Inverse mean curvature flow in quaternionic hyperbolic space

Abstract

In this paper we complete the study started in [Pi2] of evolution by inverse mean curvature flow of star-shaped hypersurface in non-compact rank one symmetric spaces. We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the quaternionic 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 defined on a codimension 3 distribution. Finally we show that there exists a family of examples such that the qc-scalar curvature of this sub-Riemannian limit is not constant.