On weakly harmonic maps from Finsler to Riemannian manifolds (Q1001977)

In this paper, the authors deal with harmonic maps from Finsler manifolds to Riemannian manifolds. Let \(\varphi : M^n \to N^m\) be a smooth mapping from an \(n\)-dimensional Finsler manifold \((M, F)\) with Finsler structure \(F\) into an \(m\)-dimensional Riemannian manifold \((N^m, h)\) without boundary. The energy density \(e(\varphi): SM \to [0, \infty)\) is defined by \(e(\varphi) (x, [y]) = \frac{1}{2}g^{ij}(x,y) \frac{\partial \varphi^\alpha}{\partial x^i} \frac{\partial \varphi^\beta}{\partial x^j} h_{\alpha \beta}\), where \(SM\) is the sphere bundle defined by \(SM = \{(x, [y])\,:\, (x,y)\in TM\setminus\;0\,\}\) which consists of the rays \((x,[y]) = \{(x, ty)\,:\, t>0\,\}\) and \(\varphi^\alpha\) is the component of \(\varphi\) with respect to a local coordinate system \((u^\alpha)\) on \(N\). The energy \(E(\varphi)\) is then defined by \(E(\varphi) = \frac{1}{\text{vol}(S^{n-1})}\int_{SM} e(\varphi)\) and the localized energy for a bounded domain \(\Omega \subset M\) is defined by \(E_{\Omega}(\varphi) = E(\varphi|_{\Omega})\). A mapping \(\varphi \in W^{1,2}_{\text{loc}}(\Omega, N)\cap L^\infty(\Omega, N)\) is said to be weakly harmonic on \(\Omega \Subset M\) if the first variation of \(E_\Omega\) vanishes at \(\varphi\). Suppose that \(\chi : \Omega \to B_0(4r) \subset {\mathbb R}^n\) is a local coordinate chart of \(M\) and suppose that the components of the Finsler metric \(g_{ij}(x,y)\) satisfy \(\lambda|\xi|^2 \leq g_{ij}(x,y)\xi^i \xi^j \leq \mu |\xi|^2\) for all \(\xi\in{\mathbb R}^n\) and all \((x, y) \in T\Omega\setminus \{0\}\cong B_0(4r)\times {\mathbb R}^n\) with constants \(0< \lambda \leq \mu < \infty\). Assume that \(\varphi : M \to N\) is a weakly harmonic map with \(\varphi(\Omega) \subset {\mathcal B}_L(q)\), where \({\mathcal B}_L(q)\) is a geodesic ball in \(N\) which is regular, that is, it does not intersect the cut-locus of \(q\) and \(L < \frac{\pi}{2\sqrt{\kappa}}, \, \kappa = \max\{0, \sup_{{\mathcal B}_L(q)} K_N\,\}\). With these hypotheses, the authors proved that the map \(\varphi\) is Hölder continuous and satisfies the following estimate: \(\sup_{x, y\in B_0(r)} \frac{|\varphi(x)-\varphi(y)|}{|x-y|^\alpha} \leq C r^{-\alpha}\) with constants \(0 < \alpha< 1\) and \(C\) depending only on \(n, \lambda, \mu, L, \kappa\) and the lower curvature bound on \(N\), but not on \(r >0\). As a consequence, when the Finsler manifold \(M\) admits a global coordinate \(\chi : M \to {\mathbb R}^n\) satisfying the Finsler metric bound mentioned above, the authors show a Liouville type theorem which says that any harmonic map \(\varphi : M \to N\) with \(\varphi(M) \subset {\mathcal B}_L(q)\) is constant. Also the authors prove an existence theorem for harmonic maps from Finsler manifolds into regular balls of a Riemannian manifold by using global \(C^{2,\alpha}\)-estimates obtained from extending the Hölder continuous estimates to the boundary, and combining them with well-known gradient estimates.