Bubbles with constant mean curvature, and almost constant mean curvature, in the hyperbolic space

TL;DR

This paper investigates spheres with constant or nearly constant mean curvature in hyperbolic space, establishing nondegeneracy results and conditions for the existence of embedded spheres with prescribed mean curvature variations.

Contribution

It proves a nondegeneracy result for a family of constant mean curvature spheres and provides conditions for constructing spheres with nearly prescribed mean curvature.

Findings

Nondegeneracy of the manifold of constant mean curvature spheres

Existence of a curve of embedded spheres with prescribed mean curvature perturbations

Conditions ensuring the existence of spheres with almost constant mean curvature

Abstract

Given a constant $k>1$, let $Z$ be the family of round spheres of radius $\textrm{artanh}(k^{-1})$ in the hyperbolic space $\mathbb{H}^3$, so that any sphere in $Z$ has mean curvature $k$. We prove a crucial nondegeneracy result involving the manifold $Z$. As an application, we provide sufficient conditions on a prescribed function $\phi$ on $\mathbb{H}^3$, which ensure the existence of a ${\cal C}^1$-curve, parametrized by $\varepsilon\approx 0$, of embedded spheres in $\mathbb{H}^3$ having mean curvature $k +\varepsilon\phi$ at each point.