Strong maximum principle for mean curvature operators on subriemannian manifolds

TL;DR

This paper establishes a strong maximum principle for horizontal mean curvature and p-Laplacian operators on subriemannian manifolds, including Heisenberg groups, with applications to minimal hypersurfaces and geometric rigidity results.

Contribution

It extends the strong maximum principle to subriemannian settings with singular points and provides new rigidity and halfspace theorems in Heisenberg groups.

Findings

Maximum principle holds at nonsingular and isolated singular points.

Rigidity of horizontal minimal hypersurfaces in Heisenberg cylinders.

Pseudo-halfspace theorem for Heisenberg groups.

Abstract

We study the strong maximum principle for horizontal (p-) mean curvature operator and p-(sub)laplacian operator on subriemannian manifolds including, in particular, Heisenberg groups and Heisenberg cylinders. Under a certain Hormander type condition on vector fields, we show the strong maximum principle holds in higher dimensions for two cases: (a) the touching point is nonsingular; (b) the touching point is an isolated singular point for one of comparison functions. For a background subriemannian manifold with local symmetry of isometric translations, we have the strong maximum principle for associated graphs which include, among others, intrinsic graphs with constant horizontal (p-) mean curvature. As applications, we show a rigidity result of horizontal (p-) minimal hypersurfaces in any higher dimensional Heisenberg cylinder and a pseudo-halfspace theorem for any Heisenberg group.