Quasiconformal harmonic maps into negatively curved manifolds (Q5954273)

Let \(F: (M,g) \to (N,h)\) be a differentiable map. For \(x\in M\), denote the eigenvalues of the operator \((dF)^*dF: T_xM \to T_xM\) by \(\lambda_1(x) \geq \lambda_2(x) \geq \cdots \lambda_n(x) \geq 0\), \(n=\dim M\). If \(F\) is a diffeomorphism and \(\lambda_1(x) \leq c\lambda_n(x)\) for some constant \(c\), then \(F\) is called a quasiconformal diffeomorphism; if \(\lambda_1(x) \leq c\lambda_2(x)\), then \(F\) is said to have bounded dilatation. NEWLINENEWLINENEWLINEThe main result is (Theorem 1.1): Suppose that \((N,h)\) is complete and simply connected with sectional curvature bounded between two negative constants, \((M,g)\) be any complete Riemannian manifold and \(F: (M,g) \to (N,h)\) a quasiconformal diffeomorphism. Then \(M\) admits a natural compactification \(\bar{M}\) so that the Dirichlet problem (of harmonic functions) at infinity is solvable. NEWLINENEWLINENEWLINEThis generalizes a well known theorem in [\textit{M. T. Anderson} and \textit{R. Schoen}, Ann. Math. 121, 429-461 (1985; Zbl 0587.53045)] which corresponds to \(M=N\) and \(F=id\). NEWLINENEWLINENEWLINEThe author also gives a vanishing result on harmonic maps with bounded dilatation (Theorem 1.2).