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Seiberg-Witten Equations and Einstein Metrics on Finite Volume 4-Manifolds with Asymptotically Hyperbolic Ends

Alex Xu

Abstract

We construct infinitely many examples of finite volume 4-manifolds with $T^3$ ends that do not admit any cusped asymptotically hyperbolic Einstein metrics yet satisfy a strict logarithmic version of the Hitchin-Thorpe inequality due to Dai-Wei. This is done by using estimates from Seiberg-Witten theory due to LeBrun as well as a method for constructing solutions to the Seiberg-Witten equations on noncompact manifolds due to Biquard. We also use constructions coming from the $Pin^-(2)$ monopole equations to obtain a larger class of manifolds where these techniques apply.

Seiberg-Witten Equations and Einstein Metrics on Finite Volume 4-Manifolds with Asymptotically Hyperbolic Ends

Abstract

We construct infinitely many examples of finite volume 4-manifolds with ends that do not admit any cusped asymptotically hyperbolic Einstein metrics yet satisfy a strict logarithmic version of the Hitchin-Thorpe inequality due to Dai-Wei. This is done by using estimates from Seiberg-Witten theory due to LeBrun as well as a method for constructing solutions to the Seiberg-Witten equations on noncompact manifolds due to Biquard. We also use constructions coming from the monopole equations to obtain a larger class of manifolds where these techniques apply.
Paper Structure (11 sections, 18 theorems, 98 equations)

This paper contains 11 sections, 18 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Approximating Metrics and Pertubations
  4. $L^2$ Cohomology
  5. $L^2$ Chern-Weil theory
  6. Apriori Estimates on 1-forms
  7. The Seiberg-Witten Equations on Asymptotically Hyperbolic Manifolds
  8. $Pin^-(2)$ Monopoles
  9. The $Pin^-(2)$ Monopole Equations
  10. Relation with the Seiberg-Witten equations
  11. Scalar Curvature Estimates

Key Result

Theorem 1.1

Suppose that $(X,g)$ is a complete noncompact Einstein 4-manifold with cylindrical ends all of the form $T^3 \times [0,\infty)$. Furthermore, suppose that $g$ is an asymptotically hyperbolic metric. Then $X$ satisfies

Theorems & Definitions (29)

  • Theorem 1.1: Dai-Wei
  • Theorem 1.2
  • Remark
  • Theorem 1.3: Di Cerbo
  • Theorem 3.1: Zucker
  • Proposition 3.2
  • Lemma 3.3
  • Lemma 4.1
  • proof
  • Corollary 4.2
  • ...and 19 more