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Pairs of Embedded Spheres with Pinched Prescribed Mean Curvature

Liam Mazurowski, Xin Zhou

TL;DR

This work proves that on the unit 3-sphere $S^3$, any admissible prescribed mean curvature function $h$ with the pinching bound $|h|<h_0$ (where $h_0$ is the real root of $\pi h_0^3 + 2h_0^2 + 4\pi h_0 - 8 = 0$, $h_0\approx 0.547$) admits at least two smoothly embedded $S^2$ with mean curvature $h$. The authors adapt the Simon–Smith min-max theory for the prescribed mean curvature functional $\mathcal{A}^h$ and exploit the Smale conjecture to identify the space of embedded spheres with $S^3$, then run a one-parameter and a four-parameter min-max to produce two distinct candidates and obtain a contradiction unless a second solution exists. An interpolation theorem combining Smale-type results with a filigree retraction controls deformations in the embedding space, allowing a Lusternik–Schnirelmann–type argument to complete the bifurcation into at least two solutions. The result advances topological existence theory for prescribed mean curvature surfaces and aligns with twin-bubble conjectures in three dimensions.

Abstract

Assume $h$ is a positive function on the unit three-sphere which satisfies the pinching condition $h < h_0 \approx 0.547$. We prove the existence of at least two embedded two-spheres with prescribed mean curvature $h$. The same result holds for sign-changing functions $h$ satisfying $\vert h\vert < h_0$ under a mild assumption on the zero set.

Pairs of Embedded Spheres with Pinched Prescribed Mean Curvature

TL;DR

This work proves that on the unit 3-sphere , any admissible prescribed mean curvature function with the pinching bound (where is the real root of , ) admits at least two smoothly embedded with mean curvature . The authors adapt the Simon–Smith min-max theory for the prescribed mean curvature functional and exploit the Smale conjecture to identify the space of embedded spheres with , then run a one-parameter and a four-parameter min-max to produce two distinct candidates and obtain a contradiction unless a second solution exists. An interpolation theorem combining Smale-type results with a filigree retraction controls deformations in the embedding space, allowing a Lusternik–Schnirelmann–type argument to complete the bifurcation into at least two solutions. The result advances topological existence theory for prescribed mean curvature surfaces and aligns with twin-bubble conjectures in three dimensions.

Abstract

Assume is a positive function on the unit three-sphere which satisfies the pinching condition . We prove the existence of at least two embedded two-spheres with prescribed mean curvature . The same result holds for sign-changing functions satisfying under a mild assumption on the zero set.

Paper Structure

This paper contains 14 sections, 15 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Sketch of Proof
  4. Organization
  5. Acknowledgment
  6. Preliminaries
  7. The Space of Embedded Two-Spheres
  8. Geometric measure theory
  9. Almost-Embeddings
  10. Min-Max Theory
  11. An Interpolation Theorem
  12. Proof of Main Result
  13. One-Parameter Min-Max
  14. Four-Parameter Min-Max

Key Result

Theorem 1.1

Let $S^3$ denote the unit three-sphere. Assume that $h\colon S^3\to \mathbb{R}$ is admissible for the PMC min-max theory and that $\vert h\vert < h_0$. Then there exist at least two distinct two-spheres smoothly embedded in $S^3$ with mean curvature given by $h$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Conjecture 1.7
  • Conjecture 1.8: Twin Bubble Conjecture
  • Conjecture 1.9
  • Theorem 2.1
  • ...and 24 more