}.$

Here in the second equation we used $v_{11}=-b,\ v_{i1}=0$ for $i=2,\cdots,n$.
We therefore arrive at

$\displaystyle 0$ $\displaystyle\geq\Delta(v_{11}+b)(p)$
$\displaystyle=-2b^{2}+2\langle\nabla b,\nabla
v\rangle-2\sum_{i,j}v_{i}v_{j}R_{j1i1}-2\sum_{i,j}v_{ij}R_{1j1i}-2b\operatorname{Ric}_{11}-\sum_{j}v_{j}\operatorname{Ric}_{11,j}$
$\displaystyle\hskip
252.94499pt+2\sum_{j}v_{j}\operatorname{Ric}_{1j,1}+\Delta b(p)$
$\displaystyle=-2b^{2}+2\langle\nabla b,\nabla
v\rangle-2\sum_{i,j}R_{j1i1}(v_{i}v_{j}+v_{ij})-2b\operatorname{Ric}_{11}-\nabla_{\nabla
v}\operatorname{Ric}(e_{1},e_{1})$ $\displaystyle\hskip
252.94499pt+2\nabla_{e_{1}}\operatorname{Ric}(e_{1},\nabla v)+\Delta b(p)$

which contradicts (2.1). ∎

###  2.1. Application to Round Spheres

Although the inequality (2.1) seems quite hard to check in general, when $b$
is a constant and the manifold has constant sectional curvature, it simplifies
dramatically.

Let $M^{n}=\mathbb{S}^{n}$, and take $b$ to be constant. Then
$\operatorname{Ric}\equiv n-1$, $R_{X}(Y)=Y-\langle X,Y\rangle X$. Therefore
(2.1) becomes

$0>2b^{2}+2\sum_{i=2}^{n}(v_{ii}+v_{i}^{2})+2(n-1)b=2b^{2}+2(n-1)b+2\left(\Delta
v+|\nabla v|^{2}-v_{11}-v_{1}^{2}\right).$

Using equation (2.6) and that at maximal $v_{11}=-b$, this is equivalent to

(2.7) $0>2b^{2}-2nb-2\lambda_{1}(\Omega(t))-2v_{1}^{2}.$

This is satisfied for all $t\in[0,1]$ when $b=0$, which recovers Lee-Wang
estimate [LW87] $\operatorname{Hess}v\leq 0.$ As
$\Omega(t)\subset\Omega(1)=\Omega$ for all $t\in[0,1]$, by domain monotonicity
$\lambda_{1}(\Omega)\leq\lambda_{1}(\Omega(t))$. Hence (2.7) is satisfied for
all $t\in[0,1]$ when $b$ satisfies

$0>2b^{2}-2nb-2\lambda_{1}(\Omega).$

This is the case whenever $b<\sqrt{n^{2}+4\lambda(\Omega)}-n$. This recovers
F.Y. Wang’s estimate [Wan00]

(2.8) $\operatorname{Hess}v\leq n-\sqrt{n^{2}+4\lambda_{1}(\Omega)}.$

It is worth noting that in equation (2.7), there is an additional negative
term $-2v_{1}^{2}$, which we have discarded. This term will be very important
when we consider surfaces with variable curvature.

This log-concavity estimate on the first eigenfunction implies the following
gap estimate.

######  Corollary 2.2.

For $\Omega\subset\mathbb{S}^{n}$ a convex domain, its fundamental gap has the
following lower bound

(2.9)
$\lambda_{2}-\lambda_{1}>\frac{\pi^{2}}{D^{2}}+\sqrt{n^{2}+4\lambda_{1}}-\frac{n+1}{2},$

where $D$ is the diameter of the domain $\Omega$.

In [Wan00, Theorem 1.2] the weaker estimate,
$\lambda_{2}-\lambda_{1}>\frac{\pi^{2}}{D^{2}}+(1-\frac{2}{\pi})\left(\sqrt{n^{2}+4\lambda_{1}}-1\right)$
was stated.

