Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The large scale structure of complete $4$-manifolds with nonnegative Ricci curvature and Euclidean volume growth

Daniele Semola

Abstract

We survey the implications of our joint work with E. Bruè and A. Pigati on the structure of blow-downs for a smooth, complete, Riemannian $4$-manifold with nonnegative Ricci curvature and Euclidean volume growth. Very imprecisely, any such manifold looks like a cone over a spherical space form at infinity. We present some open questions and discuss possible future directions along the way.

The large scale structure of complete $4$-manifolds with nonnegative Ricci curvature and Euclidean volume growth

Abstract

We survey the implications of our joint work with E. Bruè and A. Pigati on the structure of blow-downs for a smooth, complete, Riemannian -manifold with nonnegative Ricci curvature and Euclidean volume growth. Very imprecisely, any such manifold looks like a cone over a spherical space form at infinity. We present some open questions and discuss possible future directions along the way.
Paper Structure (26 sections, 33 theorems, 39 equations)

This paper contains 26 sections, 33 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Sharpness of \ref{['thm:mainIntro']}
  3. Realizing all possible topologies
  4. Realizing all possible metrics
  5. Failure of topological regularity and uniqueness for $n>4$
  6. Some previous developments
  7. Dimensions $n\le 3$
  8. The Ricci-flat case
  9. Kähler surfaces
  10. Nonnegative sectional curvature
  11. $\rm{CAT}(0)$ four-manifolds
  12. On the proof of \ref{['thm:mainIntro']}
  13. The "no $C(\mathbb{RP}^2)$" \ref{['thm:noRP2BPS']}
  14. The manifold recognition \ref{['thm:manrecRCD']}
  15. Local simple-connectedness
  16. ...and 11 more sections

Key Result

Theorem 1.1

CheegerNaber15 Let $(M^4,g)$ be a Ricci-flat $4$-manifold satisfying eq:EVG. There exists a finite group $\Gamma<\mathrm{O}(4)$ acting freely on $S^3$ such that $(M^4,r^{-2}g,p)\to (\mathbb{R}^4/\Gamma,g_{\rm{eucl}},o)$ as $r\to \infty$ in the pointed Gromov-Hausdorff sense and in $C^{\infty}_{{\rm

Theorems & Definitions (85)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1.2
  • Proposition 1
  • Lemma 1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 75 more