Nonhomogeneous expanding flows in hyperbolic spaces

TL;DR

This paper extends the study of nonhomogeneous expanding curvature flows from Euclidean space to hyperbolic spaces of various types, analyzing their long-term behavior and geometric limits influenced by ambient space curvature.

Contribution

It investigates how the geometry of real, complex, and quaternionic hyperbolic spaces affects the evolution of star-shaped hypersurfaces under nonhomogeneous expanding flows, including long-term existence and asymptotic behavior.

Findings

Flow preserves initial conditions in hyperbolic spaces.

Rescaled metrics converge to conformal multiples of standard metrics.

Existence of examples with non-constant scalar curvature limits.

Abstract

A recent paper [CGT] studies the evolution of star-shaped mean convex hypersurfaces of the Euclidean space by a class of nonhomogeneous expanding curvature flows. In the present paper we consider the same problem in the real, complex and quaternionic hyperbolic spaces, investigating how the richer geometry of the ambient space affects the evolution. In every case the initial conditions are preserved and the long time existence of the flow is proven. The geometry of the ambient space influences the asymptotic behaviour of the flow: after a suitable rescaling the induced metric converges to a conformal multiple of the standard Riemannian round metric of the sphere if the ambient manifold is the real hyperbolic space, otherwise it converges to a conformal multiple of the standard sub-Riemannian metric on the odd-dimensional sphere. Finally, in every cases, we are able to construct infinitely many examples such that the limit does not have constant scalar curvature.