Convexity at infinity in Cartan-Hadamard manifolds and applications to the asymptotic Dirichlet and Plateau problems

TL;DR

This paper investigates the convexity at infinity in Cartan-Hadamard manifolds, establishing conditions under which the asymptotic Dirichlet and Plateau problems are solvable, thereby extending previous results to broader curvature bounds and manifold classes.

Contribution

It verifies the Strict Convexity condition on manifolds with various curvature bounds and solves the asymptotic Dirichlet and Plateau problems under these conditions, generalizing prior work.

Findings

Verified SC condition on manifolds with curvature bounds going to -infinity and 0.

Solved asymptotic Plateau problem for currents with Z_2 and Z multiplicities.

Established solvability of the asymptotic Dirichlet problem for PDEs on manifolds with super-exponential curvature decay.

Abstract

We study the asymptotic Dirichlet and Plateau problems on Cartan-Hadamard manifolds satisfying the so-called Strict Convexity (abbr. SC) condition. The main part of the paper consists in studying the SC condition on a manifold whose sectional curvatures are bounded from above and below by certain functions depending on the distance to a fixed point. In particular, we are able to verify the SC condition on manifolds whose curvature lower bound can go to -infinity and upper bound to 0 simultaneously at certain rates, or on some manifolds whose sectional curvatures go to -infinity faster than any prescribed rate. These improve previous results of Anderson, Borb\'ely, and Ripoll and Telichevsky. We then solve the asymptotic Plateau problem for locally rectifiable currents with Z_2-multiplicity in a Cartan-Hadamard manifold satisfying the SC condition given any compact topologically embedded (k-1)-dimensional submanifold of \partial_{\infty}M, 2\leq k\leq n-1, as the boundary data. We also solve the asymptotic Plateau problem for locally rectifiable currents with Z-multiplicity on any rotationally symmetric manifold satisfying the SC condition given a smoothly embedded submanifold as the boundary data. These generalize previous results of Anderson, Bangert, and Lang. Moreover, we obtain new results on the asymptotic Dirichlet problem for a large class of PDEs. In particular, we are able to prove the solvability of this problem on manifolds with super-exponential decay (to -infinity) of the curvature.