# Schwarz lemma: the case of equality and an extension

Haojie Chen, Xiaolan Nie  Department of Mathematics, Zhejiang Normal
University, Jinhua Zhejiang, 321004, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We prove two results related to the Schwarz lemma in complex geometry. First,
we show that if the inequality in the Schwarz lemmata of Yau, Royden and
Tosatti becomes equality at one point, then the equality holds on the whole
manifold. In particular, the holomorphic map is totally geodesic and has
constant rank. In the second part, we study the holomorphic sectional
curvature on an almost Hermitian manifold and establish a Schwarz lemma in
terms of holomorphic sectional curvatures in almost Hermitian setting.

Dedicated to Professor Peter Li on the occasion of his 70th birthday

##  1\. Introduction

The Ahlfors-Schwarz lemma [1] is one of the most important results in complex
analysis and differential geometry. It says that if $f$ is a holomorphic map
from the unit disk $\mathbb{D}$ into a Riemann surface with a conformal metric
$ds^{2}$ with Gaussian curvature at most $-1$, then $f$ is distance decreasing
in the sense that

$\displaystyle f^{*}ds^{2}\leq ds_{\mathbb{D}}^{2},$ (1.1)

where $ds_{\mathbb{D}}^{2}$ is the Poincaré metric on $\mathbb{D}$. In [8],
Heins proved the strong form of the Ahlfors-Schwarz lemma: if the equality of
$(\ref{1.1})$ holds at one point in $\mathbb{D}$, then it holds on
$\mathbb{D}$ and $f$ is an isometric immersion. This generalizes the equality
case of the classical Schwarz lemma. Later, the result was also proved by
Royden [19] and Minda [15] using different methods (see also e.g. [4],[12],[2]
for the boundary rigidity of the Schwarz lemma). Among other applications, the
strong form of the Ahlfors-Schwarz lemma has been applied to study the Bloch
constant problem [8], which is one of the most intriguing open problems in
complex analysis of one variable.

††footnotetext: This research is partially supported by NSFC grants No.
11901530, No. 11801516, No. 12071161 and Zhejiang Provincial NSF grant No.
LY19A010017.

In complex geometry, the Ahlfors-Schwarz lemma has been generalized to higher
dimensional manifolds and to more general settings, by the work of Chern [6],
Lu [13], Yau [26], Chen-Cheng-Look [5], Royden [18], Tosatti [20] and Ni
[16][17] etc. (see also [10] and references therein). In the first part of the
paper, we study the equality case of the Schwarz lemmata obtained by Yau,
Royden and Tosatti. To be more precise, Yau [26] proves the following.

######  Theorem 1.1 (Yau).

Let $(M,g)$ be a complete Kähler manifold and $(N,h)$ be a Hermitian manifold.
Assume that the Ricci curvature of $M$ is bounded from below by $k\leq 0$ and
the holomorphic bisectional curvature of $N$ is bounded from above by $K<0$.
Then for any holomorphic map $f:M\rightarrow N$,

$\displaystyle\|df\|^{2}\leq\dfrac{k}{K},$ (1.2)

where $\|df\|^{2}=\operatorname{tr}_{g}(f^{*}h)$ is the trace of the pull back
metric $f^{*}h$.

In [18], Royden improves Yau’s Schwarz lemma when $N$ is Kähler.

######  Theorem 1.2 (Royden).

Assume that $N$ is Kähler. In Theorem 1.1, if the condition of the holomorphic
bisectional curvature of $N$ is replaced by that the holomorphic sectional
curvature of $N$ is bounded from above by $K<0$, then

$\displaystyle\|df\|^{2}\leq\dfrac{2d}{d+1}\dfrac{k}{K},$ (1.3)

where $d=\max\limits_{p\in M}rk(df(p))$ is the maximal complex rank of $df$.

In [20] Tosatti extends Yau’s Schwarz lemma from a different direction.
Namely, he considers almost Hermitian manifolds and proves the following.

######  Theorem 1.3 (Tosatti).

Let $(M,J,g)$ be a complete almost Hermitian manifold with second Ricci
curvature bounded from below by $k\leq 0$, and with torsion and $(2,0)$ part
of the curvature bounded. Let $(N,J_{N},h)$ be an almost Hermitian manifold
with holomorphic bisectional curvature bounded from above by $K<0$. Then for
any non-constant holomorphic map $f:M\rightarrow N$,

$\displaystyle\|df\|^{2}\leq\dfrac{k}{K}.$ (1.4)

We prove the following result which generalizes the equality case of the
Ahlfors-Schwarz lemma.

######  Theorem 1.4.

Assume that the equality holds at one point in
$(\ref{Yau2})or(\ref{Ro2})or(\ref{to3})$. Then it holds on the whole manifold
$M$ and $f$ is a totally geodesic map with constant rank.

The proof of the result shares some spirit of Minda in [15]. Namely, by using
the known bound of $\|df\|^{2}$ and analyzing the differential inequalities,
we can invoke the classical strong maximum principle of linear elliptic
operators to get the global equality. Then we obtain the geometric
implications by the global equality (see also the appendix in [17]). Using the
same technique, we also derive the result for the equality case for the volume
version of Schwarz lemma. See Proposition 3.4 in section 3. A special
consequence of Theorem 1.4 is the following.

######  Corollary 1.5.

Let $S$ be a Riemann surface with a complete conformal metric $ds^{2}$ of
Gaussian curvature at least $k<0$. Let $(N,h)$ be a Kähler manifold with
holomorphic sectional curvature bounded from above by $K<0$. For any non-
constant holomorphic map $f:S\longrightarrow N$, if
$f^{*}h=\dfrac{k}{K}ds^{2}$ at some point $p\in M$, then
$ds^{2}\equiv-\dfrac{1}{k}ds^{2}_{H}$ and $f$ is an isometric immersion (up to
a scalar) onto the totally geodesic submanifold $f(S)$ in $N$.

Here $ds^{2}_{H}$ is the unique hyperbolic metric on $S$. This corollary
generalizes the equality case of Theorem 2 in [19]. Theorem 1.4 may be
generalized to the case of non-smooth functions and then to other forms of
Schwarz lemmata in [17] .

The second part of the paper is to derive a new Schwarz lemma on almost
Hermitian manifolds, following the work of Ni [16] . Recently, by applying a
viscosity consideration from PDE theory, Ni proves a Schwarz lemma on Kähler
manifolds as follows (see also [5] [24]): any holomorphic map from a complete
Kähler manifold with holomorphic sectional curvature bounded from below by
$K_{1}\leq 0$ into a Kähler manifold with holomorphic sectional curvature
bounded from above by $K_{2}<0$ is distance decreasing up to a constant
$\dfrac{K_{1}}{K_{2}}$, provided that the bisectional curvature of the domain
is bounded from below. There are also several other types of Schwarz lemmata
obtained in [16] and [17]. The aforementioned result looks rather symmetric as
the distance decreasing constant only relies on the holomorphic sectional
curvatures. Along this direction, we prove the following result on almost
Hermitian manifolds.

######  Theorem 1.6.

111After this work was completed, the authors noticed that Theorem 1 and
Corollary 1.7 were also proved by Weike Yu independently in [28], using a
different method.

Let $(M,J,g)$ and $(N,J_{N},h)$ be almost Hermitian manifolds with $M$ being
complete. Suppose the holomorphic sectional curvatures of $M$ and $N$ satisfy
$H_{M}\geq K_{1}$ and $H_{N}\leq K_{2}$, with $K_{1}\leq 0$ and $K_{2}<0$. Let
$f:M\rightarrow N$ be a holomorphic map. Then

$f^{*}h\leq\dfrac{K_{1}}{K_{2}}g,$

provided that the bisectional curvature of $M$ is bounded from below, and the
torsion and $(2,0)$ part of the curvature of $M$ are bounded. If $K_{1}=0$,
then $f$ is a constant map.

To prove the theorem, we apply the local holomorphic disc technique in [18].
Namely, by choosing a suitable $J$-holomorphic disc in the maximum direction
of $df$ and pulling back the metrics, we can roughly reduce the proof in
higher dimension to the two-dimensional case, where the proof of Ahlfors-
Schwarz lemma applies. To achieve the reduction, we compare the holomorphic
sectional curvature of an almost Hermitian manifold and its almost complex
submanifolds (see also [11]). As the almost complex structure may be non-
integrable, we also need to show the existence of $J$-holomorphic discs with
prescribed conditions. This is guaranteed by the work in [9].

We remark that if $K_{1}>0$ and $K_{2}\leq 0$, the same proof of the above
result shows that $f$ is also a constant map. When $M$ is compact, the torsion
and the curvature tensor are automatically bounded. Then we get

######  Corollary 1.7.

Let $(M,J,g)$ and $(N,J_{N},h)$ be almost Hermitian manifolds with $M$ being
compact. Suppose the holomorphic sectional curvatures of $M$ and $N$ satisfy
$H_{M}\geq K_{1}$ and $H_{N}\leq K_{2}$. For any holomorphic map
$f:M\rightarrow N$,

  * (i)

if $K_{1}<0,K_{2}<0$, then $f^{*}h\leq\dfrac{K_{1}}{K_{2}}g$;

  * (ii)

if $K_{1}\geq 0,K_{2}<0$ or $K_{1}>0,K_{2}\leq 0$, then $f$ is a constant map.

Part (ii) has been obtained previously by Yang [25] on Hermitian manifolds by
a different method. Later Masood [14] generalizes it to almost Hermitian
manifolds. Another corollary of Theorem 1 is the following.

######  Corollary 1.8.

Let $(M,J,g)$ be a compact almost Hermitian manifold with negative constant
holomorphic sectional curvature. Then any biholomorphism of $(M,J)$ is an
isometry of $(M,g)$. In particular, the group of biholomorphisms of $(M,J)$ is
compact.

When $(M,J)$ is a compact Hermitian manifold with negative holomorphic
sectional curvature, Wu [23] proves that the group of biholomorphism of
$(M,J)$ is finite. To derive the finiteness in the above result, one still
need to show that the group is discrete.

The structure of the paper is as follows. In section 2, we give the notations
and background materials. In section 3, we prove Theorem 1.4. In section 4, we
discuss the holomorphic sectional curvature of the Chern connection and show
the existence of some special $J$-holomorphic discs. In section 5, we prove
Theorem 1 . Throughout the paper, we assume that $M$ is a connected manifold.
$\|df\|^{2}$ is the trace function $tr_{g}(f^{*}h)$ and $\|df\|_{0}$ is the
supreme norm of $df$.

