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Calabi-Yau type theorem for complete manifolds with nonnegative scalar curvature

Jintian Zhu

Abstract

In this paper, we are able to prove an analogy of the Calabi-Yau theorem for complete Riemannian manifolds with nonnegative scalar curvature which are aspherical at infinity. The key tool is an existence result for arbitrarily large bounded regions with weakly mean-concave boundary in Riemannian manifolds with sublinear volume growth. As an application, we use the same tool to show that a complete contractible Riemannian $3$-manifold with positive scalar curvature and sublinear volume growth is necessarily homeomorphic to $\mathbb R^3$.

Calabi-Yau type theorem for complete manifolds with nonnegative scalar curvature

Abstract

In this paper, we are able to prove an analogy of the Calabi-Yau theorem for complete Riemannian manifolds with nonnegative scalar curvature which are aspherical at infinity. The key tool is an existence result for arbitrarily large bounded regions with weakly mean-concave boundary in Riemannian manifolds with sublinear volume growth. As an application, we use the same tool to show that a complete contractible Riemannian -manifold with positive scalar curvature and sublinear volume growth is necessarily homeomorphic to .
Paper Structure (4 sections, 8 theorems, 40 equations, 1 figure)

This paper contains 4 sections, 8 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The proof
  3. Finite-volume complete manifolds with nonnegative scalar curvature
  4. $\mathbb Q$-homology vanishing theorem

Key Result

Theorem 1.1

Let $(M^n,g)$, $2\leq n\leq 4$, be a complete Riemannian manifold with nonnegative scalar curvature outside a compact subset, which is aspherical at infinity. Then it has sublinear volume growth if and only if it is closed.

Figures (1)

  • Figure 1: The separation band $V$

Theorems & Definitions (16)

  • Theorem 1.1
  • Lemma 1.2
  • Proposition 1.3
  • Theorem 1.4: Gromov-Lawson
  • Theorem 1.7
  • proof : Proof of Lemma \ref{['Lem: concave']}
  • Theorem 2.1: Calabi-Yau
  • proof : An alternative proof without using volume comparison
  • proof : Proof of Proposition \ref{['Prop: splitting']}
  • proof : Proof of Theorem \ref{['Thm: main']}
  • ...and 6 more