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Positive mass theorem for asymptotically flat manifolds with isolated conical singularities

Xianzhe Dai, Yukai Sun, Changliang Wang

Abstract

We prove the positive mass theorem for asymptotical flat (AF for short) manifolds with finitely many isolated conical singularities. We do not impose the spin condition. Instead we use the conformal blow up technique which dates back to Schoen's final resolution of the Yamabe conjecture.

Positive mass theorem for asymptotically flat manifolds with isolated conical singularities

Abstract

We prove the positive mass theorem for asymptotical flat (AF for short) manifolds with finitely many isolated conical singularities. We do not impose the spin condition. Instead we use the conformal blow up technique which dates back to Schoen's final resolution of the Yamabe conjecture.
Paper Structure (9 sections, 19 theorems, 83 equations)

This paper contains 9 sections, 19 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. AF manifolds with isolated conical singularities
  3. Analysis on AF manifolds with isolated conical singularities
  4. Weighted Sobolev spaces
  5. Elliptic estimate for the Laplace operator
  6. Fredholm property of Laplace operator
  7. Solving the Laplace equation
  8. Nonnegativity of mass
  9. Rigidity

Key Result

Theorem 1.1

Let $(M^n, g)$ be an AF manifold with finitely many isolated conical singularity and $n \geq 3$. If the scalar curvature is nonnegative on the smooth part, then the ADM mass $m(g)$ is nonnegative. Furthermore, the mass $m(g) = 0$ if and only if $(M^n, g)$ is isometric to $(\mathbb{R}^n, g_{\mathbb{R

Theorems & Definitions (34)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • ...and 24 more