Acknowledgements. We are very grateful to Professor Jiaping Wang for helpful
suggestions and his support. We want to express our gratitude to Professors
Kefeng Liu, Lei Ni and Fangyang Zheng for useful communications and
discussions. We also thank Professor Jianfei Wang for helpful communications
and the warm hospitality when we visited Huaqiao University in July 2021 and
thank Professor Yibin Ren for some nice discussions.

##  2\. Preliminaries

In this section, we give some background materials, including almost Hermitian
manifolds, the Chern connection and Kähler metrics, etc. We refer to [20],
[21] and [29] for more details.

Let $(M,J)$ be a $2m$-dimensional almost complex manifold and $g$ be a
Riemannian metric on $M$. $(M,J,g)$ is called an almost Hermitian manifold if
$g(JX,JY)=g(X,Y)$ for any $X,Y\in TM$. It is known that there is a unique
connection $\nabla$ on $(M,J,g)$ (see e.g. [21]) satisfying

$\nabla J=0,\ \ \nabla g=0$

and $T(X,JX)=0$ for all $X\in TM$, where

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

is the torsion tensor. It is called the Chern connection or canonical
connection. The last requirement is equivalent to that the (1,1) part of the
torsion vanishes. The Nijenhuis tensor $N_{J}$ of $J$ is defined by

$N_{J}(X,Y)=[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY]$

for $X,Y\in TM$. The Kähler form $\omega$ is defined by $\omega(X,Y)=g(JX,Y)$.
An almost Hermitian manifold is a Kähler manifold if $N_{J}=0$ and
$d\omega=0$. It is known that $(M,J,g)$ is Kähler if and only if $T=0$ [29].

On an almost Hermitian manifold $(M,J,g)$, denote $T^{1,0}M$ and $T^{0,1}M$
the $\sqrt{-1}$ and $-\sqrt{-1}$ eigenbundles of $J$ in
$TM\otimes_{\mathbb{R}}\mathbb{C}$. Then
$TM\otimes_{\mathbb{R}}\mathbb{C}=T^{1,0}M\oplus T^{0,1}M$. For any $\xi\in
T^{1,0}M$, we can write $\xi=\dfrac{1}{\sqrt{2}}(X-\sqrt{-1}JX)$ for some
$X\in TM$. This gives an isomorphism between $T^{1,0}M$ and $TM$. Given a
smooth function $\phi$ on $M$, the complex Hessian of $\phi$ is defined to be
$(\nabla^{2}\phi)^{(1,1)}$, i.e., the $(1,1)$ part of the Hessian
$\nabla^{2}\phi$. For instance, for $\xi,\eta\in T^{1,0}M$,

$(\nabla^{2}\phi)^{(1,1)}(\xi,\bar{\eta})=\nabla^{2}\phi(\xi,\bar{\eta})=\xi(\bar{\eta}\phi)-(\nabla_{\xi}\bar{\eta})\phi.$

Let $\\{e_{1},e_{2},...,e_{m}\\}$ be a local unitary (1,0) frame on $M$ and
$\\{\theta^{1},\theta^{2},...,\theta^{m}\\}$ be the dual frame so that
$g=\sum_{i=1}^{m}\theta^{i}\otimes\bar{\theta}^{i}+\bar{\theta}^{i}\otimes\theta^{i}.$
Let

$\phi_{i\bar{j}}=\nabla^{2}\phi(e_{i},\overline{e_{j}})=e_{i}\overline{e_{j}}\phi-(\nabla_{e_{i}}\overline{e_{j}})\phi.$

As the $(1,1)$ part of the torsion of $\nabla$ vanishes,
$\nabla_{e_{i}}\overline{e_{j}}-\nabla_{\overline{e_{j}}}e_{i}=[e_{i},\overline{e_{j}}].$
Then $\phi_{i\bar{j}}=\phi_{\bar{j}i}$. Using Einstein summation convention,
locally write

$(\nabla^{2}\phi)^{(1,1)}=\phi_{i\bar{j}}(\theta^{i}\otimes\bar{\theta}^{j}+\bar{\theta}^{j}\otimes\theta^{i}).$

The complex Laplacian of $\phi$ is defined to be

$\displaystyle\Delta\phi=tr_{g}(\nabla^{2}\phi)^{(1,1)}=2\sum_{i=1}^{m}\phi_{i\bar{i}}.$
(2.1)

It is known that $\Delta$ is an elliptic operator which differs from the
Laplacian of the Levi-Civita connection by some first order differential
operators (Lemma 3.2 in [20]). Use $\langle\ ,\ \rangle$ for the inner product
$g$ and denote

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

to be the curvature tensors of $\nabla$.

######  Definition 2.1.

Given two nonzero $(1,0)$ vectors
$\xi=\frac{1}{\sqrt{2}}(X-\sqrt{-1}JX),\eta=\frac{1}{\sqrt{2}}(Y-\sqrt{-1}JY)$,
the holomorphic bisectional curvature of $\nabla$ in directions $\xi,\eta$ is
defined to be

$B(\xi,\eta)=\dfrac{R(\xi,\bar{\xi},\eta,\bar{\eta})}{|\xi|^{2}|\eta|^{2}}=\dfrac{R(X,JX,JY,Y)}{|X|^{2}|Y|^{2}},$

where $|\xi|^{2}=|X|^{2}=\langle X,X\rangle$ and $|\eta|^{2}=|Y|^{2}=\langle
Y,Y\rangle$. The holomorphic sectional curvature in the direction $\xi$ or $X$
is defined to be

$H(\xi)=B(\xi,\xi)=R(X,JX,JX,X)/|X|^{4}.$

We also use $H(X)$ for $H(\xi)$ in the context. The first and second Ricci
curvatures of $(J,g)$ are defined by

$Ric^{(1)}(\xi,\bar{\eta})=\sum_{i=1}^{m}R(\xi,\bar{\eta},e_{i},\overline{e_{i}}),\
\
Ric^{(2)}(\xi,\bar{\eta})=\sum_{i=1}^{m}R(e_{i},\overline{e_{i}},\xi,\bar{\eta}).$

The Chern scalar curvature is defined to be
$R=\sum_{i,j=1}^{m}R(e_{i},\overline{e_{i}},e_{j},\overline{e_{j}})$.

We remark that if $(M,J,g)$ is Kähler, then $Ric^{(1)}=Ric^{(2)}$ by the
Kähler symmetry. It is called the Ricci curvature in short. We have the
following notions of boundedness on an almost Hermitian manifold (cf. [20]).

######  Definition 2.2.

  * (i)

The first (resp. second) Ricci curvature is said to be bounded from below if
there is a constant $C$ such that $Ric^{(1)}(\xi,\bar{\xi})\geq C|\xi|^{2}$
(resp. $Ric^{(2)}(\xi,\bar{\xi})\geq C|\xi|^{2}$) for any $\xi\in T^{1,0}M$.

  * (ii)

The holomorphic bisectional curvature of $M$ is said to be bounded from below
if there is a constant $C_{1}$ such that $B(\xi,\eta)\geq C_{1}$ for all
$\xi,\eta\in T^{1,0}M$.

  * (iii)

The torsion tensor $T$ is said to be bounded if there is a constant $C_{2}>0$
such that $|T(\xi,\eta)|\leq C_{2}|\xi||\eta|$ for any $\xi,\eta\in T^{1,0}M$.

  * (iv)

The $(2,0)$ part of the curvature $R$ is said to be bounded if there is a
constant $C_{3}>0$ such that $|R(\xi,\eta,\xi,\bar{\eta})|\leq
C_{3}|\xi|^{2}|\eta|^{2}$ for any $\xi,\eta\in T^{1,0}M$.

Note that the $(2,0)$ part of the curvature tensor vanishes when $M$ is
Hermitian, i.e. $N_{J}=0$. If $M$ is compact, then it clearly satisfies all
the boundedness conditions. For a fixed point $o\in M$, denote $r(q)=d(o,q)$
the distance from $o$ to $q\in M$. It is smooth away from $o$ and its cut
locus. The following complex Hessian comparison theorem on almost Hermitian
manifolds due to Tosatti [20] will be used in the proof of Theorem 1 (also see
[27]).

######  Theorem 2.3 (Tosatti).

Let $(M,J,g)$ be an almost Hermitian manifold with bisectional curvature
bounded from below by $C_{1}$, torsion bounded by $C_{2}$ and $(2,0)$ part of
the curvature bounded by $C_{3}$. Then at any point where $r$ is smooth,

$r_{i\bar{j}}\leq(\dfrac{1}{r}+C)g_{i\bar{j}},$

where $C$ depends on $C_{1},C_{2},C_{3}$ and the dimension of $M$.

Here $A_{i\bar{j}}\leq B_{i\bar{j}}$ means that $(B_{i\bar{j}}-A_{i\bar{j}})$
is positive semi-definite. The proof of the above theorem is contained in [20]
during the proof of the Laplacian comparison theorem. Explicitly, it follows
from the expressions (4.30), (4.31), (4.32) and the argument after (4.32)
there. As $r_{i\bar{j}}=\dfrac{1}{2}\sqrt{-1}d(Jdr)(e_{i},\overline{e_{j}})$,
the factor $2$ does not appear in the complex Hessian comparison.

In Proposition 2 in [18], an existence result of supporting functions is
stated. As in non-Kähler setting, it relies on the complex Hessian comparison,
it seems that the condition of Riemmanian sectional curvature bounded from
below there should be replaced by conditions on bisectional curvature and
torsion or other related conditions.

Let $(N,J_{N},h)$ be a $2n$-dimensional almost Hermitian manifold.

######  Definition 2.4.

A smooth map $f:M\longrightarrow N$ is called a $(J,J_{N})$ holomorphic map
(or holomorphic map in short) if it satisfies

$df\circ J=J_{N}\circ df.$

Let $T^{1,0}M$ and $T^{1,0}N$ be the bundle of $(1,0)$ tangent vectors of $M$
and $N$. By definition, $df(T^{1,0}M)\subset T^{1,0}N$. For any $p\in M$,
choose a local unitary frame $\\{\eta_{1},\eta_{2},...,\eta_{n}\\}$ near
$q=f(p)\in N$ such that $df$ is given by
$df(e_{i})=f_{i}^{\alpha}\eta_{\alpha}$. The norm of $df$ is defined to be

$\|df\|^{2}=\sum_{i=1}^{m}\sum_{\alpha=1}^{n}|f_{i}^{\alpha}|^{2}.$

It is easy to show that $\|df\|^{2}$ is independent of the chosen frames and
is globally defined. It is also equal to the trace function $tr_{g}(f^{*}h)$.

