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Desingularizing positive scalar curvature 4-manifolds

Demetre Kazaras

Abstract

We show that the bordism group of closed 3-manifolds with positive scalar curvature (psc) metrics is trivial by explicit methods. Our constructions are derived from scalar-flat K{ä}hler ALE surfaces discovered by Lock-Viaclovsky. Next, we study psc 4-manifolds with metric singularities along points and embedded circles. Our psc null-bordisms are essential tools in a desingularization process developed by Li-Mantoulidis. This allows us to prove a non-existence result for singular psc metrics on enlargeable 4-manifolds with uniformly Euclidean geometry. As a consequence, we obtain a positive mass theorem for asymptotically flat 4-manifolds with non-negative scalar curvature and low regularity.

Desingularizing positive scalar curvature 4-manifolds

Abstract

We show that the bordism group of closed 3-manifolds with positive scalar curvature (psc) metrics is trivial by explicit methods. Our constructions are derived from scalar-flat K{ä}hler ALE surfaces discovered by Lock-Viaclovsky. Next, we study psc 4-manifolds with metric singularities along points and embedded circles. Our psc null-bordisms are essential tools in a desingularization process developed by Li-Mantoulidis. This allows us to prove a non-existence result for singular psc metrics on enlargeable 4-manifolds with uniformly Euclidean geometry. As a consequence, we obtain a positive mass theorem for asymptotically flat 4-manifolds with non-negative scalar curvature and low regularity.

Paper Structure

This paper contains 11 sections, 10 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction and Main Results
  2. Structure of the paper and further discussion
  3. Acknowledgments
  4. Background
  5. Enlargeable manifolds
  6. The conformal Laplacian with minimal boundary conditions
  7. Psc $3$-manifolds
  8. ALE manifolds and mass
  9. Proof of Theorem B
  10. Proof of Theorem A
  11. Proof of Corollaries A and B

Key Result

Proposition 1

GL80 Suppose $M^n$ is a closed enlargeable $n$-manifold. If $N^n$ is a closed $n$-manifold and there exists a map $N\to M$ of non-zero degree, then $N$ is also enlargeable.

Figures (2)

  • Figure 1: The desingularization $F:\overline{M}\to M$ in Theorem A.
  • Figure 2: The null-cobordism we construct of a psc $(M^3,g)$

Theorems & Definitions (20)

  • Conjecture 1
  • Definition
  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Proposition 2
  • Theorem 3
  • Definition
  • Theorem 4
  • Corollary 1
  • ...and 10 more