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Examples of open manifolds with almost quadratic volume growth and infinite Betti numbers

Huihong Jiang

TL;DR

The paper constructs a family of complete $(2+n)$-dimensional open manifolds with positive Ricci curvature and a uniform lower bound on sectional curvature, yet with infinite Betti numbers $b_2$ and $b_n$. Central to the method is Perelman’s gluing criterion, which allows attaching necks between carefully designed pieces to preserve positivity of Ricci curvature. The base region $Q$ is endowed with an almost warped product metric, and infinitely many $S^2 imes D^n$ blocks are glued in to produce an open manifold whose volume growth can be made arbitrarily close to quadratic while maintaining nontrivial topology. The resulting geometry exhibits diameter growth $O(t^{(1+ ext{γ})/2})$ and volume growth $O(t^{2+ ext{γ}})$ for any small $ ext{γ}>0$, highlighting a gap between curvature/topology restrictions and near-quadratic volume growth. These results provoke questions about whether complete open manifolds with nonnegative Ricci curvature, lower sectional curvature bounds, and at most quadratic volume growth must have finite topological type.

Abstract

We construct a family of examples of complete $(2+n)-$dimensional ($n\ge 2$) open manifolds with positive Ricci curvature, sectional curvature bounded from below and infinite Betti numbers $b_2,b_n$, moreover its volume growth can be arbitrarily close to quadratic volume growth. Compared with some known result of finite topology for manifolds with nonnegative Ricci curvature and lower sectional curvature bound, it makes sense to ask whether complete manifolds with such curvature bounds must be of finite topological type or not provided with at most quadratic volume growth.

Examples of open manifolds with almost quadratic volume growth and infinite Betti numbers

TL;DR

The paper constructs a family of complete -dimensional open manifolds with positive Ricci curvature and a uniform lower bound on sectional curvature, yet with infinite Betti numbers and . Central to the method is Perelman’s gluing criterion, which allows attaching necks between carefully designed pieces to preserve positivity of Ricci curvature. The base region is endowed with an almost warped product metric, and infinitely many blocks are glued in to produce an open manifold whose volume growth can be made arbitrarily close to quadratic while maintaining nontrivial topology. The resulting geometry exhibits diameter growth and volume growth for any small , highlighting a gap between curvature/topology restrictions and near-quadratic volume growth. These results provoke questions about whether complete open manifolds with nonnegative Ricci curvature, lower sectional curvature bounds, and at most quadratic volume growth must have finite topological type.

Abstract

We construct a family of examples of complete dimensional () open manifolds with positive Ricci curvature, sectional curvature bounded from below and infinite Betti numbers , moreover its volume growth can be arbitrarily close to quadratic volume growth. Compared with some known result of finite topology for manifolds with nonnegative Ricci curvature and lower sectional curvature bound, it makes sense to ask whether complete manifolds with such curvature bounds must be of finite topological type or not provided with at most quadratic volume growth.

Paper Structure

This paper contains 13 sections, 4 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. The construction of the metrics
  3. The construction of $Q$
  4. Construction of $u(t)$
  5. Construction of $f(t,x)$
  6. Ricci curvature and principle curvature for $Q\setminus\left(\coprod\limits_{i=1}^{+\infty}B_{\frac{4}{5}}(o_i)\right)$
  7. Gluing along the boundary with Necks and $Q\setminus\left(\coprod\limits_{i=1}^{+\infty}B_{\frac{4}{5}}(o_i)\right)$
  8. The construction of $(S^2\times D^n)_i$
  9. The boundary and Principle curvature
  10. Sectional curvature and Ricci curvature
  11. Gluing along the boundary with $Q'\times_{c_0}S^{n-1}$ and $(S^2\times D^n)_i$
  12. Sectional curvature, diameter growth and volume growth
  13. The construction of Neck

Key Result

Theorem 1.1

For all integers $n\ge2$, there exists a complete $(2+n)-$dimensional Riemannian manifold $M$ with a base point $p_0$, which admits positive Ricci curvature, sectional curvature bounded from below and infinite Betti number $b_2,b_n$. Moreover, it satisfies where $\gamma$ can be chosen to be a positive constant arbitrarily close to $0$.

Theorems & Definitions (4)

  • Theorem 1.1
  • Corollary 1.1.1
  • Proposition 2.1
  • Proposition 2.2