##  3\. The equality case of Schwarz lemmata

In this section, we study the equality case of the Schwarz lemmata of Yau,
Royden and Tosatti. First, the following strong maximum principle of Hopf is
known (cf. Theorem 3.5 in [7] for example).

######  Proposition 3.1.

Let $L$ be a uniformly elliptic linear differential operator of the form

$Lu=a^{ij}D_{ij}u+b^{i}(x)D_{i}u+c(x)u,\ \ a^{ij}=a^{ji}$

in a domain $\Omega\subset\mathbb{R}^{n}$. Let $\lambda(x)$ be the minimum
eigenvalues of $(a^{ij})$. Suppose that $c\leq 0$ and $\dfrac{c}{\lambda}$ is
bounded. If $u\in C^{2}(\Omega)$ satisfies $Lu\geq 0$, then $u$ cannot achieve
a non-negative maximum in the interior of $\Omega$ unless it is constant.

We then apply the differential inequalities in [18][20] to prove Theorem 1.4.

###### Proof of Theorem 1.4.

Assume that we are in the setting of Theorem 1.2 and let $v=\|df\|^{2}$.
Suppose at some $p\in M$, $v(p)=\frac{2d}{d+1}\frac{k}{K}>0$ with $k<0,K<0$
(the case of $k=0$ or $d=0$ is trivial). By Proposition 4 in [18], at any
$q\in M$,

$\displaystyle\Delta\log v\geq k-\dfrac{rk(q)+1}{2\ rk(q)}Kv\geq
k-\dfrac{d+1}{2d}Kv,$ (3.1)

where $rk(q)$ is the complex rank of $df$ at $q\in M$ and $\Delta$ is the
complex Laplacian defined in $(\ref{lap})$. By (1.3), the following holds on
$M$,

$\displaystyle 0\leq\dfrac{K}{2k}\dfrac{d+1}{d}v\leq 1.$

Also, in a small neighbourhood $U$ of $p$, $v>0$. Let
$\dfrac{K}{2k}\dfrac{d+1}{d}=C_{1}$. As $0<C_{1}v\leq 1$ in $U$, the
elementary inequality $1-C_{1}v\leq-\log(C_{1}v)$ holds. Denote
$u=\log(C_{1}v)\leq 0$. Then we have

$\displaystyle\Delta\log v$ $\displaystyle\geq k-\dfrac{d+1}{2d}Kv$
$\displaystyle\geq-k\log(C_{1}v),$

on $U$, which gives that

$\displaystyle\Delta u+ku\geq 0.$

The complex Laplacian $\Delta$ naturally satisfies the conditions in
Proposition 3.1 in $U$. As $u\leq 0$ with $u(p)=0$ and $k<0$, by Proposition
3.1, $u$ must be constant zero and $v=\frac{2d}{d+1}\frac{k}{K}$ in $U$. It
follows that the set $\\{q\in M|v(q)=\frac{2d}{d+1}\frac{k}{K}\\}$ is open and
closed. By the connectedness of $M$, $v$ is constant on $M$ and $rk(q)=d$ for
any $q\in M$.

For the inequalities (1.2),(1.4), as Theorem 1.3 extends Theorem 1.1, we only
need to discuss the case of (1.4). Assume that we are in the setting of
Theorem 1.3 and $K<0$. Let $v=\|df\|^{2}$. By (5.41) and Theorem 4.1 in [20],
$\Delta v\geq 2(kv-Kv^{2}),$ and $v\leq\dfrac{k}{K}$. Then

$\displaystyle\Delta v$ $\displaystyle\geq 2Kv\left(\dfrac{k}{K}-v\right)$
$\displaystyle\geq 2k\left(\dfrac{k}{K}-v\right).$

Let $u=v-\dfrac{k}{K}$, then $\Delta u\geq-2ku$. Note that $u\leq 0$. The same
argument as the above shows that if the equality of (1.4) holds at some $p\in
M$, then $u=0$ in a neighborhood of $p$. It follows that $v=\dfrac{k}{K}$ in a
neighborhood of $p$. By the connectedness of $M$, $v$ is constant on the whole
manifold.

Once we have the global equality of $v$, by the formulas in the proof of
Proposition 4 in [18] (see also inequality (A.3) in [16]) and (5.41) in [20],
then $\nabla df=0$. So $f$ is totally geodesic and the rank of $df$ is
constant in all cases [22]. ∎

######  Remark 3.2.

From the above proof and Proposition 3.1, we can obtain a special strong
maximum principle as follows. Let $L$ be a uniformly elliptic linear
differential operator of the form

$Lu=a^{ij}D_{ij}u+b^{i}(x)D_{i}u,\ \ a^{ij}=a^{ji}$

in a domain $\Omega\subset\mathbb{R}^{n}$. Suppose $u\leq C$ and $Lu\geq f(u)$
for some function $f$ with $f(C)\geq 0$ and $f^{\prime}(x)\leq M,\forall\
x\leq C$ for some $M>0$. Then $u$ can not equal $C$ in the interior of
$\Omega$ unless it is constant.

In [26], Yau also obtained the volume form version of the Schwarz lemma for
holomorphic maps under weaker curvature conditions. It was extended by Tosatti
[20] to almost Hermitian manifolds as follows.

######  Theorem 3.3 (Tosatti).

Let $(M,J,g)$ be a $2m$-dimensional complete almost Hermitian manifold with
Chern scalar curvature bounded from below by $mK_{1}<0$. Let $(N,J_{N},h)$ be
another $2m$-dimensional almost Hermitian manifold with the first Ricci
curvature bounded from above by $K_{2}<0$. If the second Ricci curvature of
$(M,g)$ is bounded from below and with torsion and $(2,0)$ part of the
curvature bounded, then for any non-degenerate holomorphic map $f:M\rightarrow
N$,

$\displaystyle f^{*}dV_{h}\leq(\dfrac{K_{1}}{K_{2}})^{m}dV_{g}.$ (3.2)

Here $dV_{g}$ and $dV_{h}$ are the volume forms on $(M,g)$ and $(N,h)$
respectively and the map $f$ is non-degenerate if $f^{*}dV_{h}$ is nowhere
vanishing. Applying Remark 3.2, we prove the following result. We would like
to thank Professor Kefeng Liu for pointing it out to us.

######  Proposition 3.4.

Assume that the equality holds at one point in $(\ref{to4})$, then it holds on
the whole manifold $M$.

###### Proof.

Let $v=\dfrac{\det f^{*}h}{\det g}$. Then $v\leq(\dfrac{K_{1}}{K_{2}})^{m}$ by
Theorem 3.3. By the Chern-Lu formula for the volume form (see Page 1081 in
[20]), we have

$\Delta v\geq-2mK_{2}v^{1+\frac{1}{m}}+2mK_{1}v.$

Let $f(v)=-2mK_{2}v^{1+\frac{1}{m}}+2mK_{1}v$. Then $f^{\prime}(v)\leq-2K_{1}$
as $v\leq(\dfrac{K_{1}}{K_{2}})^{m}$ . The result then follows from Remark
3.2. ∎

###### Proof of Corollary 1.5.

By Theorem 1.4, $tr_{g}f^{*}h\equiv\frac{k}{K}$ and $f$ is totally geodesic.
Then the image $f(S)$ is a totally geodesic submanifold of $N$. As $f$ is non-
constant and $m=1$, then $f$ is an immersion and $f^{*}h=\frac{k}{K}ds^{2}$.
By the proof of Proposition 4 in [18], the equality forces that the Gaussian
curvature of $ds^{2}$ equals $k$. Therefore, $ds^{2}=-\dfrac{1}{k}ds^{2}_{H}$
by the uniqueness of the hyperbolic metric on $S$. ∎

######  Remark 3.5.

Recently in [3], Broder extends Schwarz type inequalities to more general
settings on Hermitian manifolds. He informed us that our proof may treat the
equality case of the inequalities in his paper. Indeed, the differential
inequalities derived there satisfy the conditions of Remark 3.2 and the
analogous equality result follows (c.f. Remark 1.10 in [3]). We would like to
thank him for the communications.

##  4\. Holomorphic sectional curvature and $J$-holomorphic disc

In this section, we compare the holomorphic sectional curvatures of an almost
Hermitian manifold and its almost complex submanifolds. Then we discuss the
existence of some special $J$-holomorphic discs through a point in any
direction on an almost Hermitian manifold, based on the work of [9].

######  Definition 4.1.

Let $(S,i)$ be an immersed submanifold of $(M,J)$ with $i:S\longrightarrow M$
being the immersion. For any $p\in S$, identify $T_{p}S$ with $i_{*}(T_{p}S)$
in $T_{i(p)}M$. If $J(T_{p}S)\subset T_{p}S$, then we say $(S,i)$ is an almost
complex submanifold of $(M,J)$.

The pull back of $(J,g)$ through $i$ induces an almost Hermitian structure on
$S$ which we still denote by $(J,g)$. As an immersion is locally embedding, to
focus on local properties, from now on we assume that $S$ is an embedded
almost complex submanifold and identify $S$ with $i(S)$.

Denote $\nabla$ and $\hat{\nabla}$ the Chern connections on $M$ and $S$
respectively. Let $NS$ be the normal bundle of $S$ and $\pi^{\top}$ and
$\pi^{\bot}$ be the tangential and normal projections:

$\displaystyle\pi^{\top}:\ TM|_{S}\rightarrow TS,\ \ \ \pi^{\bot}:\
TM|_{S}\rightarrow NS.$

Then $\nabla$ and $\hat{\nabla}$ are related as follows.

