A Schwarz lemma of harmonic maps into metric spaces

1.   Introduction

The classical Schwarz-Pick lemma states that any holomorphic map from unit disk in $\mathbb{C}^{2}$ into itself decreases the Poincaré-Bergman metric. Ahlfors ^[1], Chern et al.^[2], and Lu ^[3] extended this result to more general domains and targets with curvature conditions.

In 1978, Yau ^[4] used the so-called Omori-Yau maximum principle to prove the distance decreasing property for holomorphic maps from K$\ddot{\text{a}}$hler manifolds to Hermitian manifolds under suitable curvature conditions. Later, in the Riemannian settings, the Schwarz type lemma was studied extensively; see, e.g., ^[5,6]. There are also many generalizations for generalized harmonic maps; readers can refer to ^{[7,8,9,10,11]} and references therein.

There is also growing interest on harmonic maps with singular targets. Gromov and Schoen ^[12] first developed a theory of harmonic maps in which the target spaces can be taken as metric spaces of curvature bounded from above. This theory has further generalized in Korevaar and Schoen ^[13], and also Jost ^[14,15,16]. These investigations have deeply revealed the structure of harmonic maps with singular targets. Thus, it is natural to consider establishing a Schwarz type lemma for harmonic maps in this broader context.

In this note, the target spaces $(N, d)$ we considered are $\operatorname{CAT}(-k)$ $(k > 0)$ spaces, which is a class of metric spaces with curvature bounded above by $-k$ and the curvature condition is given by Toponogov's triangle comparison.

A map $u:M\longrightarrow N$ is called harmonic if $u$ is a local minimizer of the energy functional in the sense of Korevaar and Schoen. For a detailed definition and its properties, we refer to Section 2 below.

Our main result states the following.

Theorem 1.1. Let $u:M\longrightarrow N$ be a harmonic map from an $m$-dimensional complete Riemannian manifold with $\operatorname{Ric}_{M}\geq -A$ into a $\operatorname{CAT}(-k)$ space $N$, where $A\geq 0$ and $k > 0$ are both constants. Suppose $u$ has generalized dilatation of order $\beta$, then

where $\pi$ is the pull-back inner product (cf. Section 2).

Remark 1.2. The notion of harmonic maps with bounded dilatation was originated from Shen ^[6], where he proved a related Schwarz type lemma. Our result can be viewed as a generalization in the singular targets setting.

Corollary 1.3. If $\; \operatorname{Ric}_{M}\geq 0$, then any harmonic map of generalized dilatation from $M$ into a $\operatorname{CAT}(-k)$ $(k > 0)$ space is constant.

Corollary 1.4. If $M$ is a hyperbolic surface and $N$ is a $\operatorname{CAT}(-1)$ space, then the energy of conformal harmonic map $u:M\longrightarrow N$ satisfies

Remark 1.5. Freidin ^[17] proved Corollary 1.4 under the additional assumption that $\Sigma$ is closed. Our result improves his.

Owing to the lack of smoothness, one cannot employ the usual argument of the maximum principle directly in this setting. Instead, we will make use of an approximating version of the maximum principle established by Zhang and Zhu ^[18]. The similar idea has been successfully applied by Zhang et al. ^[19] to obtain gradient estimates of harmonic maps in the setting of singular targets.

The rest is organized as follows: In Section 2, we recall some basic and known results on $\operatorname{CAT}(\kappa)$ spaces and harmonic maps. In Section 3, we prove the main results.

2.   Preliminaries

2.1.   Alexandrov curvature bound

Give a metric space $(N, d)$. We assume our metric spaces to be length spaces, i.e., for each $P, Q\in N$, there is a curve, which we denote $[P, Q]$, such that the length of $[P, Q]$ is exactly $d(P, Q)$. We call $[P, Q]$ a geodesic between $P$ and $Q$. We say $N$ is a $\operatorname{CAT}(0)$ space (see ^[20]) if any geodesic triangles in $N$ are thinner than their comparison triangles in $\mathbb{R}^{2}$. In other words, for every $P, Q, R\in N$, and corresponding points $\bar{P}, \bar{Q}, \bar{R}\in \mathbb{R}^{2}$ with

we have

where $Q_{t}$ and $\bar{Q}_{t}$ lie a fraction $t$ of the way along the geodesic segment from $P$ to $Q$ and $\bar{P}$ to $\bar{Q}$, respectively.

A $\operatorname{CAT}(-1)$ space $N$, or $N$ having curvature bounded from above by -1 in the sense of Alexandrov, is simply a length space with a stronger comparison principle. Instead of constructing comparison triangles in $\mathbb{R}^{2}$, one constructs them in $\mathbb{H}^{2}$, and the $\operatorname{CAT}(-1)$ space has the same comparison inequality.

