Exponentially Harmonic Maps into Spheres

Abstract

We study smooth exponentially harmonic maps from a compact, connected, orientable Riemannian manifold M into a sphere $S^m\subset {\mathbb{R}}^{m+1}$. Given a codimension two totally geodesic submanifold $\Sigma \subset S^m$, we show that every nonconstant exponentially harmonic map $\varphi :M\to S^m$ either meets or links $\Sigma$. If $H^1(M,\mathbb{Z})=0$ then $\varphi \left(M\right)\cap \Sigma \ne \varnothing$.

1. Introduction

Let M be a compact, connected, orientable n-dimensional Riemannian manifold, with the Riemannian metric g. Let $\varphi :M\to N$ be a $C^{\infty }$ map into another Riemannian manifold $(N,h)$. The Hilbert-Schmidt norm of $d\varphi$ is $\parallel d\varphi \parallel ={\left[{\mathrm{trace}}_g\left({\varphi }^{\ast }h\right)\right]}^{1/2}:M\to \mathbb{R}$. Let us consider the functional

$$E:C^{\infty }(M,N)\to \mathbb{R},E\left(\varphi \right)={\int }_{M}exp\left(\frac{1}{2}{\parallel d\varphi \parallel }^2\right)d{\mathrm{v}}_g.$$

A $C^{\infty }$ map $\varphi :M\to N$ is exponentially harmonic if it is a critical point of E i.e., ${\left\{dE\left({\varphi }_s\right)/ds\right\}}_{s=0}=0$ for any smooth 1-parameter variation ${\left\{{\varphi }_s\right\}}_{\left|s\right|<ϵ}\subset C^{\infty }(M,N)$ of ${\varphi }_0=\varphi$. Exponentially harmonic maps were first studied by J. Eells & L. Lemaire [1], who derived the first variation formula

$$\frac{d}{ds}{\left\{E\left({\varphi }_s\right)\right\}}_{s=0}=-{\int }_{M}exp\left[e\left(\varphi \right)\right]h^{\varphi }\left(V,\tau \left(\varphi \right)+{\varphi }_{\ast }\nabla e\left(\varphi \right)\right)d{\mathrm{v}}_g$$

where $e\left(\varphi \right)=\frac{1}{2}{\parallel d\varphi \parallel }^2$ and $\tau \left(\varphi \right)\in C^{\infty }\left({\varphi }^{-1}TN\right)$ is the tension field of $\varphi$ (cf. e.g., [2]). Also $V={\left(\partial {\varphi }_s/\partial s\right)}_{s=0}$ is the infinitesimal variation induced by the given 1-parameter variation. In particular, the Euler-Lagrange equations of the variational principle $\delta E\left(\varphi \right)=0$ are

$$-\Delta {\varphi }^i+\left({\Gamma }_{jk}^i\circ \varphi \right)\frac{\partial {\varphi }^j}{\partial x^{\alpha }}\frac{\partial {\varphi }^k}{\partial x^{\beta }}{g}^{\alpha \beta }+\frac{\partial {\varphi }^i}{\partial x^{\alpha }}\frac{\partial e\left(\varphi \right)}{\partial x^{\beta }}{g}^{\alpha \beta }=0$$

(1)

where

$$\Delta u=-\frac{1}{\sqrt{G}}\frac{\partial }{\partial x^{\alpha }}\left(\sqrt{G}{g}^{\alpha \beta }\frac{\partial u}{\partial x^{\beta }}\right),G=det\left[g_{\alpha \beta }\right],$$

is the Laplace-Beltrami operator and ${\Gamma }_{jk}^i$ are the Christoffel symbols of $h_{ij}$. The (partial) regularity of weak solutions to (1) was investigated by D.M. Duc & J. Eells (cf. [3]) when $N=\mathbb{R}$ and by Y-J. Chiang et al. (cf. [4]) when $N=S^m$. Differential geometric properties of exponentially harmonic maps, including the second variation formula for E, were found by M-C. Hong (cf. [5]), J-Q. Hong & Y. Yang (cf. [6]), L-F. Cheung & P-F. Leung (cf. [7]), and Y-J. Chiang (cf. [8]).

