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Strong maximum principle for mean curvature operators on subriemannian manifolds

Research paper by Jih-Hsin Cheng, Hung-Lin Chiu, Jenn-Fang Hwang, Paul Yang

Indexed on: 07 Nov '16Published on: 07 Nov '16Published in: arXiv - Mathematics - Differential Geometry



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.