Comparing the gap estimate $\lambda_{2}-\lambda_{1}>\frac{3\pi^{2}}{D^{2}}$ in
[DSW21] to (2.9), one finds that (2.9) is a stronger estimate when the first
eigenvalue is large but the diameter is not too small. It shows that for
convex domains of $\mathbb{S}^{n}$, the fundamental gap is large whenever the
first eigenvalue is large, independent of the diameter. In particular, for any
convex thin strip, the gap goes to infinity whenever the inscribed radius $r$
goes to zero since $\lambda_{1}>\frac{\pi^{2}}{4r^{2}}+\frac{n-1}{2}$ [Lin06].
Now this is not true for thin convex domains on $\mathbb{R}^{n}$, as for thin
rectangles the gaps stay bounded in term of the diameter. On the other hand,
in the hyperbolic space or a space a mixed or negative curvature, the gap may
go to zero as the inscribed radius $r$ goes to zero [BCN+22, KN22].

###### Proof of Corollary 2.2.

Let $w=\frac{u_{2}}{u_{1}}$, where $u_{1},\ u_{2}$ are the first and second
eigenfunctions of the Laplacian on $\Omega$ with Dirichlet boundary condition.
Then $w$ can be extended to a smooth function on $\bar{\Omega}$ and satisfies

(2.10) $\Delta w+2\langle\nabla\log u_{1},\nabla
w\rangle=-(\lambda_{2}-\lambda_{1})w.$

with Neumann boundary condition on $\partial\Omega$ [SWYY85], see also [SWW19,
(4.7)]. Consider $\mathbb{S}^{n}$ with weighted measure
$u_{1}^{2}dvol=e^{-(-2\log u_{1})}$. The Bakry-Emery Ricci curvature of this
metric-measure space is $n-1-2\operatorname{Hess}\log u_{1}$, which is larger
than $2\sqrt{n^{2}+4\lambda_{1}}-n-1$ by (2.8). Furthermore, the weighted
Laplacian acting on $w$ is exactly the left hand side of (2.10). By [BQ00],
the first nonzero Neumann eigenvalue of a weighted Laplacian is
$\geq\bar{\mu}_{1}(n,D,a)$, where $a>0$ is a lower bound on the Bakry-Emery
Ricci curvature, see [AN12, Theorem 1,1]. By [AN12, Propsition 3.1],
$\bar{\mu}_{1}(n,D,a)\geq\frac{\pi^{2}}{D^{2}}+\frac{a}{2}$. Now the result
follows as $a>2\sqrt{n^{2}+4\lambda_{1}}-n-1$. ∎

###  2.2. Barrier functions which do not satisfy the boundary conditions

In Theorem 2.1, we have assumed that $b$ is bounded, and all the barriers that
we consider in this paper will be bounded. However, it is possible to weaken
this assumption. In particular, whenever $b$ satisfies the boundary growth
condition

(2.11) $\lim_{z\to\partial\Omega}b(z)dist(z,\partial\Omega)=0,$

the concavity assumption $\operatorname{Hess}v+bg<0$ is satisfied near the
boundary.

One point of interest is that there are barrier functions which satisfy
inequality (2.1) but do not satisfy the boundary requirements. For instance,
on any rank-one symmetric space of compact type, the barrier $b=|\nabla
v|^{2}$ satisfies inequality (2.1). However, this barrier corresponds to the
principal eigenfunction being concave, which cannot hold (except in dimension
$1$).

More interestingly, on a space whose curvature is positively pinched, the
barrier

(2.12) $b=\frac{\alpha}{n}|\nabla v|^{2}$

satisfies Inequality (2.1) when $\lambda_{1}$ is sufficiently large and
$\alpha$ is between $0$ and $1$. The precise values of $\alpha$ and
$\lambda_{1}$ which make this function a solution to (2.1) depend on the
curvature pinching and the derivative of the Ricci curvature in a fairly
delicate way.111The derivation to show this solves (2.1) incorporates ideas
from the Li-Yau gradient estimate [LY86], but is a fairly lengthy calculation,
which is why we have omitted it. However, it is possible to show that this
barrier fails at every point in the boundary under Dirichlet conditions (see
Lemma 3.4 of [SWW19]).