The purpose of the present paper is to further study exponentially harmonic maps $\varphi$ winding in $N=S^m$, a situation previously attacked in [4], though confined to the case where M is a Fefferman space-time (cf. [9]) over the Heisenberg group ${\mathbb{H}}_n$ and $\varphi :M\to S^m$ is $S^1$ invariant. Fefferman spaces are Lorentzian manifolds and exponentially harmonic maps of this sort are usually referred to as exponential wave maps (cf. e.g., Y-J. Chiang & Y-H. Yang, [10]). Base maps $f:{\mathbb{H}}_n\to S^m$ associated (by the $S^1$ invariance) to $\varphi :M\to S^m$ turn out to be solutions to degenerate elliptic equations [resembling (cf. [11]) the exponentially harmonic map system (1)] and the main result in [4] is got by applying regularity theory within subelliptic theory (cf. e.g., [12]).

Through this paper, M will be a compact Riemannian manifold and $\varphi :M\to S^m$ an exponentially harmonic map. Although the properties of an exponentially harmonic map may differ consistently from those of ordinary harmonic maps (see the emphasis by Y-J. Chiang, [13]), we succeed in recovering, to the setting of exponentially harmonic maps, the result by B. Solomon (cf. [14]) that for any nonconstant harmonic map $\varphi :M\to S^m$ from a compact Riemannian manifold either $\varphi \left(M\right)\cap \Sigma \ne \varnothing$ or $\varphi :M\to S^m\setminus \Sigma$ isn’t homotopically null. Here $\Sigma \subset S^m$ is an arbitrary codimension 2 totally geodesic submanifold.

The ingredients in the proof of the exponentially harmonic analog to Solomon’s theorem (see [14]) are (i) setting the Equation (1) in divergence form

$$-{\nabla }^{\ast }\left(exp\left[e\left(\varphi \right)\right]\nabla {\varphi }^i\right)+2e\left(\varphi \right)exp\left[e\left(\varphi \right)\right]{\varphi }^i=0$$

(got by a verbatim repetition of arguments in [4]), (ii) observing that $S^m\setminus \Sigma$ is isometric to the warped product manifold $S_{+}^{m-1}{\times }_w{S}^1$, and (iii) applying the Hopf maximum principle (to conclude that there are no nonconstant exponentially harmonic maps into hemispheres).

2. Exponentially Harmonic Maps into Warped Products

Let $S=L\times \mathbb{R}$, where L is a Riemannian manifold with the Riemannian metric $g_L$. Let $w\in C^{\infty }\left(S\right)$ such that $w\left(y\right)>0$ for any $y\in S$ and let us endow S with the warped product metric

$$h={\Pi }_1^{\ast }{g}_L+w^{2}dt\otimes dt,$$

where $t=\tilde{t}\circ {\Pi }_2$, $\tilde{t}$ is the Cartesian coordinate on $\mathbb{R}$, and

$${\Pi }_1:S\to L,{\Pi }_2:S\to \mathbb{R},$$

are projections. The Riemannian manifold $(S,h)$ is customarily denoted by $L{\times }_w\mathbb{R}$. Let $\varphi :M\to S$ be an exponentially harmonic map and let us set

$$F={\Pi }_1\circ \varphi ,u={\Pi }_2\circ \varphi .$$

We need to establish the following

Lemma 1.

Let M be a compact, connected, orientable Riemannian manifold and$\varphi =(F,u):M\to S=L{\times }_w\mathbb{R}$a nonconstant exponentially harmonic map. Then u is a solution to

$$\left(w\circ \varphi \right)\Delta u+\left(\frac{\partial w}{\partial t}\circ \varphi \right){\parallel \nabla u\parallel }^2$$

(2)

$$=\left(w\circ \varphi \right)(\nabla u)e\left(\varphi \right)+2(\nabla u)(w\circ \varphi ).$$

If additionally$\partial w/\partial t=0$ then $\varphi \left(M\right)\subset L\times \left\{t_{\varphi }\right\}$ for some $t_{\varphi }\in \mathbb{R}$.