######  Proposition 4.2.

Let $X,Y$ be in $\Gamma(TS)$ which are extended to vector fields on $M$. Then
we have

$\hat{\nabla}_{X}Y=(\nabla_{X}Y)^{\top}.$

###### Proof.

Define $\nabla^{\top}_{X}Y:=(\nabla_{X}Y)^{\top}$. It is clear that
$\nabla^{\top}$ is a connection on $S$. By the uniqueness of Chern connection,
to show that $\nabla^{\top}=\hat{\nabla}$, it suffices to show that
$\nabla^{\top}g=0,\ \nabla^{\top}J=0$ and its torsion has vanishing $(1,1)$
part. Let $X,Y,Z\in\Gamma(TS)$ which are extended to vector fields on $M$. At
any $p\in S$, we have

$\displaystyle X\langle Y,Z\rangle=$
$\displaystyle\langle\nabla_{X}Y,Z\rangle+\langle Y,\nabla_{X}Z\rangle$
$\displaystyle=$ $\displaystyle\langle(\nabla_{X}Y)^{\top},Z\rangle+\langle
Y,(\nabla_{X}Z)^{\top}\rangle$ $\displaystyle=$
$\displaystyle\langle\nabla^{\top}_{X}Y,Z\rangle+\langle
Y,\nabla^{\top}_{X}Z\rangle$

Thus $\nabla^{\top}g=0$. Also, as $JX\in T_{p}S$, for any $W\in\Gamma(NS)$,
$\langle X,JW\rangle=-\langle JX,W\rangle=0.$ So $JW\in\Gamma(NM)$. Then

$\displaystyle\nabla_{X}^{\top}(JY)$
$\displaystyle=(\nabla_{X}(JY))^{\top}=(J\nabla_{X}Y)^{\top}$
$\displaystyle=(J\nabla^{\top}_{X}Y+J(\nabla_{X}Y)^{\bot})^{\top}$
$\displaystyle=J\nabla^{\top}_{X}Y.$

So $\nabla^{\top}J=0$. Finally, let $X,Y$ be (1,0) and (0,1) vector fields of
$S$ respectively. Then

$\displaystyle\nabla^{\top}_{X}Y-\nabla_{Y}^{\top}X$
$\displaystyle=(\nabla_{X}Y-\nabla_{Y}X)^{\top}$
$\displaystyle=[X,Y]^{\top}=[X,Y].$

So the torsion of $\nabla^{\top}$ has vanishing (1,1) part. Therefore,
$\nabla^{\top}=\hat{\nabla}$. ∎

For any $X,Y\in TS$, the second fundamental form of $S$ is given by

$\textit{II}\ (X,Y):=(\nabla_{X}Y)^{\bot}.$

By Proposition 4.2 and the metric compatibility of $\nabla$, we have

######  Proposition 4.3.

Let $X,Y\in\Gamma(TS)$ and $W\in\Gamma(NS)$ which are extended to vector
fields on $M$. Then

$\displaystyle\nabla_{X}Y$ $\displaystyle=\hat{\nabla}_{X}Y+\mathit{II}(X,Y)$
$\displaystyle\langle\nabla_{X}W,Y\rangle$ $\displaystyle=-\langle
W,\mathit{II}(X,Y)\rangle.$

Denote $R$ and $\hat{R}$ the curvature for $\nabla$ and $\hat{\nabla}$. With
Proposition 4.3, we get the Gauss type equation by direct calculation.

######  Proposition 4.4.

For any $X,Y,Z,W\in T_{p}S$,

$R(X,Y,Z,W)=\hat{R}(X,Y,Z,W)-\langle\mathit{II}(X,W),\mathit{II}(Y,Z)\rangle+\langle\mathit{II}(X,Z),\textit{II}(Y,W)\rangle.$

We are then able to compare the holomorphic sectional curvature of $\nabla$
and $\hat{\nabla}$.

######  Corollary 4.5.

$H(X)=\hat{H}(X)+2|\mathit{II}(X,X)|^{2}/|X|^{4},$

for $X\in TS$. Hence, $\hat{H}(X)\leq H(X)$ with equality holding if and only
if $\mathit{II}(X,X)=0$.

The above inequality was obtained previously in [11] by using local frames and
structure equations.

###### Proof.

By $\nabla J=0$, we get $\nabla_{X}(JY)=J(\nabla_{X}Y)$. So
$\mathit{II}(X,JY)=J\mathit{II}(X,Y)$. As $T(X,JX)=0$ gives

$\nabla_{JX}X-\nabla_{X}(JX)=[JX,X],$

and $JX\in TS$, then

$\displaystyle\textit{II}(JX,X)=(\nabla_{JX}X)^{\bot}=(\nabla_{X}(JX)+[JX,X])^{\bot}=\textit{II}(X,JX).$

By Proposition 4.4, we have

$\displaystyle H(X)$ $\displaystyle=R(X,JX,JX,X)/|X|^{4}$
$\displaystyle=\hat{H}(X)+\dfrac{1}{|X|^{4}}(\langle\mathit{II}(X,X),\mathit{II}(X,X)\rangle+\langle\mathit{II}(X,JX),\textit{II}(JX,X)\rangle)$
$\displaystyle=\hat{H}(X)+2|\mathit{II}(X,X)|^{2}/|X|^{4}$

∎

Next, we apply Corollary 4.5 to study the curvature of the pull back metric on
a $J$-holomorphic disc on $M$. Denote
$\mathbb{D}=\\{z=x+\sqrt{-1}y\in\mathbb{C}||z|<1\\}$ the unit disc associated
with the standard complex structure $J_{st}$.

######  Definition 4.6.

A $J$-holomorphic disc on $M$ is a smooth map
$\varphi:\mathbb{D}\longrightarrow M$ such that $J\circ d\varphi=d\varphi\circ
J_{st}$.

By the work of Nijenhuis-Woolf, given any point $p\in M$ and $(1,0)$ vector
$\xi$ in $T^{1,0}_{p}M$ with $|\xi|$ small enough, there exists a
$J$-holomorphic disc on $M$ such that $\varphi(0)=p$ and
$\varphi_{*}(\frac{\partial}{\partial z})(p)=\xi$. Equivalently, for a
sufficiently small real vector $v\in T_{p}M$, there exists a $J$-holomorphic
disc with $\varphi(0)=p$ and $\dfrac{\partial\varphi}{\partial
x}(0)=\varphi_{*}(\frac{\partial}{\partial x})(p)=v$. The following more
general existence result is proved in [9] (Proposition 1.1) using the implicit
function theorem.

######  Theorem 4.7 (Ivashkovich-Rosay).

Let $k\in\mathbb{N}$, $k\geq 1$. Let $J$ be a smooth almost complex structure
near $0$ in $\mathbb{R}^{2n}$. For any $p\in\mathbb{R}^{2n}$ close enough to
0, and every $\\{v_{1},\cdots,v_{k}\\}\in(\mathbb{R}^{2n})^{k}$ small enough,
there exist a smooth $J$-holomorphic disc
$\varphi:\mathbb{D}\longrightarrow\mathbb{R}^{2n}$ such that $\varphi(0)=p$
and $\dfrac{\partial^{l}\varphi}{\partial x^{l}}(0)=v_{l}$, for any $1\leq
l\leq k$.

Then we can prove

######  Proposition 4.8.

Let $(M,J,g)$ be an almost Hermitian manifold. For any $p\in M$ and $v\in
T_{p}M$ with $|v|$ small enough, there exists a $J$-holomorphic disc
$\varphi:\mathbb{D}\longrightarrow M$ such that
$\varphi(0)=p,\dfrac{\partial\varphi}{\partial x}(0)=v$ and the Gaussian
curvature of $\varphi^{*}g$ at $0$ identifies with the holomorphic sectional
curvature $H(v)$.

###### Proof.

Let $t^{1},t^{2},...,t^{2m}$ be local coordinates of $M$ at $p$. Let
$\Gamma_{ij}^{k}$ be the real Christoffel symbols of the Chern connection
$\nabla$. Write $v=a^{i}\dfrac{\partial}{\partial t^{i}}$. For $|v|$ small
enough, by Theorem 4.7, there exists a $J$-holomorphic disc
$\varphi:\mathbb{D}\longrightarrow M$ such that $\varphi(0)=p$,
$\dfrac{\partial\varphi}{\partial x}(0)=v$ and

$\dfrac{\partial^{2}\varphi^{k}}{\partial
x^{2}}(0)+a^{i}a^{j}\Gamma_{ij}^{k}(p)=0,$

where $\varphi=(\varphi^{1},\cdots,\varphi^{2n}).$ If $v$ is nonzero, then
$\varphi$ is an embedding in a neighborhood $U$ of $0$. Denote
$X=\varphi_{*}(\dfrac{\partial}{\partial x})$ the vector field near $p$, which
extends to a local vector field on $M$. Then

$\displaystyle\nabla_{X}X(p)=\nabla_{\varphi_{*}(\frac{\partial}{\partial
x})}(\dfrac{\partial\varphi^{i}}{\partial x}\dfrac{\partial}{\partial
t^{i}})(p)=(\dfrac{\partial^{2}\varphi^{k}}{\partial
x^{2}}(0)+a^{i}a^{j}\Gamma_{ij}^{k}(p))\dfrac{\partial}{\partial t^{k}}=0$

Therefore, $\mathit{II}(X,X)(p)=(\nabla_{X}X)^{\bot}(p)=0$. Denote $K(z)$ the
Gaussian curvature of $\varphi^{*}g$ in $U\subset\mathbb{D}$, which is equal
to the holomorphic sectional curvature of $(\varphi(U)$. Therefore, by
Proposition 4.5, $K(0)=H(v)$. ∎

##  5\. A Schwarz lemma on almost Hermitian manifolds

In this section, we prove Theorem 1. Let $(M,J,g)$ and $(N,J_{N},h)$ be two
almost Hermitian manifolds and $f:M\longrightarrow N$ a $(J,J_{N})$
holomorphic map. Denote $H_{M}$ and $H_{N}$ the holomorphic sectional
curvatures of $M$ and $N$ and assume $H_{M}\geq K_{1},H_{N}\leq K_{2}$ for
some constants $K_{1}\leq 0,K_{2}<0$. Define the supreme norm of $df$ as
$\|df\|_{0}$ by

$\displaystyle\|df\|_{0}(p)=\sup_{\xi\neq 0}\dfrac{|df(\xi)|_{h}}{|\xi|_{g}},\
\ \ p\in M,\ \xi\in T_{p}^{1,0}M.$

Our main object is to estimate $||df||_{0}$ under the conditions in Theorem 1,
since $||df||^{2}_{0}\leq\dfrac{K_{1}}{K_{2}}$ implies that
$f^{*}h\leq\dfrac{K_{1}}{K_{2}}g$. First by definition, for any $p\in M$ and
any $\xi\in T_{p}^{1,0}M$, we have $|df(\xi)|_{h}\leq\|df\|_{0}(p)|\xi|_{g}$.
By change of unitary frames, we may assume that there are local unitary (1,0)
frames $\\{e_{1},e_{2},...,e_{m}\\}$ near $p\in M$ and
$\\{\eta_{1},\eta_{2},...,\eta_{n}\\}$ near $q=f(p)\in N$ such that $df$ is
given by $df(e_{i})=f_{i}^{\alpha}\eta_{\alpha}$ with
$f_{i}^{\alpha}=\lambda_{i}\delta_{i}^{\alpha}$ and
$\lambda_{1}\geq\lambda_{2}\geq...\geq\lambda_{r}>\lambda_{r+1}=...=0$, and
$r$ being the rank of $(f_{i}^{\alpha})$. Then $\|df\|_{0}=\lambda_{1}$ and
$|df(e_{1})|_{h}=\|df\|_{0}|e_{1}|_{g}$. We estimate $\lambda_{1}$ in the
following.

