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On conical asymptotically flat manifolds

Mingyang Li, Song Sun

Abstract

We prove a conjecture of Petrunin and Tuschmann on the non-existence of asymptotically flat 4-manifolds asymptotic to the half plane. We also survey recent progress and questions concerning gravitational instantons, which serve as our motivation for studying this question.

On conical asymptotically flat manifolds

Abstract

We prove a conjecture of Petrunin and Tuschmann on the non-existence of asymptotically flat 4-manifolds asymptotic to the half plane. We also survey recent progress and questions concerning gravitational instantons, which serve as our motivation for studying this question.

Paper Structure

This paper contains 7 sections, 14 theorems, 31 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proof of the main result
  3. Further discussion
  4. A model flat metric
  5. Simply-connected domains collapsing to
  6. No gap for asymptotic curvature
  7. Gravitational instantons

Key Result

Theorem 1.2

Conjecture conj:PT holds.

Figures (4)

  • Figure 1: Cover $\mathbb{H}^\circ$ by $\mathbb{U}_1$ and $\mathbb{U}_2$
  • Figure 2: Singular fibration over $\mathbb{U}_i$
  • Figure 3: Intermediate covers $\widehat{V}_{1,j},\widehat{V}_{2,j}$ and the universal cover $\widecheck{V}_j$
  • Figure 4: Images of $\mathbb{G}_{\sigma,\tau}^\alpha$

Theorems & Definitions (30)

  • Conjecture 1.1: Petrunin-Tuschmann PT
  • Theorem 1.2
  • Lemma 2.1
  • Remark 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 20 more