Also for an arbitrary test function $\phi \in C^{\infty }\left(M\right)$ we set

$${\varphi }_s\left(x\right)=\left(F\left(x\right),u\left(x\right)+s\phi \left(x\right)\right),x\in M,\left|s\right|<ϵ,$$

so that ${\left\{{\varphi }_s\right\}}_{\left|s\right|<ϵ}$ is a 1-parameter variation of $\varphi$. For each $x_0\in M$ let $\{E_{\alpha }:1\le \alpha \le n\}\subset C^{\infty }(U,T\left(M\right))$ be a local g-orthonormal (i.e., $g(E_{\alpha },E_{\beta })={\delta }_{\alpha \beta }$) frame, defined on an open neighborhood $U\subset M$ of $x_0$. Then

$${\parallel d{\varphi }_s\parallel }^2={\mathrm{trace}}_g\left({\varphi }_s^{\ast }h\right)=\sum _{\alpha =1}^n\left({\varphi }_s^{\ast }h\right)(E_{\alpha },E_{\alpha })$$

on U. On the other hand

$$\left({\varphi }_s^{\ast }h\right)(X,X)=\left(F^{\ast }{g}_L\right)(X,X)+{(w\circ {\varphi }_s)}^2{\left[X\left(u\right)+sX\left(\phi \right)\right]}^2$$

(3)

for every tangent vector field $X\in \mathfrak{X}\left(M\right)$. Formula (3) for $X=E_{\alpha }$ yields

$${\parallel d{\varphi }_s\parallel }^2={\parallel dF\parallel }^2+{\left(w\circ {\varphi }_s\right)}^2\left[{\parallel \nabla u\parallel }^2+2sg(\nabla u,\nabla \phi )+s^2{\parallel \nabla \phi \parallel }^2\right].$$

Hence (differentiating with respect to s)

$$\frac{d}{ds}{\left\{E\left({\varphi }_s\right)\right\}}_{s=0}={\int }_{M}exp\left[e\left(\varphi \right)\right]\left\{{\left(w\circ \varphi \right)}^{2}g(\nabla u,\nabla \phi )\right.$$

(4)

$$\left.+\left(w\circ \varphi \right)\left(w_t\circ \varphi \right)\phi {\parallel \nabla u\parallel }^2\right\}d{\mathrm{v}}_g$$

where $w_t=\partial w/\partial t$. Moreover

$$exp\left[e\left(\varphi \right)\right]{\left(w\circ \varphi \right)}^{2}g\left(\nabla u,\nabla \phi \right)$$

(5)

$$=\mathrm{div}\left(\phi exp\left[e\left(\varphi \right)\right]{(w\circ \varphi )}^2\nabla u\right)$$

$$+\phi \left\{exp\left[e\left(\varphi \right)\right]{\left(w\circ \varphi \right)}^2\Delta u-(\nabla u)\left(exp\left[e\left(\varphi \right)\right]{(w\circ \varphi )}^2\right)\right\}$$

where $\mathrm{div}:\mathfrak{X}\left(M\right)\to C^{\infty }\left(M\right)$ is the divergence operator with respect to the Riemannian volume form

$$d{\mathrm{v}}_g=\sqrt{G}d{x}^1\wedge \cdots \wedge d{x}^n$$

i.e., ${\mathcal{L}}_{X}d{\mathrm{v}}_g=\mathrm{div}\left(X\right)d{\mathrm{v}}_g$ and $\Delta$ is the Laplace-Beltrami operator (on functions) i.e., $\Delta u=-\mathrm{div}(\nabla u)$. Substitution from (5) into (4) together with Green’s lemma yields [by ${\{dE\left({\varphi }_s\right)/ds\}}_{s=0}=0$ and the density of $C^{\infty }\left(M\right)$ in $L^2\left(M\right)$]

$$\left(w\circ \varphi \right)\Delta u+\left(w_t\circ \varphi \right){\parallel \nabla u\parallel }^2$$

(6)

$$=\left(w\circ \varphi \right)(\nabla u)e\left(\varphi \right)+2(\nabla u)(w\circ \varphi )$$

which is (2) in Lemma 1. When $w_t=0$ Equation (6) is

$$\mathrm{div}\left\{exp\left[e\left(\varphi \right)\right]{(w\circ \varphi )}^2\nabla u\right\}=0.$$

(7)

