On the asymptotic Plateau problem in Cartan--Hadamard manifolds

TL;DR

This paper solves the asymptotic Plateau problem for constant extrinsic curvature surfaces in Cartan--Hadamard manifolds, establishing dynamical stability for a class of hypersurface laminations.

Contribution

It provides a complete solution to the asymptotic Plateau problem in Cartan--Hadamard manifolds and extends it to higher dimensions, advancing understanding of geometric stability.

Findings

Proves dynamical stability of hypersurface laminations.

Provides a complete solution to the asymptotic Plateau problem.

Extends results to higher-dimensional manifolds.

Abstract

We prove dynamical stability of a natural class of hypersurface laminations defined over Cartan--Hadamard manifolds of pinched curvature. We achieve this by providing a complete solution to the asymptotic Plateau problem for immersed surfaces of constant extrinsic curvature in Cartan--Hadamard manifolds, proposed by Labourie in [25], together with its natural higher-dimensional generalisation.