Therefore, these barriers cannot be used directly to study the fundamental gap
problem. However, they can be used to establish log-concavity properties for
eigenfunctions which satisfy other boundary conditions (e.g., non-homogeneous
positive Dirichlet conditions).

##  3\. Surfaces with non-constant positive curvature

In this section we apply Theorem 2.1 to surfaces to prove Theorem 1.1. The
crux is to find an appropriate barrier function, which will not be constant.

Before we present the proof, note that by [Gag90, Prop. 1.4], a strictly
convex domain $\Omega\subset M^{2}$ is still strictly convex when the boundary
is deformed by curve shortening flow. Moreover when $M^{2}$ has positive
curvature, the minimum of the geodesic curvature of the curve is nondecreasing
(i.e. the $\alpha$ in the proof of Prop. 1.4 in [Gag90] can be chosen to be
zero). Hence we have a family of strictly convex domain $\Omega(t)$ such that
$\Omega(1)=\Omega$ and $\Omega(0)$ is very close to a small ball with
$\Omega(t)\subset\Omega$ as needed for Theorem 2.1.

###### Proof of Theorem 1.1.

When $n=2$, $\operatorname{Tr}[(\operatorname{Hess}v+\nabla v\otimes\nabla
v)\circ R_{X}]$ simplifies considerably. As before, at the maximal $X_{p}$,
denote $e_{1}=X_{p}$ and $e_{2}$ its orthogonal complement so that
$\\{e_{1},e_{2}\\}$ form an orthonormal basis. Then

$\displaystyle\operatorname{Tr}[(\operatorname{Hess}v+\nabla v\otimes\nabla
v)\circ R_{X}]$ $\displaystyle=\kappa(v_{2}^{2}+v_{22})$
$\displaystyle=\kappa(-\lambda_{1}-v_{1}^{2}-v_{11})=\kappa(-\lambda_{1}-v_{1}^{2}+b).$

Here we used (2.6) in the second equality. Also
$\operatorname{Ric}=\kappa\,g$, hence $(\nabla_{\nabla
v}\operatorname{Ric})(X,X)-2(\nabla_{X}\operatorname{Ric})(X,\nabla
v)=v_{1}\kappa_{1}+v_{2}\kappa_{2}-2v_{1}\kappa_{1}=v_{2}\kappa_{2}-v_{1}\kappa_{1}.$
Therefore (2.1) becomes

$\displaystyle\Delta b$ $\displaystyle>2b^{2}-2\langle\nabla b,\nabla
v\rangle-2\kappa(\lambda_{1}+v_{1}^{2})+4b\kappa+v_{2}\kappa_{2}-v_{1}\kappa_{1}.$

Note that if $b=0$, the problem terms are $v_{2}\kappa_{2}-v_{1}\kappa_{1}$.
One can use $-2\kappa v_{1}^{2}$ to control $v_{1}\kappa_{1}$, so one would
like to choose $b$ suitably to cancel $v_{2}\kappa_{2}$. Now if we choose

(3.1) $b=\tfrac{\kappa}{2}+C,$

where $C\in\mathbb{R}$ is a constant (we add the constant to allow more
flexibility), this does the job. The expression above becomes

$\displaystyle\Delta\tfrac{\kappa}{2}$
$\displaystyle>2(\tfrac{\kappa}{2}+C)^{2}-2v_{1}\kappa_{1}-2\kappa(\lambda_{1}+v_{1}^{2})+4\kappa(\tfrac{\kappa}{2}+C)$
$\displaystyle=2(\tfrac{\kappa}{2}+C)(\tfrac{5\kappa}{2}+C)-2\kappa\lambda_{1}-2\kappa(v_{1}+\tfrac{\kappa_{1}}{2\kappa})^{2}+\tfrac{\kappa_{1}^{2}}{2\kappa}.$