Equation (7) is part of the Euler-Lagrange system associated to the variational principle $\delta E\left(\varphi \right)=0$. Next (by (7))

$$\mathrm{div}\left\{{\left(w\circ \varphi \right)}^{2}uexp\left[e\left(\varphi \right)\right]\nabla u\right\}=exp\left[e\left(\varphi \right)\right]{\left(w\circ \varphi \right)}^2{\parallel \nabla u\parallel }^2.$$

(8)

Let us integrate over M in (8) and use Green’s lemma. We obtain

$${\int }_{M}exp\left[e\left(\varphi \right)\right]{\left(w\circ \varphi \right)}^2{\parallel \nabla u\parallel }^{2}d{\mathrm{v}}_g=0$$

yielding (as $\varphi$ is assumed to be nonconstant) $u\left(x\right)=t_{\varphi }$ for some $t_{\varphi }\in \mathbb{R}$ and any $x\in M$. Q.e.d.

3. Exponentially Harmonic Maps Omitting a Codimension 2 Sphere Aren’t Null Homotopic

Let $\Sigma \subset S^m$ be a codimension 2 totally geodesic submanifold. A continuous map $\varphi :M\to S^m$ meets $\Sigma$ if $\varphi \left(M\right)\cap \Sigma \ne \varnothing$ and links $\Sigma$ if $\varphi \left(M\right)\cap \Sigma =\varnothing$ and $\varphi :M\to S^m\setminus \Sigma$ is not null homotopic. The purpose of the section is to establish

Theorem 1.

Let $\varphi :M\to S^m$ be a nonconstant exponentially harmonic map from a compact, connected, orientable Riemannian manifold M into the sphere $S^m\subset {\mathbb{R}}^{m+1}$. If $\Sigma \subset S^m$ is a codimension 2 totally geodesic submanifold, then φ either meets or links Σ.

Proof. 

The proof is by contradiction, i.e., we assume that $\varphi$ doesn’t meet $\Sigma$ and the map $\varphi :M\to S^m\setminus \Sigma$ is null homotpic. Let $\left({\xi }_j\right)$ be a system of coordinates on ${\mathbb{R}}^{m+1}$ such that $\Sigma$ is given by the equations ${\xi }_1={\xi }_2=0$. Let $S_{+}^{m-1}\subset {\mathbb{R}}^m$ be the hemisphere

$$S_{+}^{m-1}=\left\{y=(y^{\prime },y_m)\in {\mathbb{R}}^{m-1}\times \mathbb{R}:y\in S^{m-1},y_m>0\right\}.$$

Let us consider the map

$$I:S_{+}^{m-1}\times S^1\to S^m\setminus \Sigma ,I(y,\zeta )=\left(y_{m}u,y_{m}v,y^{\prime }\right),$$

$$y=\left(y^{\prime },y_m\right)\in S_{+}^{m-1},\zeta =u+iv\in S^1\subset \mathbb{C}.$$

Let $g_N$ denote the canonical Riemannian metric on $S^N\subset {\mathbb{R}}^{N+1}$. The map I is an isometry of $S_{+}^{m-1}{\times }_f{S}^1$ onto $(S^m\setminus \Sigma ,g_m)$ with the warping function

$$f\in C^{\infty }(S_{+}^{m-1}\times S^1),f(y,\zeta )=y_m.$$

Let us consider the map $\tilde{\psi }=I^{-1}\circ \varphi$. We need the following. □

Lemma 2.

Let S and$\overline{S}$be Riemannian manifolds,$\pi :S\to \overline{S}$a local isometry, and$\overline{f}:M\to \overline{S}$an exponentially harmonic map. Then every map$f:M\to S$such that$\pi \circ f=\overline{f}$is exponentially harmonic.

Proof. 

Let h and $\overline{h}$ be the Riemannian metrics on S and $\overline{S}$. For every 1-parameter variation ${\left\{f_s\right\}}_{\left|s\right|<ϵ}$ of $f_0=f$ we set ${\overline{f}}_s=\pi \circ f_s$ so that ${\left\{{\overline{f}}_s\right\}}_{\left|s\right|<ϵ}$ is a 1-parameter variation of ${\overline{f}}_0=\overline{f}$. A calculation relying on ${\pi }^{\ast }\overline{h}=h$ yields $E\left(f_s\right)=E\left({\overline{f}}_s\right)$ for every $\left|s\right|<ϵ$. Q.e.d.

