Proper harmonic maps between asymptotically hyperbolic manifolds (Q261406)

Let \((M^{n}, g)\) be a non-compact manifold of dimension at least two, equipped with a smooth Riemannian metric \(g\) and \(\bar{M}=M\cup\partial M\). \((M, g)\) is called a \(\mathcal{C}^{2}\) conformally compact manifold if \(r^{2}g\) extends to \(\mathcal{C}^{2}\) Riemannian metric \(\bar{g}\) on \(\bar{M}\), where \(r \in C^{\infty}(M)\) is a smooth (positive) defining function. If \(| d\log r|_{g}= | dr |_{\bar{g}}= 1\) on \( \partial M\) (equivalent, that the sectional curvature \( K_{g}\) uniformly tends to \(-1\)), then \((M^{n}, g)\) is said to be asymptotically hyperbolic (AH, for short). For two \(\mathcal{C}^{2}\) conformally compact AH manifolds, \((M^{m+1}, g)\) and \((N^{n+1}, h)\), where \(m, n \geq 1\) and for any boundary map \(f \in C^{0}(\partial M,\partial N )\), consider \(\mathcal{M}_{f}=\{ u \in C^{0}(\bar{M}, \bar{N}) | u \text{ maps } \partial M \text{ into } \partial N \}\). If \(\mathcal{M}_{f}\) is nonempty, each of its connected components is called a relative homotopy class. Under the assumptions \(|\mathrm{Ric}(g) + ng |_{g} =o(r)\) as \(r\rightarrow 0\) and \(h\) has a nonpositive sectional curvature, and if \( u_{0} \in C^{1}(\bar{M}, \bar{N}) \) satisfies \(u_{0}(\partial M) \subset \partial N\) and \(f=u_{0}| \partial M\) has nonwhere vanishing differential \(df: T\partial M \rightarrow T\partial M\), the authors prove the existence and uniqueness of a proper harmonic map (tension field \(\tau (u)=0\)) \(u \in C^{1}(\bar{M}, \bar{N}) \cap C^{\infty}(M, N) \) within the same relative homotopy class. This theorem is an extension of results proved by \textit{P. Li} and \textit{L.-f. Tam} [Indiana Univ. Math. J. 42, No. 2, 591--635 (1993; Zbl 0790.58011)] and regarded as the non-compact version of the result of \textit{J. Eells jun.} and \textit{J. H. Sampson} in [Am. J. Math. 86, 109--160 (1964; Zbl 0122.40102)].