Existence of H-bubbles in a perturbative setting

Abstract

Given a $C^1$ function $H:{\mathbb{R}}^3\to \mathbb{R}$, we look for $H$-bubbles, i.e., surfaces in ${\mathbb{R}}^3$ parametrized by the sphere ${\mathbb{S}}^2$ with mean curvature $H$ at every regular point. Here we study the case $H(u)=H_0(u)+ϵH_1(u)$ where $H_0$ is some “good” curvature (for which there exist $H_0$-bubbles with minimal energy, uniformly bounded in $L^{\infty }$), $ϵ$ is the smallness parameter, and $H_1$ is any $C^1$ function.