By Lemma 2 the map $\tilde{\psi }=I^{-1}\circ \varphi$ is exponentially harmonic. Let us set

$$F={\pi }_1\circ \tilde{\psi },\tilde{u}={\pi }_2\circ \tilde{\psi },$$

where ${\pi }_1:S_{+}^{m-1}\times S^1\to S_{+}^{m-1}$ and ${\pi }_2:S_{+}^{m-1}\times S^1\to S^1$ are projections. Next let us consider a point $x_0\in M$ and set ${\zeta }_0=\tilde{u}\left(x_0\right)\in S^1$. Also, considered the covering map $p:\mathbb{R}\to S^1$, $p\left(t\right)=exp\left(2\pi it\right)$, pick $t_0\in \mathbb{R}$ such that $p\left(t_0\right)={\zeta }_0$. As $\varphi$ is null homotopic, the map $\tilde{\psi }$ is null homotopic as well. Then

$${\tilde{u}}_{\ast }{\pi }_1(M,x_0)=0$$

where ${\pi }_1(M,x_0)$ is the first homotopy group of M. Consequently there is a unique smooth function $u:M\to \mathbb{R}$ such that $p\circ u=\tilde{u}$ and $u\left(x_0\right)=t_0$. The map

$$\psi =(F,u):M\to S_{+}^{m-1}{\times }_w\mathbb{R}$$

is exponentially harmonic [because $\psi =\pi \circ \tilde{\psi }$ and

$$\pi =\left(1_{S_{+}^{m-1}},p\right):S_{+}^{m-1}{\times }_w\mathbb{R}\to S_{+}^{m-1}{\times }_f{S}^1$$

is a local isometry, where $w\in C^{\infty }\left(S_{+}^{m-1}\right)$ is given by $w\left(y\right)=y_m$]. We may then apply Lemma 1 to the map $\psi$ with $L=S_{+}^{m-1}$ to conclude that

$$\psi \left(M\right)\subset S_{+}^{m-1}\times \left\{t_{\psi }\right\}$$

for some $t_{\psi }\in \mathbb{R}$. It follows that $F={\pi }_1\circ \psi :M\to S_{+}^{m-1}$ is exponentially harmonic. We shall close the proof of Theorem 1 by showing that exponentially harmonic mappings into $S_{+}^{m-1}$ are constant. □

4. Exponentially Harmonic Map System in Divergence Form

Let us consider the $L^2$ inner products

$${(u,v)}_{L^2}={\int }_{M}uvd{\mathrm{v}}_g,{(X,Y)}_{L^2}={\int }_{M}g(X,Y)d{\mathrm{v}}_g.$$

Let us think of the gradient ∇ as a first order differential operator $\nabla :C^1\left(M\right)\to C\left(T\left(M\right)\right)$ and let ${\nabla }^{\ast }$ be its formal adjoint, i.e.,

$${\left({\nabla }^{\ast }X,u\right)}_{L^2}={\left(X,\nabla u\right)}_{L^2}$$

for any $X\in C^1\left(T\left(M\right)\right)$ and $u\in C^1\left(M\right)$. Ordinary integration by parts shows that ${\nabla }^{\ast }X=-\mathrm{div}\left(X\right)$. Let $\rho =exp\left[e\left(F\right)\right]\in C^{\infty }\left(M\right)$. Starting from $\Delta u=-\mathrm{div}(\nabla u)$ one has

$${\left(\rho \Delta u,\phi \right)}_{L^2}={\left({\nabla }^{\ast }\nabla u,\rho \phi \right)}_{L^2}={\left(\nabla u,\nabla \left(\rho \phi \right)\right)}_{L^2}$$

$$={\left({\nabla }^{\ast }(\rho \nabla u),\phi \right)}_{L^2}+{\int }_M\phi g(\nabla u,\nabla \rho )d{\mathrm{v}}_g$$

for any $\phi \in C^{\infty }\left(M\right)$, that is

$$exp\left[e\left(F\right)\right]\Delta u={\nabla }^{\ast }\left(exp\left[e\left(F\right)\right]\nabla u\right)$$

(9)

$$+exp\left[e\left(F\right)\right]g\left(\nabla u,\nabla e\left(F\right)\right).$$

Lemma 3.

