Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Curvature estimates for steady and expanding solitons in higher dimensions

Pak-Yeung Chan, Ming Hsiao

TL;DR

This work establishes sharp curvature control and asymptotic geometry for complete non-compact steady and expanding gradient Ricci solitons in higher dimensions. It proves that expanding solitons with finite asymptotic Ricci curvature ratio have curvature decays |Rm| = O(log r / r^2) for n≥5 and O(1/r^2) for n=4, leading to a unique conical structure at infinity and pGH convergence to a cone; for steady solitons, a weighted integral curvature framework yields polynomial curvature growth under linear Ricci decay and bounded curvature under super-linear decay, with cylindrical end behavior in the latter case. The results extend previous bounds by removing nonnegative Ricci curvature assumptions and provide a unified analysis via level-set flows, Moser iteration, and weighted curvature estimates. Collectively, they clarify the precise asymptotic geometry of solitons and the rigidity of their ends, with concrete implications for conical and cylindrical structures at infinity.

Abstract

In this paper, we demonstrate certain curvature estimates on complete non-compact steady and expanding gradient Ricci solitons in higher dimensions. In the expanding case, we prove that if the Ricci curvature decays at least quadratically, then the curvature operator decays at the rate $\BigO(1/r^{2})$ when $n=4$ and $\BigO((\log r)/r^{2})$ when $n\geq5$. This refines the curvature bounds in a previous result by Cao-Liu-Xie, and removes the nonnegative Ricci curvature assumption in the estimates by Cao-Liu and Cao-Liu-Xie. As a geometric application, we establish the existence and uniqueness of $C^{1,α}$ conical structure at infinity of Ricci expander with finite Ricci curvature ratio. In the steady case, using an integral estimate of the curvature, we prove that the curvature operator has at most polynomial growth when the potential function is proper and the Ricci curvature has linear decay. Moreover, we also confirm that the curvature is bounded if we further assume the Ricci curvature has super-linear decay $\BigO(r^{-1-\varepsilon})$. As an application, we prove the existence and uniqueness of cylindrical structure at infinity of steady soliton with super-linear Ricci curvature decay and proper potential function.

Curvature estimates for steady and expanding solitons in higher dimensions

TL;DR

This work establishes sharp curvature control and asymptotic geometry for complete non-compact steady and expanding gradient Ricci solitons in higher dimensions. It proves that expanding solitons with finite asymptotic Ricci curvature ratio have curvature decays |Rm| = O(log r / r^2) for n≥5 and O(1/r^2) for n=4, leading to a unique conical structure at infinity and pGH convergence to a cone; for steady solitons, a weighted integral curvature framework yields polynomial curvature growth under linear Ricci decay and bounded curvature under super-linear decay, with cylindrical end behavior in the latter case. The results extend previous bounds by removing nonnegative Ricci curvature assumptions and provide a unified analysis via level-set flows, Moser iteration, and weighted curvature estimates. Collectively, they clarify the precise asymptotic geometry of solitons and the rigidity of their ends, with concrete implications for conical and cylindrical structures at infinity.

Abstract

In this paper, we demonstrate certain curvature estimates on complete non-compact steady and expanding gradient Ricci solitons in higher dimensions. In the expanding case, we prove that if the Ricci curvature decays at least quadratically, then the curvature operator decays at the rate when and when . This refines the curvature bounds in a previous result by Cao-Liu-Xie, and removes the nonnegative Ricci curvature assumption in the estimates by Cao-Liu and Cao-Liu-Xie. As a geometric application, we establish the existence and uniqueness of conical structure at infinity of Ricci expander with finite Ricci curvature ratio. In the steady case, using an integral estimate of the curvature, we prove that the curvature operator has at most polynomial growth when the potential function is proper and the Ricci curvature has linear decay. Moreover, we also confirm that the curvature is bounded if we further assume the Ricci curvature has super-linear decay . As an application, we prove the existence and uniqueness of cylindrical structure at infinity of steady soliton with super-linear Ricci curvature decay and proper potential function.

Paper Structure

This paper contains 7 sections, 18 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Steady solitons
  4. Expanding soliton
  5. The integral estimate on steady soliton
  6. Main Theorems - expander
  7. Main Theorems - steady soliton

Key Result

Theorem 1

Let $(\mathcal{M}^{n},g,f)$ be an $n$-dimensional complete gradient expanding Ricci soliton with finite asymptotic Ricci curvature ratio where $n\ge 4$ and $r:=r(x)$ is the distance function to some fixed point $x_{0}\in\mathcal{M}^{n}$. Then for any $n\ge 5$, there is a constant $C>0$ such that when $r(x)\gg 1$ is sufficiently large. Moreover, when $n=4$, the asymptotic curvature ratio is finit

Theorems & Definitions (42)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Remark 4
  • ...and 32 more