Survey on the asymptotic Dirichlet problem for the minimal surface equation

TL;DR

This survey reviews the development of the asymptotic Dirichlet problem for the minimal surface equation on Cartan-Hadamard manifolds, highlighting key results, methods, and unsolvability issues.

Contribution

It provides a comprehensive overview of the progress and techniques in solving the asymptotic Dirichlet problem for minimal surfaces on these manifolds.

Findings

Development of existence results for the problem

Discussion of methods used in proofs

Identification of cases where the problem is unsolvable

Abstract

We give a survey on the development of the study of the asymptotic Dirichlet problem for the minimal surface equation on Cartan-Hadamard manifolds. Part of this survey is based on the introductory part of the doctoral dissertation of the author. The paper is organised as follows. First we introduce Cartan-Hadamard manifolds and the concept of asymptotic Dirichlet problem, then discuss about the development of the results and describe the methods used in the proofs. In the end we mention some results about the nonsolvability of the asymptotic Dirichlet problem.