Multiplying the above by $\tfrac{2}{\kappa}$ gives

$\displaystyle\tfrac{\Delta\kappa}{k}-\tfrac{\kappa_{1}^{2}}{\kappa^{2}}$
$\displaystyle>-4\lambda_{1}+(1+\tfrac{2C}{\kappa})(5\kappa+2C)-2\kappa(v_{1}+\tfrac{\kappa_{1}}{2\kappa})^{2}$
$\displaystyle=-4\lambda_{1}+5\kappa+4C(3+\tfrac{C}{\kappa})-2\kappa(v_{1}+\tfrac{\kappa_{1}}{2\kappa})^{2}.$

This is satisfied when

$\Delta\log\kappa>-4\lambda_{1}(\Omega(t))+5\kappa+4C(3+\tfrac{C}{\kappa}).$

Again since $\Omega(t)\subset\Omega(1)$, above is satisfied for all
$t\in[0,1]$ if it is satisfied at $t=1$. Applying Theorem 2.1 finishes the
proof. ∎

It should be noted that there are two algebraic cancellations in the preceding
argument. Because the curvature can be expressed in terms of a single
function, it is possible to build a barrier which eliminates the problem
$v_{2}\kappa_{2}$ term. It is remarkable that after completing the square to
control the $v_{1}\kappa_{1}$ term, the extra term is controlled when we
rewrite the inequality so that the leading term is $\Delta\log\kappa$.

With the log-concavity estimate in Theorem 1.1, we obtain the gap estimate in
Corollary 1.2 as in the proof of Corollary 2.2. Namely, the Bakry-Emery Ricci
is $\geq\inf_{\Omega}\kappa-2(-C-\tfrac{\kappa}{2})\geq
2C+2\inf_{\Omega}\kappa$, and the fundamental gap
$\lambda_{2}-\lambda_{1}\geq\frac{\pi^{2}}{D^{2}}+\inf_{\Omega}\kappa+C$.

###  3.1. Examples

In this subsection, we specialize condition (1.1) for $C=0,-\kappa_{\min}$ and
find examples of surfaces which satisfy these conditions for all convex
domains. In order to do so, let us first prove a corollary which gives a way
to verify this condition solely by computing the curvature and its
derivatives.

######  Corollary 3.1.

If the sectional curvature $\kappa$ of $M^{2}$ satisfies
$\kappa\geq\kappa_{\min{}}>0$ and

(3.2) $-\Delta\log\kappa\leq 6\kappa_{\min{}}-5\kappa\ \ \ \ \mbox{or}\ \ \
-\Delta\log\kappa\leq 11\kappa_{\min{}}-5\kappa,$

then for any convex domain $\Omega\subset M$, its first Dirichlet
eigenfunction $u_{1}$ satisfies

(3.3) $\operatorname{Hess}\,(\log u_{1})\leq-\frac{\kappa}{2}\ \ \mbox{or}\ \
\ \operatorname{Hess}\,(\log u_{1})\leq 0,$

respectively. In particular, when $M^{2}$ is positively pinched with
$0<\kappa_{\min{}}\leq\kappa\leq\alpha\kappa_{\min{}}$, then assumptions (3.2)
are satisfied if $1\leq\alpha<\tfrac{6}{5}$, and
$-\Delta\log\kappa<(6-5\alpha)\kappa_{\min{}},$ or
$1\leq\alpha<\tfrac{11}{5}$, and
$-\Delta\log\kappa<(11-5\alpha)\kappa_{\min{}}$.

