# Inverse mean curvature flow in quaternionic hyperbolic space

## TL;DR

This paper studies the inverse mean curvature flow of star-shaped hypersurfaces in quaternionic hyperbolic space, showing long-term existence, preservation of shape properties, and convergence to a sub-Riemannian metric with non-constant qc-scalar curvature.

## Contribution

It extends inverse mean curvature flow analysis to quaternionic hyperbolic space, demonstrating convergence and properties of the limit metric, including non-constant qc-scalar curvature.

## Key findings

- Flow exists for all positive time
- Hypersurfaces remain star-shaped and mean convex
- Limit metric converges to a conformal sub-Riemannian structure

## 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.