2.2.   Harmonic maps into metric spaces

In this subsection, we define harmonic maps from an $m$-dimensional complete Riemannian manifold $(M, g)$ to a general metric space $(N, d)$. Let $\Omega\subset M$ be a relatively compact domain. We denote by $L^{2}(\Omega, N)$ the space of all Borel maps $u:\Omega\longrightarrow N$, (i.e., measurable with respect to $dV_{g}$) such that for some and, thus, every $p\in N$, we have

The (Korevaar-Schoen) energy of $u\in L^{2}(\Omega, N)$ is defined as follows. For $\varepsilon > 0$, we set

whenever $x\in \Omega$ satisfies $d(x, \partial \Omega) > \varepsilon$ and $e_{\varepsilon}(x) = 0$ otherwise. Here, $S(x, \varepsilon)$ is the sphere centered at $x$ with radius $\varepsilon$. A map $u$ is said to be in $W^{1, 2}(\Omega, N)$ if its energy, defined by

is finite. If $u \in W^{1, 2}(\Omega, N)$, then there exists a function $e(u) \in L^1(\Omega)$, called the energy density function of $u$, such that $e_{\varepsilon}dV_{g} \rightharpoonup e(u) dV_{g}$ as $\varepsilon \rightarrow 0$ and $E(u) = \int_{\Omega}e(u)dV_{g}$; see ^[13]. We often write $|\nabla u|^{2}(x)$ in place of $e(u)(x)$. In the case that $N$ is a Riemannian manifold and $u$ is smooth, the energy defined in (2.1) coincides with the usual energy.

Definition 2.1. A map $u \in W^{1, 2}(\Omega, N)$ is said to be energy minimizing harmonic if $E(u) \leq E(v)$ for all $v \in W^{1, 2}(\Omega, N)$. A map $u: M \rightarrow N$ is called energy minimizing harmonic if its restriction to every relatively compact domain is energy minimizing harmonic.

Let $\overline{x}(x, t)$ be the flow generated by a unit vector field $\omega$ on $M$, that is,

The directional energy density $|u_{\ast}(\omega)|^{2}$ is defined by

and the energy density satisfies

where $\mathbb{S}^{n-1}\subset T_{x}M$.

The $\operatorname{CAT}(-k)$ hypothesis implies that we can make sense of the notion of the pull-back inner product.

Definition 2.2 (^[13]). The pull-back inner product $\pi:\Gamma(TM)\times \Gamma(TM)\longrightarrow L^{1}(M)$ is defined by

Proposition 2.3 (Theorem 2.3.2 in ^[13]). For the operator $\pi$ defined above, we have

(1) $\pi$ is continuous, symmetric, bilinear, nonnegative and tensorial.

(2) If we write $\pi_{ij} = \pi(e_{i}, e_{j})$ for a local frame $\{e_{i}\}$ on $M$, then for $Z = Z^{i}e_{i}$ and $W = W^{j}e_{j}$, we have $\pi(Z, W) = \pi_{ij}Z^{i}W^{j}$.

(3) The energy density is the trace of $\pi$ with respect to $g$, i.e., $|\nabla u|^{2} = g^{ij}\pi_{ij} = \operatorname{tr}_{g}(\pi)$.

3.   The Schwarz lemma

Let $(M^{m}, g)$ be a complete Riemannian manifold with $\operatorname{Ric}_{M}\geq -A$, where $A\geq 0$ is a constant, and suppose that $u:M\longrightarrow N$ is a harmonic map into a $\operatorname{CAT}(-k)$ $(k > 0)$ space $N$. Under a local frame $\{e_{i}\}$ on $M$, the pull-back tensor $\pi$ can be expressed as a matrix

From Proposition 2.3, it is clear that the eigenvalus of $\pi$ are nonnegative, say,

Hence, we can introduce the notion of generalized dilatation in this context.

Definition 3.1. Let $(M, g)$ be a Riemannian manifold and $N$ be a $\operatorname{CAT}(0)$ space. A map $u\in W^{1, 2}(M, N)$ is said to have generalized dilatation of order $\beta$, if there is a positive number $\beta$ such that $\lambda_{1}(x)\leq \beta^{2}(\lambda_{2}(x)+\cdots+\lambda_{m}(x))$ for $a.e.$ $x\in M$.

Let us recall the proof of Schwarz type lemma in the smooth context. We refer to ^[7] and ^[8]. There are two main ingredients: the Bochner formula and a maximum principle. When the target space is of $\operatorname{CAT}(-1)$ type, we have the following Bochner inequality.

Lemma 3.2 (Theorem 1 in ^[17]). For a harmonic map $u:M\longrightarrow N$ from a Riemannian manifold $M$ into a $\operatorname{CAT}(-k)$ $(k > 0)$ metric space, $|\nabla u|^{2}$ satisfies

in the sense of distributions. Here, $\langle \operatorname{Ric}_{M}, \pi \rangle$ denotes the inner product on symmetric $2$-tensors and $|\pi|^{2} = \langle \pi, \pi \rangle$.

We remark that the energy density $|\nabla u|^{2}$ is not smooth generally. Moreover, it even may not be continuous. That presents a problem to carry the gradient estimates argument in ^[7] or Omori-Yau maximum principle in ^[8] directly due to the lack of smoothness. This can be overcome by making use of the following Zhang-Zhu's maximum principle.

