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Index estimates for constant mean curvature surfaces in three-manifolds by energy comparison

Luca Seemungal, Ben Sharp

TL;DR

This work proves a linear-in-area Morse index bound for closed constant mean curvature surfaces in orientable 3-manifolds by establishing a precise comparison between second variations of area and Dirichlet energy at conformal maps. By extending the Ejiri–Micallef framework to branched CMC immersions and exploiting a conformal-defect quantity μ, the authors derive i_{\mathcal{A}_h}+n_{\mathcal{A}_h} ≤ i_{\mathcal{E}_h}+n_{\mathcal{E}_h}+r, with a topological term r depending on genus and branch points, and an ambient-geometry–dependent constant C multiplying the Willmore-type energy terms. In particular, they obtain explicit linear bounds in terms of area and curvature data, and show improved bounds in the totally umbilic case; under negative ambient curvature with h^2≤4|κ_0|, they obtain either trivial index or strong rigidity. The results bridge area-variational and energy-variational viewpoints for CMC surfaces, providing quantitative links between index, genus, branching, Willmore-type energy, and ambient geometry, with practical implications for understanding the landscape of CMC surfaces in 3-manifolds.

Abstract

We prove a linear upper bound on the Morse index of closed constant mean curvature (CMC) surfaces in orientable three-manifolds in terms of genus, number of branch points and a Willmore-type energy.

Index estimates for constant mean curvature surfaces in three-manifolds by energy comparison

TL;DR

This work proves a linear-in-area Morse index bound for closed constant mean curvature surfaces in orientable 3-manifolds by establishing a precise comparison between second variations of area and Dirichlet energy at conformal maps. By extending the Ejiri–Micallef framework to branched CMC immersions and exploiting a conformal-defect quantity μ, the authors derive i_{\mathcal{A}_h}+n_{\mathcal{A}_h} ≤ i_{\mathcal{E}_h}+n_{\mathcal{E}_h}+r, with a topological term r depending on genus and branch points, and an ambient-geometry–dependent constant C multiplying the Willmore-type energy terms. In particular, they obtain explicit linear bounds in terms of area and curvature data, and show improved bounds in the totally umbilic case; under negative ambient curvature with h^2≤4|κ_0|, they obtain either trivial index or strong rigidity. The results bridge area-variational and energy-variational viewpoints for CMC surfaces, providing quantitative links between index, genus, branching, Willmore-type energy, and ambient geometry, with practical implications for understanding the landscape of CMC surfaces in 3-manifolds.

Abstract

We prove a linear upper bound on the Morse index of closed constant mean curvature (CMC) surfaces in orientable three-manifolds in terms of genus, number of branch points and a Willmore-type energy.

Paper Structure

This paper contains 12 sections, 11 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. The case $m=2$
  3. Terminology
  4. Acknowledgements
  5. Comparison of second variations of area and energy at a conformal map
  6. Infinitesimal conformal variations:
  7. Term (i)
  8. Term (ii)
  9. The CMC index bounds: proofs of the main Theorems
  10. Proofs of Theorems \ref{['thm:energy-bound']} and \ref{['thm:index-bound-curvature-assumptions']}
  11. Area, conformally invariant variational problems, enclosed volume and their variations
  12. Conformally invariant variational problems and enclosed volume

Key Result

Theorem 1.1

Let $u:\Sigma_g\to N$ be a (possibly branched) minimal immersion of a closed Riemann surface of genus $g$ into a Riemannian manifold $N^n$. Then and $b$ is the total number of branch points of $u$ counted with multiplicity, and $[x]=\max_{\mathbb{Z}}\{k: k\leq x\}$.

Theorems & Definitions (21)

  • Theorem 1.1: Ejiri-Micallef: Theorem 1.1 in EM08
  • Theorem 1.2
  • Remark 1.3
  • Conjecture 1.4
  • Theorem 1.5: cf. Corollary 4.1 in EM08
  • Theorem 2.1: cf. Theorem 2.1 in EM08
  • Corollary 2.2
  • proof : Proof of Theorem \ref{['thm:second-var-area-energy']}
  • Theorem 3.1
  • proof
  • ...and 11 more