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Notes on scalar curvature lower bounds of steady gradient Ricci solitons

Shota Hamanaka

TL;DR

The paper investigates lower bounds and decay for the scalar curvature on complete non-compact steady gradient Ricci solitons and proves a diameter bound under $\infty$-Bakry–Émery type Ricci lower bounds. It introduces a new decay-type estimate for $R_g$ at infinity by employing warped $\mu$-bubbles together with Green's function techniques, linking curvature to the soliton potential and to Green's function behavior. The authors establish three main results: (i) a universal bound on $\inf_M R_g$ under a Ricci decay condition, (ii) $\inf_M R_g = 0$ in the nonparabolic setting with a $-C d^{-2} G$ type Ricci bound, and (iii) a precise liminf control of $R_g d_g^{\alpha}$ for $\alpha \in (0,1]$, plus a Bonnet–Myers–type diameter bound in the appendix. The work highlights how nonparabolicity and Green's function decay constrain curvature and demonstrates the efficacy of the warped $\mu$-bubble framework in deriving geometric-analytic bounds for solitons.

Abstract

We provide new type of decay estimate for scalar curvatures of steady gradient Ricci solitons. We also give certain upper bound for the diameter of a Riemannian manifold whose $\infty$-Bakry--Emery Ricci tensor is bounded by some positive constant from below. For the proofs, we use $μ$-bubbles introduced by Gromov.

Notes on scalar curvature lower bounds of steady gradient Ricci solitons

TL;DR

The paper investigates lower bounds and decay for the scalar curvature on complete non-compact steady gradient Ricci solitons and proves a diameter bound under -Bakry–Émery type Ricci lower bounds. It introduces a new decay-type estimate for at infinity by employing warped -bubbles together with Green's function techniques, linking curvature to the soliton potential and to Green's function behavior. The authors establish three main results: (i) a universal bound on under a Ricci decay condition, (ii) in the nonparabolic setting with a type Ricci bound, and (iii) a precise liminf control of for , plus a Bonnet–Myers–type diameter bound in the appendix. The work highlights how nonparabolicity and Green's function decay constrain curvature and demonstrates the efficacy of the warped -bubble framework in deriving geometric-analytic bounds for solitons.

Abstract

We provide new type of decay estimate for scalar curvatures of steady gradient Ricci solitons. We also give certain upper bound for the diameter of a Riemannian manifold whose -Bakry--Emery Ricci tensor is bounded by some positive constant from below. For the proofs, we use -bubbles introduced by Gromov.
Paper Structure (10 sections, 11 theorems, 84 equations)

This paper contains 10 sections, 11 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Propositions
  3. Preliminaries
  4. Green's functions on a Riemannian manifold with Ricci lower bound
  5. Warped $\mu$-bubbles
  6. Proofs of Propositions \ref{['main-1']}, \ref{['main-2']} and Theorem \ref{['main-3']}
  7. Proof of Proposition \ref{['main-1']}
  8. Proof of Proposition \ref{['main-2']}
  9. Proof of Theorem \ref{['main-3']}
  10. Appendix

Key Result

Theorem 1.1

Let $n \ge 2$ and $0 < \alpha \le 1$. Suppose that $(M^{n}, g, f)$ is an $n$-dimensional complete non-compact nonparabolic steady gradient Ricci soliton. Assume that where $G(\cdot)$ is the minimal positive Green's function with the pole at $p \in M.$ Moreover we assume that there is a constant $C \ge 0$ such that for all $x \in M$ with $d_{g} (p, x) >> 1.$ Then there is a positive constant $C =

Theorems & Definitions (32)

  • proof
  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.1
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.1
  • ...and 22 more