Let$F:M\to S_{+}^{m-1}$be an exponentially harmonic map and$\mathbf{F}=j\circ F$where$j:S^{m-1}\hookrightarrow {\mathbb{R}}^m$is the inclusion. If$\mathbf{F}=\left(F^1,\cdots ,F^m\right)$then

$$-{\nabla }^{\ast }\left(exp\left[e\left(F\right)\right]\nabla F^i\right)+2e\left(F\right)exp\left[e\left(F\right)\right]F^i=0$$

(10)

for any$1\le i\le m$.

Proof. 

Let $y=\left(y^1,\cdots ,y^{m-1}\right):S_{+}^{m-1}\to {\mathbb{B}}^{m-1}$ be the projection, where ${\mathbb{B}}^{m-1}\subset {\mathbb{R}}^{m-1}$ is the open unit ball. With respect to this choice of local coordinates, the standard metric $g_{m-1}$ and its Christoffel symbols are

$$h_{ij}={\delta }_{ij}+\frac{y^i{y}^j}{1-{\left|y\right|}^2},{\left|y\right|}^2=\sum _{i=1}^{m-1}{\left(y^i\right)}^2,$$

(11)

$$h^{ij}={\delta }^{ij}-y^i{y}^j,$$

(12)

$${\Gamma }_{jk}^i=y^i{h}_{jk}.$$

(13)

Let us substitute from (13) into (1) [with ${\varphi }^i=F^i$] and take into account

$$e\left(F\right)=\frac{1}{2}{g}^{\alpha \beta }\frac{\partial F^j}{\partial x^{\alpha }}\frac{\partial F^k}{\partial x^{\beta }}\left(h_{jk}\circ F\right).$$

(14)

The exponentially harmonic map system (1) becomes

$$-\Delta F^i+2e\left(F\right)F^i+g\left(\nabla e\left(F\right),\nabla F^i\right)=0,1\le i\le m-1.$$

(15)

Multiplication of (15) by $exp\left[e\left(F\right)\right]$ and subtraction from (9) [with $u=F^i$] yields (10) for any $1\le i\le m-1$.

To see that (15) (and therefore (10)) holds for $i=m$ as well, one first exploits the constraint ${\left(F^m\right)}^2=1-{\sum }_{i=1}^{m-1}{\left(F^i\right)}^2$ together with (11) and (14) to show that

$$e\left(F\right)=\frac{1}{2}\sum _{j=1}^m{\parallel \nabla F^j\parallel }^2.$$

Finally, one contracts (15) by $F^i$ and uses once again the constraint together with $\Delta \left(u^2\right)=2\left\{u\Delta u-{\parallel \nabla u\parallel }^2\right\}$. Q.e.d.

We may now end the proof of Theorem 1 as follows. Let $F:M\to S_{+}^{m-1}$ be an exponentially harmonic map. Let us integrate over M in (10) for $j=m$. Then (by Green’s lemma)

$${\int }_{M}e\left(F\right)exp\left[e\left(F\right)\right]F^{m}d{\mathrm{v}}_g=0$$

and $F^m>0$ so that

$$0=e\left(F\right)=\frac{1}{2}\sum _{j=1}^m{\parallel \nabla F^j\parallel }^2$$

yielding $F^j=$ constant. So $\varphi$ is constant as well, a contradiction. □

As well known $S_{+}^{m-1}\times S^1$ and $S^1$ are homotopically equivalent. Therefore a continuous map $\varphi :M\to S_{+}^{m-1}\times S^1$ is null homotopic if and only if ${\pi }_2\circ \varphi :M\to S^1$ is null homotopic. The homotopy classes of continuous maps $M\to S^1$ form an abelian group ${\pi }^1\left(M\right)$ (the Bruschlinski group of M) naturally isomorphic to $H^1(M,\mathbb{Z})$. We may conclude that

Corollary 1.

Let M be a compact, orientable, connected Riemannian manifold with$H^1(M,\mathbb{Z})=0$. Then every nonconstant exponentially harmonic map$\varphi :M\to S^m$meets Σ.