Let $\xi_{1}=te_{1}\in T_{p}^{1,0}M$ with $t>0$ small enough. By Proposition
4.8, there is a $J$-holomorphic disc $\varphi:\mathbb{D}\rightarrow M$ with
$\varphi(0)=p,\ \dfrac{\partial\varphi}{\partial x}(0)=Re(\xi_{1})$ and the
Gaussian curvature of $\varphi^{*}g$ at $0$ is $H_{M}(\xi_{1})$. Denote

$\varphi^{*}g=u^{2}dzd\bar{z}$

for some positive function $u$ near $0$. As $H_{M}\geq K_{1}$, and the
Gaussian curvature for $\varphi^{*}g$ is
$-\dfrac{4}{u^{2}}\dfrac{{\partial}^{2}\log u}{\partial z\partial\bar{z}}$, we
have

$\displaystyle-\dfrac{4}{u^{2}}\dfrac{{\partial}^{2}\log u}{\partial
z\partial\bar{z}}(0)\geq K_{1}.$

Since both $\varphi$ and $f$ are holomorphic, if $\lambda_{1}(p)\neq 0$, there
is a neighbourhood $U$ of $0$ in $\mathbb{D}$ such that$(U,f\circ\varphi)$ is
an almost complex submanifold of $(N,J_{N})$. Then we pull back $h$ to $U$ to
get

$\displaystyle\varphi^{*}(f^{*}h)=v^{2}dzd\bar{z}.$

We have

$\displaystyle v(z)=|f_{*}\varphi_{*}(\frac{\partial}{\partial
z})|_{h}\leq\|df\|_{0}|\varphi_{*}(\frac{\partial}{\partial
z})|_{g}=\lambda_{1}(\varphi(z))u(z).$ (5.1)

As $H_{N}\leq K_{2}$, by Corollary 4.5, we get

$-\dfrac{4}{v^{2}}\dfrac{{\partial}^{2}\log v}{\partial
z\partial\bar{z}}(0)\leq K_{2}.$

Now let $\phi$ be an arbitrary smooth function on $M$. Recall that the complex
Hessian of $\phi$ is

$(\nabla^{2}\phi)^{(1,1)}=\phi_{i\bar{j}}\theta^{i}\otimes\bar{\theta}^{j}+\phi_{\bar{i}j}\bar{\theta}^{i}\otimes\theta^{j},$

where

$\phi_{i\bar{j}}=\nabla^{2}\phi(e_{i},\bar{e}_{j})=e_{i}\bar{e}_{j}\phi-\nabla_{e_{i}}\bar{e}_{j}\phi=\phi_{\bar{j}i}.$

We pull back $\phi$ to $\mathbb{D}$ by $\varphi$ and compute its
differentials. As $\varphi$ is holomorphic, we may write
$\varphi_{*}(\frac{\partial}{\partial z})=\varphi^{i}_{z}e_{i}$ and then
$\varphi_{*}(\frac{\partial}{\partial\bar{z}})=\overline{\varphi}^{i}_{z}\bar{e}_{i}$,
where $\overline{\varphi}^{i}_{z}$ is the complex conjugate of
$\varphi^{i}_{z}$. Then

$\displaystyle\dfrac{\partial\phi}{\partial\bar{z}}=$
$\displaystyle\varphi_{*}(\frac{\partial}{\partial\bar{z}})\phi=\overline{\varphi}^{j}_{z}\bar{e}_{j}\phi$
$\displaystyle\dfrac{\partial^{2}\phi}{\partial z\partial\bar{z}}=$
$\displaystyle\varphi_{*}(\frac{\partial}{\partial
z})(\bar{\varphi}^{j}_{z}\overline{e}_{j}\phi)$ $\displaystyle=$
$\displaystyle\varphi^{i}_{z}\bar{\varphi}^{j}_{z}e_{i}e_{\bar{j}}\phi+\varphi^{i}_{z}(e_{i}\bar{\varphi}^{j}_{z})\overline{e}_{j}\phi.$
(5.2)

Since $[\varphi_{*}\frac{\partial}{\partial
z},\varphi_{*}\frac{\partial}{\partial\bar{z}}]=\varphi_{*}[\frac{\partial}{\partial
z},\frac{\partial}{\partial\bar{z}}]=0,$ we have

$\displaystyle\varphi^{i}_{z}(e_{i}\overline{\varphi}^{j}_{z})\overline{e}_{j}+\varphi^{i}_{z}\overline{\varphi}^{j}_{z}[e_{i},\overline{e}_{j}]-\overline{\varphi}^{j}_{z}(\overline{e}_{j}\varphi^{i}_{z})e_{i}=0.$

As $\nabla_{e_{i}}\bar{e}_{j}-\nabla_{\bar{e}_{j}}e_{i}=[e_{i},\bar{e}_{j}]$,
then $\nabla_{e_{i}}\bar{e}_{j}=[e_{i},\bar{e}_{j}]^{(0,1)}$. It follows that

$\displaystyle\varphi^{i}_{z}\overline{\varphi}^{j}_{z}\nabla_{e_{i}}\bar{e}_{j}=$
$\displaystyle\varphi^{i}_{z}\overline{\varphi}^{j}_{z}[e_{i},\overline{e}_{j}]^{(0,1)}=-\varphi^{i}_{z}(e_{i}\overline{\varphi}^{j}_{z})\overline{e}_{j}$

Putting it into (5.2), we get

$\displaystyle\dfrac{\partial^{2}\phi}{\partial
z\partial\bar{z}}=\varphi^{i}_{z}\bar{\varphi}^{j}_{z}\phi_{i\bar{j}}.$ (5.3)

Now we finish the proof of Theorem 1.

###### Proof of Theorem 1.

Fix a point $o\in M$. Let $r(q)=d(o,q)$ be the distance function on $M$. For
$a>0$, denote

$D_{a}=\\{q\in M\ |\ r(q)<a\\},\ \ \ \ \overline{D}_{a}=\\{q\in M\ |\ r(q)\leq
a\\}.$

Define a function $\rho$ on $D_{a}$ as

$\rho=(1-\dfrac{r^{2}}{a^{2}})\|df\|_{0}.$

Then $\rho\geq 0$ on $\overline{D}_{a}$ and achieves its maximum at some $p\in
D_{a}$. If $p$ is not a cut point of $o$, then $r$ is smooth at $p$.
Otherwise, we can use the Calabi’s trick to deal with it. Indeed, let $\gamma$
be a minimal unit speed geodesic from $o$ to $p$. Take a point $q_{0}$ on
$\gamma$ which is not a cut point of $p$ and denote $b=d(o,q_{0})$. For any
$q\in M$, $\widetilde{r}(q)=b+d(q_{0},q)$ is smooth near $p$. Note that
$\widetilde{r}\geq r$ and $\widetilde{r}(p)=r(p)$. Choose $\delta$ small
enough such that the open geodesic ball $B_{p}(\delta)\subset D_{a}$ and
$\widetilde{r}(q)<a$ is smooth in $B_{p}(\delta)$. It follows that
$\widetilde{\rho}=(1-\dfrac{\widetilde{r}^{2}}{a^{2}})||df||_{0}\leq\rho$ and
achieves the same maximum at $p$. Then we may use $\widetilde{\rho}$ to
replace $\rho$ and apply the maximum principle to $\widetilde{\rho}$. So we
just assume that $r$ is smooth at $p$.

Assume that $\|df\|_{0}(p)=\lambda_{1}(p)>0$ (otherwise, there is nothing to
prove). Here $\lambda_{1}$ is the greatest eigenvalue of $df$ introduced
previously. Let $\xi_{1}$ be an eigenvector of $\lambda_{1}$ at $p$ with
$|\xi_{1}|_{g}$ small enough. By the previous argument, there is a
$J$-holomorphic disc $\varphi:\mathbb{D}\rightarrow M$ such that
$\varphi(0)=p,\ \dfrac{\partial\varphi}{\partial x}(0)=Re(\xi_{1})$ and the
Gaussian curvature of $\varphi^{*}g$ at $0$ is $H_{M}(\xi_{1})$. Consider the
function

$\displaystyle\hat{\rho}(z)=\left(1-\dfrac{r^{2}(\varphi(z))}{a^{2}}\right)\dfrac{v(z)}{u(z)}$
(5.4)

on $\mathbb{D}$. By (5.1), $v(z)\leq\lambda_{1}(\varphi(z))u(z)$ and
$v(0)=\lambda_{1}(p)u(0)$. So $\hat{\rho}(z)\leq\rho(\varphi(z))$ and
$\hat{\rho}(0)=\rho(\varphi(0))$. Therefore, $\hat{\rho}$ achieves its maximum
at $z=0$. We have at $z=0$,

$\displaystyle 0\geq$ $\displaystyle\frac{\partial^{2}}{\partial
z\partial\bar{z}}\log\hat{\rho}$ $\displaystyle=$
$\displaystyle-2\frac{a^{2}+r^{2}}{(a^{2}-r^{2})^{2}}\left|\frac{\partial
r}{\partial\bar{z}}\right|^{2}-\frac{2r}{a^{2}-r^{2}}\frac{\partial^{2}r}{\partial
z\partial\bar{z}}+\frac{\partial^{2}}{\partial
z\partial\bar{z}}\log\frac{v(z)}{u(z)}.$

Denote $r_{i}=e_{i}r,r_{\bar{i}}=\bar{e}_{i}r$. As $1=|\nabla
r|^{2}_{g}=\sum_{i=1}^{m}|r_{i}|^{2}$, we have

$\displaystyle\left|\frac{\partial
r}{\partial\bar{z}}\right|^{2}=|\sum_{i=1}^{m}\overline{\varphi}^{i}_{z}r_{i}|^{2}\leq(\sum_{i=1}^{m}|\varphi^{i}_{z}|^{2})(\sum_{i=1}^{m}|r_{i}|^{2})=u^{2}(z).$

By (5.3) and Theorem 2.3,

$\displaystyle\frac{\partial^{2}r}{\partial
z\partial\bar{z}}=\varphi^{i}_{z}\bar{\varphi}^{j}_{z}r_{i\bar{j}}\leq(\frac{1}{r}+C)g_{i\bar{j}}\varphi^{i}_{z}\bar{\varphi}^{j}_{z}=(\frac{1}{r}+C)u^{2}(z).$

