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Homological Filling and Minimal Varifolds in Four-Dimensional Einstein Manifolds

Wenjie Fu, Zhifei Zhu

Abstract

We study the smallest area $A(M,g)$ of a 2-dimensional stationary integral varifold in a closed Einstein 4-manifold $(M^4,g)$ with $Ric_g = λg, |λ|\leq 3, Vol(M,g)\geq v>0, diam(M,g)\leq D, H_1(M;\mathbb{Z})=0.$ Building on the previous work on homological filling functions, we show that for every $(M^4,g)$ in this Einstein class, there is an upper bound $A(M,g)\leq F_{Ein}(v,D),$ where $F_{Ein}$ depends only on $(v,D)$ and on quantitative Sobolev and $\varepsilon$-regularity constants for Einstein metrics.

Homological Filling and Minimal Varifolds in Four-Dimensional Einstein Manifolds

Abstract

We study the smallest area of a 2-dimensional stationary integral varifold in a closed Einstein 4-manifold with Building on the previous work on homological filling functions, we show that for every in this Einstein class, there is an upper bound where depends only on and on quantitative Sobolev and -regularity constants for Einstein metrics.

Paper Structure

This paper contains 12 sections, 17 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Sobolev inequality and Einstein $\varepsilon$-regularity
  3. Global Sobolev inequality
  4. $L^2$ curvature and the curvature equation
  5. Einstein $\varepsilon$-regularity
  6. Bubble--tree decomposition
  7. Filling functions and stationary varifolds
  8. Homological filling functions and stationary varifolds
  9. A homological filling inequality for Einstein manifolds
  10. Local fillings
  11. Approximation by $\Gamma$ and improved combinatorial filling
  12. Decomposition and the Einstein filling inequality

Key Result

Theorem 1.1

For every $v,D>0$ there exists a constant $F_{\mathop{\mathrm{Ein}}\nolimits}(v,D)>0$ with the following property. If $(M^4,g)\in \mathcal{E}_1(4,v,D)$, then the area $A_{\min}(M,g)$ of a smallest $2$-dimensional stationary integral varifold in $(M,g)$ satisfies Moreover, the dependence of $F_{\mathop{\mathrm{Ein}}\nolimits}(v,D)$ is reduced to the Sobolev constant $C_S(v,D)$, the global $L^2$-cu

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Sobolev inequality
  • proof
  • Remark 2.2
  • Theorem 2.3: $L^2$ curvature bound
  • proof
  • Theorem 2.4: Einstein $\varepsilon$-regularity
  • proof
  • Remark 2.5
  • ...and 21 more