Existence of constant mean curvature surfaces with controlled topology in 3-manifolds

TL;DR

This paper proves the existence of constant mean curvature surfaces with controlled topology in any closed 3-manifold using a min-max approach and varifold convergence, for almost every prescribed mean curvature value.

Contribution

It establishes the existence of non-trivial, branched CMC surfaces in closed 3-manifolds with genus bounded by the Heegaard genus, extending previous results to a broader setting.

Findings

Existence of branched CMC surfaces for almost every H

Bound on surface genus by Heegaard genus

Application of min-max and varifold techniques

Abstract

We establish the existence of a non-trivial, branched immersion of a closed Riemann surface $\Sigma$ with constant mean curvature (CMC) $H$ into any closed, orientable 3-manifold $\mathcal{M}$, for almost every prescribed value of $H$. The genus of the surface $\Sigma$ is bounded from above by the Heegaard genus $h$ of $\mathcal{M}$. Starting from a family of sweep-outs of $\mathcal{M}$ by surfaces of genus $h$, we apply a min-max construction for a family $\{E_{H,\sigma}\}_\sigma$ of perturbations of the energy involving the second fundamental form of the immersions to produce almost-critical points $u_k$ of $E_{H,\sigma}$. We then show, following ideas developed by Pigati and Rivi\`ere, that the maps $u_k$ converge to a "CMC-parametrized varifold". This limiting object is then shown to be a smooth, branched immersion with the prescribed mean curvature $H$.