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Essential minimal volume of Einstein 4-manifolds

Antoine Song

Abstract

The minimal volume of a closed manifold $M$ is the infimum of the volume of $(M,g)$ over all metrics $g$ with sectional curvature between $-1$ and $1$. We introduce a variant called the essential minimal volume, $\mathrm{ess-Minvol}(M)$, which is the limit, as $δ>0$ goes to $0$, of the infimum of the volume of the $δ$-thick part of $(M,g)$ over all metrics $g$ with sectional curvature between $-1$ and $1$. We show that, for some universal constant $C>0$, any closed Einstein 4-manifold $M$ with Euler characteristic $e(M)$ satisfies $$C^{-1}e(M) \leq \mathrm{ess-Minvol}(M) \leq Ce(M).$$ As a corollary, these inequalities are true for the essential minimal volume of closed complex surfaces of nonnegative Kodaira dimension. We conjecture that those linear bounds in fact hold for the minimal volume.

Essential minimal volume of Einstein 4-manifolds

Abstract

The minimal volume of a closed manifold is the infimum of the volume of over all metrics with sectional curvature between and . We introduce a variant called the essential minimal volume, , which is the limit, as goes to , of the infimum of the volume of the -thick part of over all metrics with sectional curvature between and . We show that, for some universal constant , any closed Einstein 4-manifold with Euler characteristic satisfies As a corollary, these inequalities are true for the essential minimal volume of closed complex surfaces of nonnegative Kodaira dimension. We conjecture that those linear bounds in fact hold for the minimal volume.

Paper Structure

This paper contains 19 sections, 16 theorems, 157 equations.

Table of Contents

  1. Definitions and first properties
  2. Thick/thin decomposition
  3. F-structures and N-structures
  4. Definition of the essential minimal volume
  5. First properties
  6. Uniform local collapsing in dimension 4
  7. A practical bound on $\mathop{\mathrm{ess-MinVol}}\nolimits$ in dimension 4
  8. Essential minimal volume of Einstein $4$-manifolds
  9. Preliminaries
  10. The linear bound
  11. Thick-thin decomposition for Einstein $4$-manifolds
  12. Some covering lemmas
  13. Metric deformations
  14. End of the proof
  15. An extension to $4$-orbifolds with isolated singularities
  16. ...and 4 more sections

Key Result

Theorem 1

There is a constant $C>0$ such that the following holds. Let $M$ be any closed $4$-manifold admitting an Einstein metric, or any closed complex surface with nonnegative Kodaira dimension. Then where $\mathop{\mathrm{\textbf{e}}}\nolimits(M)$ is the Euler characteristic of $M$.

Theorems & Definitions (31)

  • Theorem 1
  • Definition 1.1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • ...and 21 more