Asymptotic Dirichlet problems for Laplace's and minimal equations on Hadamard manifolds

TL;DR

This paper proves the existence of solutions to Laplace's and minimal hypersurface PDEs on Hadamard manifolds under specific curvature conditions, extending previous results and allowing curvature to degenerate at infinity.

Contribution

It extends and improves existing theorems on asymptotic Dirichlet problems for these PDEs on Hadamard manifolds, especially regarding curvature decay and degeneracy.

Findings

Existence of solutions under exponential decay of Ricci curvature.

Extension of previous theorems to cases with curvature degenerating to zero at infinity.

Improved conditions for solvability of PDEs on Hadamard manifolds.

Abstract

It is proved the existence of entire solutions of the Laplace's and minimal hypersurface's PDEs on a Hadamard manifold $M$ under certain curvature conditions by investigating the asymptotic Dirichlet's problems for these PDEs. In the harmonic case it is obtained an existence result which assumes the same growth condition on the sectional curvature as of Theorem 1.2 of E. Hsu \cite{Hsu} but that contemplates cases having Ricci curvature with exponential decay. It is also obtained a result which extends and improves Theorem 3.6 of Choi \cite{Choi}. In the minimal case one obtains an extension and an improvement of Theorem 1 of N. do Esp\'{\i}rito-Santo, S. Fornari and J. Ripoll \cite{EFR}, and partial extensions of Theorem 5.2 of J. A. G\'alvez and H. Rosenberg \cite{GR} by allowing the sectional curvature of $M$ degenerate to 0 at infinity.