Therefore, at $z=0$,

$\displaystyle
0\geq-2\frac{a^{2}+r^{2}}{(a^{2}-r^{2})^{2}}u^{2}(0)-(\frac{1}{r}+C)\frac{2r}{a^{2}-r^{2}}u^{2}(0)-\frac{K_{2}}{4}v^{2}(0)+\frac{K_{1}}{4}u^{2}(0).$

As $K_{2}<0$,

$\displaystyle\lambda_{1}^{2}(p)=$
$\displaystyle\dfrac{v^{2}(0)}{u^{2}(0)}\leq\dfrac{K_{1}}{K_{2}}-\frac{8(a^{2}+r^{2})}{K_{2}(a^{2}-r^{2})^{2}}-\frac{8(1+Cr)}{K_{2}(a^{2}-r^{2})}.$

Thus for any $q\in D_{a}$,

$\displaystyle\rho^{2}(q)\leq$
$\displaystyle\left(\dfrac{a^{2}-r^{2}(p)}{a^{2}}\right)^{2}\left(\dfrac{K_{1}}{K_{2}}-\frac{16a^{2}}{K_{2}(a^{2}-r(p)^{2})^{2}}-\frac{8(1+Ca)}{K_{2}(a^{2}-r(p)^{2})}\right)$
$\displaystyle\leq$
$\displaystyle\dfrac{K_{1}}{K_{2}}-\frac{16}{K_{2}a^{2}}-\frac{8(1+Ca)}{K_{2}a^{2}}.$

So we obtain the estimate for $\rho$. Then for any $q\in D_{a}$, we get

$\displaystyle\|df\|_{0}^{2}(q)\leq$
$\displaystyle\left(1-\dfrac{r^{2}(q)}{a^{2}}\right)^{-2}\left(\dfrac{K_{1}}{K_{2}}-\frac{16}{K_{2}a^{2}}-\frac{8(1+Ca)}{K_{2}a^{2}}\right)$

Making $a\rightarrow+\infty$, we obtain
$\|df\|_{0}^{2}(q)\leq\dfrac{K_{1}}{K_{2}}$, for any $q\in M$. The proof is
complete. ∎

Now Corollary 1.7 directly follows from the result and proof above. For
Corollary 1.8, let $f:M\to M$ be any biholomorphism. Apply Theorem 1 to $f$
and $f^{-1}$ to get that $g\leq f^{*}g\leq g$. Therefore, $f$ is an isometry.
As the isometry group of a compact Riemannian manifold is compact, the group
of biholomorphisms is also compact, as a closed subset.

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# The Forecast Trap

Carl Boettiger<EMAIL_ADDRESS>Department of Environmental Science,
Policy, and Management, University of California, 130 Mulford Hall Berkeley,
CA 94720-3114, USA

###### Abstract

Encouraged by decision makers’ appetite for future information on topics
ranging from elections to pandemics, and enabled by the explosion of data and
computational methods, model based forecasts have garnered increasing
influence on a breadth of decisions in modern society. Using several classic
examples from fisheries management, I demonstrate that selecting the model or
models that produce the most accurate and precise forecast (measured by
statistical scores) can sometimes lead to worse outcomes (measured by real-
world objectives). This can create a forecast trap, in which the outcomes such
as fish biomass or economic yield decline while the manager becomes
increasingly convinced that these actions are consistent with the best models
and data available. The forecast trap is not unique to this example, but a
fundamental consequence of non-uniqueness of models. Existing practices
promoting a broader set of models are the best way to avoid the trap.

###### keywords:

forecasting adaptive management stochasticity uncertainty optimal control

††journal: Ecology Letters

Global change issues are complex and outcomes are difficult to predict (Clark
_et al._ 2001). To guide decisions in an uncertain world, researchers and
decision makers may consider a range of alternative plausible models to better
reflect what we do and do not know about the processes involved (Polasky _et
al._ 2011). Forecasts or predictions from possible models can indicate what
outcomes are most likely to result under what decisions or actions. This has
made model-based forecasts a cornerstone for scientifically based decision
making. By comparing outcomes predicted by a model to future observations, a
decision maker can not only _plan_ for the uncertainty, but also _learn_ which
models are most trustworthy. The value of iterative learning has long been
reflected in the theory of adaptive management (Walters & Hilborn 1978) as
well as in actual adaptive management practices such as Management Strategy
Evaluation (MSE) (Punt _et al._ 2016) used in fisheries, and is a central
tenet of a rapidly growing interest in ecological forecasting (Dietze _et al._
2018). But, do iterative learning approaches always lead to better decisions?

In this paper, I demonstrate that the model that makes the better prediction
(defined as a strictly proper score, Gneiting & Raftery (2007)) is not
necessarily the model that makes the better policy (defined in terms of
utility, e.g. expected net present value, Clark (1990)). I show that our best
methods for learning about model structure or parameters by repeatedly
comparing forecasts to observations can be counter-productive. Put another
way, the value of information (VOI, as measured by the expected utility given
that information minus the utility without it; see Howard (1966); Katz _et
al._ (1987)), can actually be negative. When VOI is negative, the decision-
maker may become trapped into accepting mediocre outcomes derived from a model
that makes accurate forecasts, even when a less accurate model that would
generate better outcomes is available. This trap is invisible to the manager
unless sufficient alternative models outside the original set are introduced.
I will present two examples of this “forecast trap” and examine how it arises
as a result of non-uniqueness of models (Oreskes _et al._ 1994; Schindler &
Hilborn 2015) with respect to either of these objectives.

The forecast trap is not the only mechanism by which some model-choice methods
lead to worse outcomes. Previous work has long acknowledged the panoply of
ways in which model-based decision making can go astray due to conflicting
incentives, implementation errors, or lack of resources for monitoring and
updating (e.g. Ludwig _et al._ 1993). Another widely recognized problem is
that of over-fitting (Burnham & Anderson 1998), in which the model that best
fits historical data fails to best predict future data (Ginzburg & Jensen
2004). Under such circumstances, it is easy to see how an over-fit model would
also lead to bad outcomes. However, over-fitting plays no role in the forecast
trap, where model predictions are assessed only using probabilistic forecasts,
and not observations which had previously been used to fit the models.
Formally, these scores satisfy the ‘proper scoring’ rule of Gneiting & Raftery
(2007), which proves no other probabilistic prediction $Q(x)$ will have a
better expected score than that of the true model (i.e. generative process),
$P(x)$. Gneiting & Raftery (2007)’s proof of proper scoring has since become a
critical tool to avoid over-fitting when choosing models to make decisions,
but as I illustrate, will not prevent the forecast trap.

First, I will introduce a motivating example in which we will consider two
reasonable process-based models, A and B. Model A will produce very accurate
forecasts, but lead to much worse outcomes than Model B. Though I will
establish that these accurate forecasts in Model A are not the result of
chance or of over-fitting the data, this example may raise more questions than
answers. To get a better understanding of when the forecast trap arises, I
will turn to a simpler ecological model, to which we may apply more
sophisticated decision tools of iterative forecasting and adaptive management.
We will see that these approaches do not avoid the forecast trap either. No
collection of such examples can establish precisely how common the forecast
trap may be in real-world applications. The examples do establish
unequivocally that achieving incrementally ever-more-accurate forecasts does
_not guarantee_ better decisions. I conclude by pointing to a range of
established and emerging approaches to quantitative decision-making which are
not based on forecasts. As sophisticated forecasting techniques become more
common-place in conservation and ecology, the forecast trap is a reminder that
we should not forget about these alternatives.

### A note on models and data

I will use the term “model” to refer to any set of equations or code that can
be used to produce a forecast. This term thus includes not only process-based
models, but could also statistical forecasting methods, non-parametric
approaches such as empirical dynamical modeling (Ye _et al._ 2015), or machine
learning. Most such models must first be calibrated to historical data before
they can produce a forecast, e.g. by parameter fitting, expert knowledge, or
some other means. Different choices for those parameters create different
forecasts, I will refer to those different parameterizations as different
models. It is of course possible for a decision-maker to consider forecasts
coming from multiple structurally different models simultaneously, and
potentially assigning different weights to each model. As more data becomes
available, it is possible to update model parameters, or equivalently, update
the weights assigned across models. I will examine such approaches for model
ensembles and model updating further on.

I shall focus on examples involving fisheries management to illustrate
principles shared in many ecological systems. Fisheries are a significant
economic and conservation concern worldwide and their management remains an
important debate (e.g. Worm _et al._ 2006, 2009; Costello _et al._ 2016).
Moreover, fisheries management has been a proving grounds for theoretical and
practical decision-making issues (e.g. Clark 1973; Reed 1979; Walters 1981;
Ludwig & Walters 1982) arising in a wide range of other contexts, including
invasive species (Boettiger 2021), infectious diseases (Shea _et al._ 2014; Li
_et al._ 2017), fire management (Richards _et al._ 1999) conservation planning
and prioritization (Wilson _et al._ 2006; Chadès _et al._ 2008) climate policy
(Nitzbon _et al._ 2017) and much else (Ludwig _et al._ 1993; Lande _et al._
1994; Polasky _et al._ 2011).

In these examples, we will focus on situations in which our ‘data’ comes from
a model simulation rather than empirical sources. Simulations are
simplifications of the real world – just because a method works in a
simulation is no guarantee that it works in reality. Conversely, if decision
methods are not reliable even when applied to simulated cases, we should be
even more cautious in how we use them. Simulations also allow us to consider
many replicates and conduct experiments that would be often impossible or
unethical to perform in the real world: for instance, does a given fishery
experience better long-term outcomes when managed according to forecasts
derived from model 1 or from model 2?

## A motivating example

To better understand how a model can produce a more accurate forecast and yet
still lead to a worse decision, it may be helpful to start with a concrete
example in which a manager faces a trade-off between cormorant conservation
and fish harvest. Fig 1 shows both the forecasts and realized management
outcomes of two alternative three-species models, “A” and “B” (see Appendix A)
in predicting the population dynamics of striped bass (an economically
important fishery) and double-crested cormorants (a target species for
conservation) which both feed on a population of river herring (whose
abundance we assume is not measured). Model A accurately forecasts the
abundance of both bass and cormorants well into the future, but the optimal
management strategy derived from its forecasts leads to steadily declining
species abundances and overall disappointing net utility. Model B produces
substantially less accurate forecasts, but nevertheless achieves better
outcomes. The net utility under model B is in fact nearly identical to the
maximum utility attainable given the true model, while management under model
A achieves only 38% of that utility.

