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3-Manifolds with positive scalar curvature and bounded geometry

Otis Chodosh, Yi Lai, Kai Xu

TL;DR

This work addresses the classification of complete 3-manifolds with nonnegative scalar curvature under bounded geometry. It develops and exploits a maximal weak inverse mean curvature flow to extract topological information, producing exhaustions by sphere- or torus-boundary surfaces and limiting the possible topology. The main results show that a complete contractible 3-manifold with $R\\ge 0$ and bounded geometry must be diffeomorphic to $\\mathbb{R}^3$, and that open handlebodies of genus $\\gamma>1$ cannot admit such metrics, with genus $\\gamma\\le 1$ possible in the borderline case. These findings advance the program of classifying noncompact 3-manifolds under nonnegative curvature hypotheses using IMCF techniques, even when the scalar curvature is merely nonnegative rather than uniformly positive.

Abstract

We show that a complete contractible 3-manifold with positive scalar curvature and bounded geometry must be $\mathbb R^3$. We also show that an open handlebody of genus larger than 1 does not admit complete metrics with positive scalar curvature and bounded geometry. Our results rely on the maximal weak solution to inverse mean curvature flow due to the third-named author.

3-Manifolds with positive scalar curvature and bounded geometry

TL;DR

This work addresses the classification of complete 3-manifolds with nonnegative scalar curvature under bounded geometry. It develops and exploits a maximal weak inverse mean curvature flow to extract topological information, producing exhaustions by sphere- or torus-boundary surfaces and limiting the possible topology. The main results show that a complete contractible 3-manifold with and bounded geometry must be diffeomorphic to , and that open handlebodies of genus cannot admit such metrics, with genus possible in the borderline case. These findings advance the program of classifying noncompact 3-manifolds under nonnegative curvature hypotheses using IMCF techniques, even when the scalar curvature is merely nonnegative rather than uniformly positive.

Abstract

We show that a complete contractible 3-manifold with positive scalar curvature and bounded geometry must be . We also show that an open handlebody of genus larger than 1 does not admit complete metrics with positive scalar curvature and bounded geometry. Our results rely on the maximal weak solution to inverse mean curvature flow due to the third-named author.

Paper Structure

This paper contains 8 sections, 22 theorems, 35 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Topological obstructions via inverse mean curvature flow
  3. Manifolds that are nested by solid tori
  4. Acknowledgements
  5. Topological preliminaries
  6. IMCF Preliminaries
  7. Instantly escaping IMCF
  8. Proof of the main theorems

Key Result

Theorem 1.1

Let $(M,g)$ be a complete, connected, contractible Riemannian 3-manifold satisfying $R\geq0$ and eq:bounded_geometry. Then $M$ is diffeomorphic to $\mathbb{R}^3$.

Figures (5)

  • Figure 1: An IMCF sweeping out the manifold at $t=T$ (left) and one that escapes at $t=T$ (right).
  • Figure 2: Iterated trefoils. The figure shows the embedding $\Omega_1\subset\Omega_2\subset\Omega_3$.
  • Figure 3: A complicated case of sweeping IMCF (grey regions represent jumps)
  • Figure 4: Examples of maximal IMCF that do not fall in Definition \ref{['def:sol_types']}.
  • Figure 5: Left: the instantly escaping maixmal IMCF. Right: the metric perturbation and the new level set (see Lemma \ref{['lemma:perturb']} below).

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 45 more