###### Proof.

Given any domain $\Omega$, Ling [Lin06] showed that when the Ricci curvature
satisfies $\operatorname{Ric}\geq(n-1)\kappa_{\min{}}$ and the boundary
$\partial\Omega$ has non-negative mean curvature, the first Dirichlet
eigenvalue satisfies

$\lambda_{1}(\Omega)\geq\frac{1}{2}(n-1)\kappa_{\min{}}+\frac{\pi^{2}}{d^{2}},$

where $d$ is the diameter of the largest inscribed ball. Furthermore, Myers’s
theorem implies that the diameter of $M$ satisfies
$\textrm{diam}(M)^{2}<\frac{\pi^{2}}{\kappa_{\min{}}}$. Since
$\textrm{diam}(M)>d(\Omega)$, we have
$\lambda_{1}(\Omega)\geq\frac{1}{2}(n+1)\kappa_{\min{}}$. Combining this with
(1.1) and letting $C=0,-\kappa_{\min}$ respectively, then (1.2) gives (3.3). ∎

With this result, we can show all relatively round ellipsoids satisfy (3.2)
for all convex domains. Hence the first eigenfunctions are all log-concave.

######  Example 3.2.

For any convex domain $\Omega$ in an ellipsoid of the form

$\frac{x^{2}}{\epsilon^{2}}+y^{2}+z^{2}=1,$

the first Dirichlet eigenfunction $u_{1}$ satisfies
$\operatorname{Hess}\,(\log u_{1})\leq 0$ whenever $\epsilon\in[.8064,1]$.

We will take for granted the fact that the quantity $\Delta\log\kappa-5\kappa$
is minimized at the point of maximal curvature for such an ellipsoid. This can
be verified in Mathematica. For such ellipsoids, the MTW curvature has been
computed explicitly in [FRV10, Section 6], in order to study the regularity of
optimal transport on these surfaces. We use some of the computations from this
paper.

###### Proof.

As in [FRV10], we can view an ellipsoid as a surface of revolution by rotating
the graph $z=F_{\epsilon}(x)$ along the $x$-axis, where
$F_{\epsilon}:[-\epsilon,\epsilon]\rightarrow\mathbb{R}_{+}$ is defined by

$F_{\epsilon}(x)=\left(1-\tfrac{x^{2}}{\epsilon^{2}}\right)^{1/2}.$

Its metric is

$dr^{2}+f^{2}(r)d\theta^{2},$

where $r(x)=\int_{-\epsilon}^{x}\sqrt{1+F^{\prime}(u)^{2}}du$,
$f(r(x))=F_{\epsilon}(x).$ Then the Gaussian curvature
$\kappa=-\frac{f^{\prime\prime}}{f}$, and the Laplace operator
$\Delta=\frac{\partial^{2}}{\partial
r^{2}}+\frac{f^{\prime}}{f}\frac{\partial}{\partial
r}+\frac{1}{f^{2}}\frac{\partial^{2}}{\partial\theta^{2}}.$ Hence

$\displaystyle\Delta\log\kappa-5\kappa$ $\displaystyle=$
$\displaystyle\tfrac{\partial^{2}}{\partial
r^{2}}\left[\log(-f^{\prime\prime})-\log
f\right]+\tfrac{f^{\prime}}{f}\tfrac{\partial}{\partial
r}\left[\log(-f^{\prime\prime})-\log
f\right]-\left(-5\tfrac{f^{\prime\prime}}{f}\right)$ $\displaystyle=$
$\displaystyle\frac{f^{\prime\prime\prime\prime}}{f^{\prime\prime}}-\left(\frac{f^{\prime\prime\prime}}{f^{\prime\prime}}\right)^{2}+\frac{f^{\prime}f^{\prime\prime\prime}}{ff^{\prime\prime}}+4\frac{f^{\prime\prime}}{f}.$

