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Complete gradient expanding Ricci solitons with finite asymptotic scalar curvature ratio

Huai-Dong Cao, Tianbo Liu, Junming Xie

Abstract

Let $(M^n, g, f)$, $n\geq 5$, be a complete gradient expanding Ricci soliton with nonnegative Ricci curvature $Rc\geq 0$. In this paper, we show that if the asymptotic scalar curvature ratio of $(M^n, g, f)$ is finite (i.e., $ \limsup_{r\to \infty} R r^2< \infty $), then the Riemann curvature tensor must have at least sub-quadratic decay, namely, $\limsup_{r\to \infty} |Rm| \ \! r^α< \infty$ for any $0<α<2$.

Complete gradient expanding Ricci solitons with finite asymptotic scalar curvature ratio

Abstract

Let , , be a complete gradient expanding Ricci soliton with nonnegative Ricci curvature . In this paper, we show that if the asymptotic scalar curvature ratio of is finite (i.e., ), then the Riemann curvature tensor must have at least sub-quadratic decay, namely, for any .
Paper Structure (4 sections, 16 theorems, 97 equations)

This paper contains 4 sections, 16 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Integral Estimate
  4. The proof of Main theorem

Key Result

Theorem 1.1

Let $(M^n, g, f)$, $n\ge 5$, be an $n$-dimensional complete gradient expanding Ricci soliton with nonnegative Ricci curvature $Rc\geq 0$ and finite asymptotic scalar curvature ratio where $r=r(x)$ is the distance function to a fixed base point $x_0\in M$. Then $(M^n, g, f)$ has finite $\alpha$-asymptotic curvature ratio for any $0<\alpha<2$, Furthermore, there exist constant $C>0$ depending on $

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 18 more