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

Filippo Gaia, Xuanyu Li

Abstract

We establish the existence of a non-trivial, branched immersion of a closed Riemann surface $Σ$ 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 $Σ$ 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 $\{E_{H,σ}\}_σ$ of perturbations of the energy involving the second fundamental form of the immersions to produce almost-critical points $u_k$ of $E_{H,σ}$. We then show, following ideas developed by Pigati and Rivière, 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 $H$.

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

Abstract

We establish the existence of a non-trivial, branched immersion of a closed Riemann surface with constant mean curvature (CMC) into any closed, orientable 3-manifold , for almost every prescribed value of . The genus of the surface is bounded from above by the Heegaard genus of . Starting from a family of sweep-outs of by surfaces of genus , we apply a min-max construction for a family of perturbations of the energy involving the second fundamental form of the immersions to produce almost-critical points of . We then show, following ideas developed by Pigati and Rivière, that the maps converge to a "CMC-parametrized varifold". This limiting object is then shown to be a smooth, branched immersion with the prescribed mean curvature .
Paper Structure (15 sections, 37 theorems, 292 equations)

This paper contains 15 sections, 37 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Related backgroud on minimal and CMC surfaces
  4. Overview of the arguments
  5. Organization of the paper
  6. Preliminaries
  7. The existence of CMC parametrized varifolds
  8. Asymptotic behavior
  9. Degeneration of the conformal structure and bubbling
  10. Regularity and multiplicity one
  11. Application: existence of CMC immersions in general manifolds
  12. Min-max setup
  13. The deformation Lemma
  14. Technical results
  15. Results from the works of Pigati and Rivière

Key Result

Theorem 1.1

Let $(\mathcal{M}^3,g)$ be a smooth, closed, oriented Riemannian manifold with Heegaard genus $h$. For almost every $H>0$ and for $H=0$, there exists a closed Riemann surface $\Sigma$ with genus $g(\Sigma)\leq h$ and a non-trivial branched $H$-constant-mean-curvature immersion $u:\Sigma\rightarrow\m

Theorems & Definitions (82)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 2.1
  • Definition 2.2
  • Lemma 2.3: Cheng-Zhou
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • ...and 72 more