At $(0,0,1)$, which achieves the maximal curvature, $F_{\epsilon}(0)=1,\
F_{\epsilon}^{\prime}(0)=F_{\epsilon}^{\prime\prime\prime}(0)=0,\
F_{\epsilon}^{\prime\prime}(0)=-\tfrac{1}{\epsilon^{2}},\
F_{\epsilon}^{\prime\prime\prime\prime}(0)=-\tfrac{3}{\epsilon^{4}}$, and
$f=1,\ f^{\prime}=f^{\prime\prime\prime}=0,\
f^{\prime\prime}=-\tfrac{1}{\epsilon^{2}},\
f^{\prime\prime\prime\prime}=F_{\epsilon}^{\prime\prime\prime\prime}(0)-4(F_{\epsilon}^{\prime\prime}(0))^{3}.$

Hence at $(0,0,1)$, $\Delta\log\kappa-5\kappa=-\epsilon^{-2}-4\epsilon^{-4}.$
Now $-\epsilon^{-2}-4\epsilon^{-4}+11\geq 0$ when
$0.80634\approx(\frac{1+\sqrt{177}}{22})^{1/2}\leq\epsilon\leq 1$. ∎

If we drop the assumption that the ellipsoid is rotationally symmetric, it is
possible to use Mathematica to minimize the quantity
$\Delta\log\kappa-5\kappa+12\kappa_{\min{}}-\frac{\kappa_{\min{}}^{2}}{\kappa}$.
Doing so, we find the following (See Definition 4.2 for a definition of
spectral dominance).

######  Example 3.3.

An ellipsoid of the form

$\frac{x^{2}}{a}+\frac{y^{2}}{b}+\frac{z^{2}}{c}=1$

with $a\leq b\leq c$ satisfies the spectral dominance property whenever

(3.4) $\frac{c}{a}\leq 1.132$

##  4\. Deformations of metrics

###  4.1. The spectral dominance and $\Gamma_{\delta}$ properties

Throughout this section, it will be convenient to have terminology to keep
track of the various gap estimates as well as the hypotheses required to
satisfy them. To this end, we provide some definitions.

######  Definition 4.1.

A space $M$ satisfies the fundamental gap $\delta$ (denoted $\Gamma_{\delta}$)
property if for _all_ convex domains $\Omega$, the following estimate holds:

$\lambda_{2}(\Omega)-\lambda_{1}(\Omega)\geq\frac{\delta\pi^{2}}{D^{2}(\Omega)},$

where $D(\Omega)$ is the diameter of $\Omega$.

In other words, a surface satisfies $\Gamma_{\delta}$ if the following
inequality is satisfied.

$\inf\\{\Gamma(\Omega)\,D^{2}(\Omega)~{}|~{}\Omega\subset M\textrm{
convex}\\}\geq\delta\pi^{2}.$

In view of Theorem 1.1, it is natural to consider surfaces which satisfy the
curvature condition (1.1) for _all_ convex domains. We take
$C=-\frac{\kappa_{\min{}}}{2}$, so that the condition holds for the widest
possible class of surfaces. However, it is possible to consider other values
of $C$ as well.

######  Definition 4.2.

A surface $M$ satisfies the spectral-dominance property if it has positive
curvature and for all convex domains $\Omega\subset M$ and points in $M$, we
have the inequality

$\Delta\log\kappa>\frac{4}{\kappa}\left(\frac{\kappa}{2}-\frac{\kappa_{\min{}}}{2}\right)^{2}-4\lambda+8\left(\frac{\kappa}{2}-\frac{\kappa_{\min{}}}{2}\right).$

In other words, a surface is spectrally dominant whenever

$\inf\\{\lambda(\Omega)~{}|~{}\Omega\subset M\textrm{ convex}\\}+\inf_{z\in
M}\left(\Delta\log\kappa(z)-\frac{(\kappa-\kappa_{\min{}})^{2}}{\kappa}+4(\kappa-\kappa_{\min{}})\right)>0,$

which is to say that the spectrum “dominates” the curvature.

###  4.2. General observations

Corollaries 3.1 and 1.2 imply that the spectral dominance property holds for
$C^{4}$-small deformations of $\mathbb{S}^{2}$, and thus that these spaces
satisfy $\Gamma_{1}$.