This management problem is motivated by a real world example of a herring
fishery as described in Brias & Munch (2021), in which our manager seeks to
balance multiple objectives of sustaining the cormorant population while
maximizing the economic value from harvesting both herring and striped bass.
In the scenarios depicted in Fig 1, I have used a the richer five-species
model introduced Brias & Munch (2021) to drive the underlying dynamics, which
includes three competing species of herring that are preyed upon by both the
bass and the cormorants. Here, the manager seeks to maximize utility given by
the weighted sum of the individual objectives (Brias & Munch 2021), placing
50% of the weight on the conservation objective and splits the remaining
weight evenly over the harvests for predator (bass) and prey (herring)
species. I have assumed a partially observable system - in this case, the
manager measures only the abundance of bass and cormorant species, and not of
the three herring species. In this scenario, I have further assumed the
manager must choose a fixed fishing effort for herring and for the bass
harvest, I will consider more dynamic decision-processes later, but it is
worth noting that in many real world conservation settings policy choices are
highly constrained and frequent adjustment of those policies may be costly or
impossible. Likewise, the assumptions of partially observable system and
imperfect models are characteristic of ecological decision-making. Equations
and code for all models in this example are presented in Appendix A.

Both models A & B can be seen as alternative attempts to approximate the
“true” model (generative process), which in real systems is always unknown and
more complex than any model thereof. Model A assumes a three-species Ricker
model which closely matches the trophic structure of the “true” five-species
model, lumping the three competing herring species into a single variable.
Because herring abundance is not observed directly, the model parameters
related to herring growth are less accurate than other parameters. Model B
also lumps herring species together into a single variable, but fails to
reflect the trophic relationship between bass and herring. Model B also
oversimplifies the relationship between cormorant and herring population. This
does not make Model B an unreasonable model out-of-hand – all models contain
such simplifications (e.g. our “true” model does not model the trophic
relationship between the herring and its food sources or environmental
conditions explicitly either). Both models are consistent with the limited
historical data available to them.

Figure 1: Forecast performance and realized outcomes from management of a
5-species system under either model. Based on forecast performance alone,
Model A clearly performs better, accurately predicting steady declines. Model
B predicts overly optimistic outcomes for Bass population levels, and overly
pessimistic outcomes for cormorants, with observed dynamics falling well
outside the predicted range. Despite this, net utility achieved under this
management regime is virtually optimal, while the declines under model A
result in net utility that is only 38% of optimal.

## An Iterative Decision Example

While this example demonstrates that the model which provides the better
forecast does not necessarily lead to a better decision, it may raise more
questions than it answers. Why does this happen? Is this an isolated example
or not? Can this forecast trap be resolved by more sophisticated approaches to
model selection and decision-making? I now consider scenarios involving
iterative forecasts and adaptive management: in which the manager monitors
outcomes, compares forecasts to observations and updates model estimates. Such
_sequential decision processes_ are not merely iterative versions of single-
decision problems, but are much more challenging. In the opening example, the
manager had to choose the harvest policy for each fish species at the start of
the scenario and stick with it. The ability to select new actions in response
to new observations turns that decision into a game of chess: each turn, the
manager must consider not only their next move but all possible series of
moves.

How do we translate a model-based forecast into a decision policy? It is
impossible to discuss outcomes associated with a forecast without first
agreeing on this process. In practice, decision-makers may use a forecast in a
wide variety of ways in selecting a course of action, including ways which may
run counter to the stated objectives of management (Ludwig _et al._ 1993). In
principle at least, the field of decision theory provides a formal mechanism
for determining the optimal strategy given a model forecast. For instance, a
wide range of ecological conservation and management problems can be expressed
as a Markov Decision Process (MDP) problems (Marescot _et al._ 2013). Existing
computer algorithms such as stochastic dynamic programming (SDP) take a
probabilistic model _forecast_ (more precisely, the probability
$P(x_{t+1}|x_{t},a_{t})$ of the system being in state $x_{t+1}$ in the next
iteration given that it was previously in state $x_{t}$ and the manager
selected action $a_{t}$) and the _desired management objective_ (i.e. the
maximize the expected biomass of species protected or the expected dollar
profit of a fishery (see Clark 1990; Halpern _et al._ 2013)) as input, and
return the _decision policy_ which maximizes that objective (Marescot _et al._
2013). This provides a principled way to associate a decision policy with any
given forecast model.

Two features of this approach are worth emphasizing. As before, the resulting
decision is derived directly from the forecast model and the desired
objective. The SDP algorithm is a reasonable description of the approach any
ideal manager would use – considering all possible outcomes from all possible
sequences of actions and selecting the best sequence. For complex models this
process is too laborious even for a computer, and is often simplified by
considering only a selection of predetermined policies (as in Management
Strategy Evaluation, MSE, Punt _et al._ (2016)), or scenarios (as in scenario
analysis, Polasky _et al._ (2011)). Such shortcuts are often necessary for
complex real-world models, but open additional room for error: the policy we
derive from a given forecast may perform poorly not because the model forecast
was at fault, but because of those simplifying assumptions about possible
policies. To ensure that the forecast trap is not a result of such assumptions
about possible policies, we will consider a problem simple enough to solve
directly with SDP. The resulting decision policy is optimal, so long as the
forecast model is correct. In this way, the SDP merely stands in for a
mathematically precise way in which forecasts are turned into decisions.
Recognizing the SDP-derived policy (A) comes directly from the forecast model,
and (B) gives the optimal policy for said forecast, seems to suggest that
whichever model makes the better forecast will surely also lead to better
outcomes (as measured in terms of whatever utility we have chosen to
maximize). While this intuition is no doubt _often_ accurate, our purpose here
is to demonstrate that it is by no means _guaranteed_ : it is also possible
for the model which makes the better forecast to lead to worse outcomes.

Let us consider the management of single species in which we seek to maximize
the long-term net harvest. In this scenario, the manager estimates the
population size each year and must set the total allowable catch (TAC) for
that season. The underlying dynamics are unknown, but the manager is presented
with any of a variety of forecast models which can predict the future stock
sizes given the current population size and proposed TAC. Our manager does not
know which of these models is the most accurate a priori. Instead, the manager
will be able to compare the population size predicted by each forecast (under
the chosen TAC) to the measurement of the population size in the following
year before coming up with the next year’s catch limit.

Faced with a collection of models, a manager can seek either to identify the
best model to use, or to consider an integrative assessment which uses the
whole ensemble of models to represent the manager’s uncertainty about the
underlying process. I will consider both approaches in turn. Figure 2 compares
forecasts generated by two of the candidate models to observations drawn from
simulations of the underlying process. As before, these are true forecasts:
the model forecasts are generated first, they have not been fit to these
observations. In the un-fished scenario (top panels), both models try to
predicting the same un-fished equilibrium dynamics. In the second scenario
(lower panel), the manager uses the optimal SDP policy derived from each
forecast to determine the TAC for the following year, and compares the
observed stock size to that which the model predicted given that fishing
quota. In both cases, model 2 provides far more accurate forecasts, as seen in
the error bars and confirmed by the distribution of proper scores [Fig 2C-D;
Gneiting & Raftery (2007)]. Model and simulation details are provided in
Appendix B.

Figure 2: Forecast performance of each model. Panels A, B: Step ahead
predictions of stock size under unfished (A) and fished (B) scenarios. Error
bars indicating the 95% confidence intervals around each prediction, while
stars denote the observed value in that year. Because the models make
different decisions each year in the fished scenario, the observed stock size
in year 2, 3, etc under the management of model 1 (blue stars) is different
from that under model 2 (red stars). Panels C, D: corresponding distribution
of proper scores across all predictions (100 replicates of 100 timesteps).
Higher scores are better, confirming that model 2 makes the better forecasts.

Despite the clearly superior predictive accuracy of model 2 in both scenarios,
the outcomes from management under model 2 are substantially worse. We can
assess such outcomes in less abstract terms than forecasting skill, such as
economic value them manager sought to optimize (in dollars) or the ecological
value (unharvested biomass) [Fig 3].

Figure 3: Ecological and economic performance of each forecast. Harvest quotas
derived from model 1 result in a significantly higher fish stock size than
under Model 2 (panel A). Economic returns under model 1 are also substantially
higher (panel B)

A manager operating under model 2 would have little indication that the model
was flawed: both future stock sizes and expected harvest yields consistently
match model predictions. This manager would be stuck in the forecast trap,
incorrectly mistaking a degraded ecological state and reduced economic
outcomes as the best that can be achieved because future observations continue
to validate this model. Had we been able to include Model 3 in our forecast
comparisons, it would equal or outperform the forecasting skill of both model
1 and model 2 (as guaranteed by the theorem of Gneiting & Raftery (2007)),
while also matching the economic utility of model 1 (as guaranteed by the
theorem of Reed (1979)). In practice, we never have access to the generating
model, so it is reasonable to expect model selection to determine the better
approximation. As we see here, the better approximation for forecasting future
states does not in fact lead to better outcomes.

### Adaptive Management

Rather than select a single best model (or best parameter value), a manager
could choose to integrate over possible outcomes generated from all candidate
models. Updating posterior distributions over parameters and/or weights
assigned to different models are examples of this kind of adaptive management
(Ludwig & Walters 1982; Punt _et al._ 2016). I illustrate the application of
such an adaptive management strategy, following classic examples for parameter
(Ludwig & Walters 1982) or structural (Smith & Walters 1981) model
uncertainty. I first consider only the same two models considered in the
previous example. I later consider a larger suite of 42 models, spanning the
parameter space of Gordon-Schaefer curves. To avoid failure to explore
sufficiently, (see exploration-exploitation trade-off, e.g. Walters 1981), I
assign prior belief of 99% weight on the optimally performing model, model 1.
For comparison, I consider the baseline case in which the manager does not
update posterior distribution over which models/parameter values are correct
(the manager still chooses a new TAC after each observation, but does not
update their belief in the model, i.e. does not _learn_ over time). The
difference between the performance with and without learning is known as the
“Value of Information” (Howard 1966). In both 2-model and 42-model scenarios,
the value of information is strongly negative. The 2-model case achieves a net
present value to -58% of the value of having used model 1 alone [Fig 4].
Including all 42 models reduces this to a value of -32%. Both harvests and
fish biomass remain significantly lower under adaptive learning scenarios.

