Inverse mean curvature flow in complex hyperbolic space

TL;DR

This paper studies the inverse mean curvature flow of star-shaped hypersurfaces in complex hyperbolic space, showing long-term existence, preservation of shape properties, and convergence to a sub-Riemannian sphere with non-constant Webster curvature in some cases.

Contribution

It establishes the long-term behavior and convergence properties of inverse mean curvature flow in complex hyperbolic space, including examples with non-constant Webster curvature.

Findings

Flow exists for all positive time.

Hypersurfaces remain star-shaped and mean convex.

Convergence to a sub-Riemannian sphere with non-constant Webster curvature.

Abstract

We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the complex hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to a conformal multiple of the standard sub- Riemannian metric on the sphere. Finally we show that there exists a family of examples such that the Webster curvature of this sub-Riemannian limit is not constant.