Eigenvalues and eigenfunctions depend on the metric in a continuous way (in a
smooth enough topology), so it may be tempting to conclude that since the
round sphere satisfies $\Gamma_{3}$, a nearly-round sphere automatically
satisfies $\Gamma_{1}$, simply by continuity. However, this continuity
argument _fails_ for Euclidean space, which satisfies $\Gamma_{3}$ but can be
deformed in $C^{\infty}$ fashion to a metric with some negative sectional
curvature. Once there is any negative sectional curvature, $\Gamma_{\delta}$
fails for all $\delta>0$ as it is possible to build small convex domains with
arbitrarily small fundamental gaps. So the fundamental gap $\delta$-property
does _not_ depend on the metric in a continuous way.

The reason is that the eigenvalues of domains do not depend on the metric in a
_uniformly_ Lipschitz way. In particular, for a given domain, the Lipschitz
constant of the first two eigenvalues depends strongly on the geometry of the
domain. For instance, if a domain is Gromov-Hausdorff close to a submanifold
of positive codimension, even small perturbations can greatly affect the
spectrum. This effect plays a central role in showing that fundamental gaps
can be arbitrarily small when there is even a single tangent plane of negative
sectional curvature.

These properties explain why we cannot expect the $\Gamma_{\delta}$ condition
to be deformation stable in general. However, the results of this paper raise
the following questions.

######  Question 1.

Under what conditions is the $\Gamma_{\delta}$ condition deformation stable,
and in what topology? For instance, do sufficiently $C^{2}$-small deformations
of a round sphere satisfy $\Gamma_{2.99}$?

Although the $\delta$ in the $\Gamma_{\delta}$ estimate will not always depend
continuously on the metric, in all the examples we have considered, it appears
to be upper-semicontinuous in the metric.

######  Question 2.

Let $(M_{i},g_{\infty})$ be a sequence of Riemannian manifolds which all
satisfy $\Gamma_{\delta}$ and converge to an Alexandrov space222Note that
$\Gamma_{\delta}$ implies nonnegative sectional curvature, so the Gromov-
Hausdorff limit of such spaces is Alexandrov. $M_{\infty}$. Does
$(M_{\infty},g_{\infty})$ also satisfy $\Gamma_{\delta}$?

At first, Gromov-Hausdorff convergence might seem to be too rough for such a
result to hold. However, there is another fourth-order curvature condition for
which this is the case, which suggests that such a result might be possible
[Vil08].

###  4.3. Fundamental gaps and Ricci flows

One question of interest is whether it is possible to weaken the assumption on
the curvature in equation (1.1) to a condition which only depends on the
curvature, and not its derivatives.

If one is willing to deform the metric by Ricci flow, it is possible to obtain
such an estimate after some amount of time. In particular, Hamilton’s Harnack
estimate can be used to gain control of the term $\Delta\log\kappa$, which
implies the following corollary.

######  Corollary 4.3.

Let $(M^{2},g_{0})$ be a Riemannian surface whose sectional curvature
positively $\frac{6}{7}$-pinched and whose average sectional curvature is $1$.
Consider the solution to the normalized Ricci flow

$\partial_{t}g=-Ric(g)+g.$

At time $t=\log(3/2)$, the manifold $(M^{2},g_{t})$ is spectrally dominant.

In fact, for any given ratio $\frac{\kappa_{\min{}}}{\kappa_{\max{}}}$ close
enough to $1$, there is a time $t$ at which the metric is spectrally dominant.
As $\frac{\kappa_{\min{}}}{\kappa_{\max{}}}$ converges to $1$, the smallest
such time for which we can establish this estimate converges to
$\log(6/5)\approx.182$, but with more careful Ricci flow analysis and sharper
eigenvalue bounds, one might be able to lower this time somewhat.

###### Proof.

From Hamilton’s Harnack estimate on surfaces (Theorem 6.3 of [Ham88]), under
the normalized Ricci flow, we have the following estimate.

