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

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.