Theorem 3.3 (Theorem 1.3 in ^[18]). Let $\Omega$ be a bounded domain in a Riemannian manifold $(M^{m}, g)$ with $\operatorname{Ric}_{M}\geq -A$ for some constant $A\geq 0$. Let $f\in W^{1, 2}_{loc}(\Omega)\cap L^{\infty}_{loc}(\Omega)$ satisfy $\Delta f$ as a signed Radon measure with $\Delta^{\operatorname{sing}}f\geq 0$, where $\Delta f = \Delta^{\operatorname{ac}}f\cdot dV_{g}+\Delta^{\operatorname{sing}}f$ is the Radon-Nikodym decomposition with respect to $dV_{g}$. Assume $f$ attains its strict maximum in $\Omega$ in the following sense: there is a neighborhood $U\subset\subset \Omega$ such that

Then, for any function $w\in W^{1, 2}(\Omega)\cap L^{\infty}(\Omega)$, there is a sequence $\{x_{j}\}_{j\in \mathbb{N}}\subset U$ satisfying that they are approximate continuity points of $\Delta^{\operatorname{ac}}f$ and $\langle \nabla f, \nabla w \rangle$, with the following properties:

In the following, $\sup_{U}f$ means $\operatorname{ess-sup}_{U}f$.

We prove the main result.

Theorem 3.4. Let $u:M\longrightarrow N$ be a harmonic map from a complete Riemannian manifold with $\operatorname{Ric}_{M}\geq -A$ into a $\operatorname{CAT}(-k)$ space $N$, where $A\geq 0$ and $k > 0$ are both constants. Suppose $u$ has generalized dilatation of order $\beta$, then

where $\pi$ is the pull-back inner product.

Proof. For simplicity, we assume $k = 1$. By the curvature condition and the Bochner inequality (3.1), we have

Note that $u$ is of bounded dilatation, thus

On the other hand,

Combining (3.2), (3.3), and (3.4), we conclude that $|\nabla u|^{2}$ satisfies

in the weak sense. For simplicity, let $F: = |\nabla u|^{2}$.

Fix a constant $\delta > 0$ to be sufficiently small and let $\eta(x): = \eta(r(x))$ be a function of the distance $r(x, x_{0})$, where $x_{0}$ is a fixed point in $M$, such that

and

where $C_{1}$ is a constant independent of $m, K, R$. We remark that the function $\eta$ can be chosen in the following ways: one first takes a function $\phi$ such that $0\leq \phi\leq 1$, $\phi = 1$ on $(0, \frac{R}{2})$, $\phi = 0$ on $(\frac{3R}{4}, R)$, $-\frac{C}{R}\leq \phi^{\prime}\leq 0$, and $|\phi^{\prime\prime}|\leq \frac{C}{R^{2}}$, and then sets $\eta: = \left(\frac{\varepsilon+\phi}{\varepsilon+1}\right)^{2}$, where $\frac{\varepsilon}{\varepsilon+1} = \sqrt{\delta}$. One can verify that $\eta$ satisfies all of the requirements by direct calculations.

Thus,

By the Laplacian comparison theorem, we also have that

in the sense of distributions. Here, $C_{2}: = C_{1}\left(R\sqrt{(m-1)A}+m\right)$.

Next, we set $G: = F\cdot \eta$. It is obvious that $G$ is in $W^{1, 2}(B_{\frac{3R}{4}}(x_{0}))\cap L^{\infty}(B_{\frac{3R}{4}}(x_{0}))$ and $G$ achieves one of its strict maxima in $B_{\frac{R}{2}}(x_{0})$ in the sense of Theorem 3.3. Then,

Let $w: = -2\log\eta$, then $w\in W^{1, 2}(B_{\frac{3R}{4}}(x_{0}))\cap L^{\infty}(B_{\frac{3R}{4}}(x_{0}))$. The above inequality reads as

where we have used $G\geq 0$, $1\leq \frac{1}{\eta}$, and $C_{3}: = C_{2}+2C_{1}^{2}$. That is, we have

in the weak sense. By Theorem 3.3, we can conclude that there exists a sequence of points $\{x_{j}\}$ such that for each $j\in \mathbb{N}$,

and

Since $\eta(x_{j})\geq \delta > 0$, we can take $j\to \infty$ to obtain

which implies

Letting $R\to \infty$, it follows that

As $|\nabla u|^{2} = \operatorname{tr}_{g}(\pi) = \sum_{i = 1}^{m}\lambda_{i}$, we have

This yields $\lambda_{1}\leq \frac{A\beta^{4}m^{2}}{2(1+\beta^{2})}$, and we have finished the proof. □

Proof of Corollary 1.4. Note that a mapping $u:M\longrightarrow N$ is called conformal if $\pi$ satisfies $\pi = \lambda g$ for some nonnegative function $\lambda$. Then, the corollary follows from Theorem 1.1 immediately. □

Use of AI tools declaration

The author declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is partially supported by National Natural Science Foundation of China (NNSFC) Key Program Grant No. 12331002 and NNSFC Grant No. 11971358.

Conflict of interest

The author declare there is no conflict of interest.