$\Delta\log\kappa+\kappa-1\geq-\frac{e^{t}}{e^{t}-1}.$

Corollary 3.1 therefore holds as soon as

$1-\frac{e^{t}}{e^{t}-1}-6\kappa_{\max}+11\kappa_{\min}>0.$

What remains is to show that this inequality holds after some amount of time.

To establish this, we consider the evolution of the Gaussian curvature under
the normalized Ricci flow, which is given by

$\frac{\partial}{\partial t}\kappa=\Delta\kappa+\kappa^{2}-\kappa.$

Applying the maximum principle, we find that

$\kappa_{\max{}}(t)<\frac{1}{1-C_{0}e^{t}},$

where $C_{0}=1-\frac{1}{\kappa_{\max{}}(0)}$ and that

$\kappa_{\min{}}(t)>\frac{1}{1-C_{1}e^{t}},$

where $C_{1}=1-\frac{1}{\kappa_{\min{}}(0)}$.

On the other hand, the pinching $\kappa_{\max}-\kappa_{\min}$ satisfies

$\displaystyle\frac{\partial}{\partial t}(\kappa_{\max}(t)-\kappa_{\min}(t))$
$\displaystyle\leq\kappa_{\max}^{2}(t)-\kappa_{\max}(t)-\kappa_{\min}^{2}(t)+\kappa_{\min}(t)$
$\displaystyle=(\kappa_{\max}+\kappa_{\min}-1)(\kappa_{\max}(t)-\kappa_{\min}(t))$
$\displaystyle<\frac{1}{1-C_{0}e^{t}}(\kappa_{\max}(t)-\kappa_{\min}(t)).$

In other words, for a short period of time, the curvature pinching grows at
most exponentially, where the exponential constant is bounded by an upper
bound on the sectional curvature.

If the initial curvature is $\alpha$-pinched and the average curvature is $1$,
then the maximum principle provides curvature bounds up to time
$t=\log(\frac{1}{1-\alpha})$. We can solve the differential equations to find
explicit upper bounds on $\kappa_{\max}$ and
$\kappa_{\max}(t)-\kappa_{\min}(t)$ as well as lower bounds on
$\kappa_{\min}$.

Doing so, we have that

$\kappa_{\max}<\frac{1}{1-C_{0}e^{t}},\quad\kappa_{\min}>\frac{1}{1+C_{1}e^{t}},$

and

$\kappa_{\max}-\kappa_{\min}<C_{1}\frac{e^{t}}{1-C_{0}e^{t}}.$

where $C_{0}=1-\alpha$ and $C_{1}=\frac{1}{\alpha}-1$. Putting this together,
we have that

$1-\frac{e^{t}}{e^{t}-1}-6\kappa+11\kappa_{\min}>1-\frac{e^{t}}{e^{t}-1}-6C_{1}\frac{e^{t}}{1-C_{0}e^{t}}+5\frac{1}{1+C_{1}e^{t}}.$

Whenever this quantity is positive, the surface is spectral dominant. For
instance, when $\alpha=\frac{6}{7}$, at $t=\log(3/2)$,

$1-\frac{e^{t}}{e^{t}-1}-6C_{1}\frac{e^{t}}{1-C_{0}e^{t}}+5\frac{1}{1+C_{1}e^{t}}=\frac{1}{11},$

which implies any surface which is $\frac{6}{7}$ pinched becomes spectral
dominant at time $t=\log\left(\frac{3}{2}\right)\approx.405$. ∎

Note that this argument does not rule out the possibility that the metrics
fail to satisfy the spectral dominance condition at some later time. For
instance, for metrics which are initially $\frac{6}{7}$-pinched, this analysis
no longer implies spectral dominance after $t\approx.67485$. Since the metric
is converging exponentially quickly to the round sphere in $C^{4}$, there is a
time $T$ after which the solution will always satisfy spectral dominance.
However, we are currently unable to make quantitative estimates on the time
$T$.

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