Figure 4: Adaptive management under model uncertainty. Solid lines trace the
trajectories of the state (fish stock, circles) and action (harvest quota,
triangles), under adaptive management (learning). Dotted lines trace the
corresponding trajectories if iterative learning is omitted, leaving the prior
belief fixed throughout the simulation (planning). Color indicates the belief
that model 1 is correct (blue), with an initial prior belief of 99%. Panel A:
Management over the two candidate models, Model 1 and Model 2. Within a single
iteration of adaptive management, the belief over models switches from a prior
belief that heavily favored model 1 to a posterior that favors model 2 with
near certainty. Future iterations reinforce the belief in model 2, resulting
in both depressed harvests and low stock sizes (solid lines). If no iterative
learning updates are performed, stock sizes and realized harvests (and thus
economic profit) are both higher. Panel B: given 42 candidate models over a
broad range of parameter values, adaptive management quickly reduces the
probability of model 1, and substantially underperforms management without
learning (dotted lines). While outcomes improve marginally relative to the
two-model case (panel A) they remain significantly worse than had no iterative
learning been included.

The reason for model 1’s seemingly contradictory ability to make good
decisions but bad forecasts becomes obvious once we compare both curves to
that of the underlying model, model 3. Looking at plots of the growth rate
curves for each model [Fig 5A], it is hardly surprising that all model
selection approaches prefer the closely overlapping curve of model 2 to the
no-where-close curve of model 1 as the better approximation of model 3.
Nevertheless, the decision policy derived from model 1 forecasts is
indistinguishable from that based on the true model [Fig 5B], while the policy
derived from model 2 forecasts lead to over-harvesting. Being closest to the
true model’s forecast skill never guarantees that we are closest to the true
model’s optimal policy.

Figure 5: Panel A: Population growth curves of each model. The positive
equilibrium of each model occurs where the curve crosses the horizontal axis.
Note that while Model 2 is a better approximation to the truth (Model 3),
Model 1 better approximates the stock size which leads to maximum growth.
Panel B: The optimal control policy under Model 1 is nearly identical to that
under the true Model 3, while the optimal policy under Model 2 suppresses
stock to a much lower escapement level.

How can the very different forecasts from model 1 and model 3 could produce
exactly the same optimal management policy (Fig 5B) under the SDP algorithm?
Analytic solutions offer more insight as to when and why very different
forecasts can generate the identical policy. Such a solution was first
provided by Reed (1979), who demonstrated the optimal policy in the case
considered here would be a so-called “bang-bang” policy. Intuitively one can
think of this as maintaining the biomass at the most productive size: the
maximum population growth rate (position of the peak of the growth curves in
Fig 5A), though this is only precisely true without discounting ($\delta=1$):
the optimal stock size $\hat{x}$ is the solution to
$f(\hat{x})=\hat{x}/\delta$ when stochasticity is sufficiently small (Reed
1979). Thus, all models in which the peak growth rate occurs at the same stock
size will have the same optimal policy. These are not merely bad models
getting lucky – all such models correctly capture the crucial feature relevant
to the decision. In more complex models, such features are more difficult or
impossible to identify analytically; but this does not mean they do not exist.
For instance, Recent mathematical breakthroughs such as Holden & Conrad
(2015), Hening _et al._ (2019), and Hening (2021) have proven that the optimal
harvest control rule in age-structured and predator-prey systems maintain
similar bang-bang dynamics. This means that the optimal policy of such very
complex models will once again be shared by infinite number of simpler models.

## Discussion

The forecast trap illustrated in both examples can best be understood as a
problem of _non-uniqueness_ (Oreskes _et al._ 1994). Even modestly complex
models can successfully predict the observed dynamics, but for wrong
mechanistic reasons (Schnute & Richards 2001; Schindler & Hilborn 2015). In
both examples, a decision-maker who accepts the model which leads to very
accurate predictions as the basis for their decision-making winds up in the
forecast trap: accepting poor ecological and economic outcomes as the best
possible option. The space of possible models is infinitely large when
measured against any a scalar metric such as mean forecast skill or expected
net utility. Perhaps it should be no surprise then that many models will
achieve the same policy outcomes or achieve comparable predictive accuracy.
Just as the forecast skill is not unique, both examples also demonstrated that
the optimal policy is not unique to the “true model” – many models will result
in the same policy and achieve the same outcomes; despite making very
different forecasts.

The forecast trap is likely to be more common in contexts in which systems are
more complex, partially observed, and available actions are constrained – all
features which are particularly common to ecological management and
conservation. Because simplicity of the second example allows analytic theory
to reveal a precise explanation, it is tempting to assume the trap is only a
consequence of examining overly simple models. In fact, the opposite is true.
In the second example, the “true” model is simple enough to be covered by a
suitable candidate set of models (e.g. a Gaussian Process, Boettiger _et al._
(2015)), which would resolve the trap given sufficient data. In reality, our
models never span the true process. Partially-observed systems increase the
space of possible models that achieve comparable predictive accuracy.
Constraints on action space such as adjustment costs (Boettiger _et al._ 2016)
or piecewise-linear control rules (Punt 2010) increase the space of models
which will result in the same policy. Both of these aspects make the forecast
trap easier to encounter in our opening example.

The forecast trap demonstrates that for certain ensembles of candidate models,
the value of information (VOI) (Howard 1966; Katz _et al._ 1987) can in fact
be negative. Consequently, methods to select models or re-estimate parameters
can lead to worse outcomes than had these new observations simply been
ignored. Crucially, a manager implementing the optimal policy from the most
predictive model sees no indication that their models are wrong – the
declining ecosystem and economic returns observed under model A in the first
example or under iterative learning in the second example are completely
consistent with and predicted by the models. Only by winding back the clock,
making decisions based over the original uncertainty without learning (Fig 4),
can we see that better outcomes could have been achieved. The policy derived
without learning reflects greater uncertainty: it is thus more _robust_.

### A way forward

In practice, managers rarely rely only on forecasting skill to assess models,
nor determine policies directly from forecasts alone. In both examples
presented above, the forecast trap is most likely in circumstances where the
collection of candidate models is insufficiently broad. Management practice
tends to emphasize approaches which _broaden_ rather than _narrow down_ this
candidate set. This reflects the view that “the primary values of ecosystem
models are as heuristic tools for communication and for developing scenarios
to express uncertainties and test policies” rather than as a source of
reliable forecasts (Schindler & Hilborn 2015). Such practices include:

  1. (a)

emphasizing a better articulation of uncertainty _a priori_ ;

  2. (b)

active exploration of alternative policies can reveal when the model set is
inadequate

  3. (c)

methods for generating strategies that are robust to the sort of uncertainty
described here.

A central role of models is to help articulate uncertainty around possible
outcomes rather than make precise predictions. As Schindler & Hilborn (2015)
notes, “current approaches to verification and validation of ecosystem models
likely produce overly optimistic impressions of the reliability of forecasts
underlying management and conservation prescriptions.” Forecasts based on non-
mechanistic models such as empirical dynamical modeling (EDM, Ye _et al._
2015; Brias & Munch 2021) may help in articulating a broader ensemble of
scenarios (Boettiger _et al._ 2015). In contrast, if such approaches are
selected solely on forecasting skill, they may increase the probability of the
forecast trap. The danger of an insufficiently broad model ensemble is well
understood in many disciplines which use _scenario planning_ to assist policy
in accounting for irreducible uncertainties (Peterson _et al._ 2003).

Second, a manager may also escape the forecast trap by exploring actions which
are never recommended from any of the available forecasts. For example, a
manager looking at the low stocks and poor harvests achieved under model 2,
could decide experimentally to reduce fishing quota. This will allow the
population to enter a range of state-space where discrepancies between model 2
and observations are more obvious. Runge _et al._ (2016) describes how such
“double-loop learning” can be used to identify when the entire model set is
inaccurate, a problem which is not solved by “single-loop” adaptive-management
in our examples above. Schindler & Hilborn (2015) also underscores the value
of flexible policies; rather than “managing solely within the range of past
variation; active probing is usually needed,” and contrasts this to a typical
interpretation of the ‘precautionary principle’ often cited as a reason to
avoid exploratory actions. However, just because active exploration can escape
the forecast trap does not mean it is always a good idea.

Third, managers may emphasize policy robustness over forecast skill (Schindler
& Hilborn 2015). In many formal treatments, this is not qualitatively
different to the analysis considered here: a manager simply chooses a
different utility function, such as minimizing ‘regret’ rather than maximizing
expected value (Polasky _et al._ 2011). Such approaches are just as vulnerable
to bad outcomes (as defined by their own utility functions) whenever models
are selected only on the basis of forecast skill. Alternative approaches may
not seek any such optimization, emphasize the viability (Aubin 1991) of
possible policy under constraints. In practice, robust design may emphasize
acceptable performance across the widest possible array of scenarios (e.g.
candidate models). This acts more like a sensitivity analysis of utility with
respect to underlying assumptions, rather than an optimization routine
(Fischer _et al._ 2009; e.g. Punt _et al._ 2016). Computationally, the former
is much simpler, allowing researchers to evaluate the performance of a policy
on more complex simulations for which calculating the optimal policy would be
prohibitively difficult.

Finally, it is worth noting that decisions do not need to be premised on a
forecast at all, but can be premised entirely on the basis of past experience:
‘If the fish stock size has gone up, increase harvest slightly, otherwise,
decrease slightly.’ Such a policy is not optimal, but it is robust across a
wide range of unimodal stock-recruitment curves without ever estimating a
predictive model. This is the basis of so-called ‘model-free’ reinforcement
learning algorithms such as DQN (Mnih _et al._ 2015) and SAC (Haarnoja _et
al._ 2018), which train deep neural networks to learn a policy without ever
attempting to predict future states of the underlying process. Training such a
artificial intelligence agents across a wide suite of simulations, a process
known as curriculum learning (Graves _et al._ 2017), mimics the scenario
analyses and search for robust policies. Such approaches have been used to
train agents to play 2600 Atari console games at superhuman ability (Mnih _et
al._ 2015), outperform race-car drivers (Wurman _et al._ 2022) and control
nuclear fusion reactions (Degrave _et al._ 2022). Such approaches are also not
yet well understood, introducing new risks as well as new possibilities(Dulac-
Arnold _et al._ 2019; Henderson _et al._ 2019).

### Acknowledgements

The author acknowledges support from NSF CAREER Award #1942280 and helpful
discussions with Melissa Chapman. The author is also deeply grateful to the
Michael Runge, Chih-hao Hsieh, Antoine Brias, and other anonymous reviewers
whose detailed feedback and insights have greatly shaped the paper and my own
understanding of these issues.

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