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Volume growth and asymptotic cones of manifolds with nonnegative Ricci curvature

Zhu Ye

TL;DR

This work investigates how volume growth on open manifolds with nonnegative Ricci curvature controls the dimension of asymptotic cones. By introducing the conic-at-infinity condition and employing a slope-lemma analysis of $f(R)=\ln\text{vol}(B_{e^R}(p))$ together with renormalized limit measures, it proves that there exists an asymptotic cone with upper box dimension bounded by the infimum volume-growth order $\text{IV}(M)$; in particular, $\text{dim}_{H}(Y)\le \text{IV}(M)$. When $\text{IV}(M)<2$ and the asymptotic cone is 1-dimensional, the paper obtains 1-dimensional blow-down limits and derives corollaries for Sormani-type linear growth results, including finite generation of the fundamental group in certain settings. It also constructs an explicit example of an open $n$-manifold with $\text{sec}_M\ge 0$ where volume growth oscillates between order $1$ and $n$ (so $\text{IV}(M)=1$, $\text{SV}(M)=n$), and provides a new proof of Sormani's sublinear diameter growth theorem, highlighting the sharp interplay between volume decay, cone-dimension, and large-scale geometry.

Abstract

Let $M$ be an open (i.e. complete and noncompact) manifold with nonnegative Ricci curvature. In this paper, we study whether the volume growth order of $M$ is always greater than or equal to the dimension of some (or every) asymptotic cone of $M$. Our first main result asserts that, under the conic at infinity condition, if the infimum of the volume growth order of $M$ equals $k$, then there exists an asymptotic cone of $M$ whose upper box dimension is at most $k$. In particular, this yields a complete affirmative answer to our problem in the setting of nonnegative sectional curvature. In the subsequent part of the paper, we extend or partially extend Sormani's results concerning $M$ with linear volume growth to more relaxed volume growth conditions. Our approach also allows us to present a new proof of Sormani's sublinear diameter growth theorem for open manifolds with $\mathrm{Ric}\geq 0$ and linear volume growth. Finally, we construct an example of an open $n$-manifold $M$ with $\mathrm{sec}_M\geq0$ whose volume growth order oscillates between 1 and $n$.

Volume growth and asymptotic cones of manifolds with nonnegative Ricci curvature

TL;DR

This work investigates how volume growth on open manifolds with nonnegative Ricci curvature controls the dimension of asymptotic cones. By introducing the conic-at-infinity condition and employing a slope-lemma analysis of together with renormalized limit measures, it proves that there exists an asymptotic cone with upper box dimension bounded by the infimum volume-growth order ; in particular, . When and the asymptotic cone is 1-dimensional, the paper obtains 1-dimensional blow-down limits and derives corollaries for Sormani-type linear growth results, including finite generation of the fundamental group in certain settings. It also constructs an explicit example of an open -manifold with where volume growth oscillates between order and (so , ), and provides a new proof of Sormani's sublinear diameter growth theorem, highlighting the sharp interplay between volume decay, cone-dimension, and large-scale geometry.

Abstract

Let be an open (i.e. complete and noncompact) manifold with nonnegative Ricci curvature. In this paper, we study whether the volume growth order of is always greater than or equal to the dimension of some (or every) asymptotic cone of . Our first main result asserts that, under the conic at infinity condition, if the infimum of the volume growth order of equals , then there exists an asymptotic cone of whose upper box dimension is at most . In particular, this yields a complete affirmative answer to our problem in the setting of nonnegative sectional curvature. In the subsequent part of the paper, we extend or partially extend Sormani's results concerning with linear volume growth to more relaxed volume growth conditions. Our approach also allows us to present a new proof of Sormani's sublinear diameter growth theorem for open manifolds with and linear volume growth. Finally, we construct an example of an open -manifold with whose volume growth order oscillates between 1 and .

Paper Structure

This paper contains 7 sections, 22 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Volume growth and asymptotic cone
  3. Volume growth order $<2$ and 1-dimensional asymptotic cone
  4. Oscillating vs stable volume growth
  5. Examples of oscillating volume growth
  6. Stable volume growth
  7. Appendix: a new proof of Sormani's sublinear diameter growth theorem

Key Result

Theorem 1.3

Let $M^n$ be an open manifold with $\mathrm{sec}_M\geq 0$. Denote by $(Y,y)$ the asymptotic cone of $M$. If $M$ also has an upper curvature bound $\mathrm{sec}_M\leq K$ for some constant $K\geq 0$, then $\mathrm{dim}_{H}(Y)\leq \mathrm{IV}(M)$.

Theorems & Definitions (56)

  • Remark 1.2
  • Theorem 1.3
  • Theorem A
  • Corollary A
  • Theorem 1.4
  • Corollary B
  • Remark 1.5
  • Theorem B
  • Remark 1.6
  • Corollary C
  • ...and 46 more