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Weyl energy and connected sums of four-manifolds

Andrea Malchiodi, Francesco Malizia

Abstract

Given two closed, oriented Riemannian four-manifolds $(M,g_M)$ and $(Z,g_Z)$, which are not locally conformally flat and not both self-dual or both anti-self-dual, we prove that there exists a metric $g_Y$ on the connected sum $Y\cong M\#Z$ such that the Weyl energy of $g_Y$ is strictly smaller than the sum of Weyl energies of $g_M$ and $g_Z$.

Weyl energy and connected sums of four-manifolds

Abstract

Given two closed, oriented Riemannian four-manifolds and , which are not locally conformally flat and not both self-dual or both anti-self-dual, we prove that there exists a metric on the connected sum such that the Weyl energy of is strictly smaller than the sum of Weyl energies of and .

Paper Structure

This paper contains 12 sections, 12 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linearization of curvature tensors
  4. Expansion of Weyl functional in TT directions at the Euclidean metric
  5. The connected sum setup
  6. Biharmonic interpolation
  7. Comparison of Weyl energies
  8. Higher order terms in energy balance: inner boundary component
  9. Higher order terms in energy balance: outer boundary component
  10. Computation of the energy balance
  11. Proof of Theorem \ref{['thm:main']}
  12. Appendix

Key Result

Theorem 1.1

Let $(M,g_M)$, $(Z,g_Z)$ be two $4$-dimensional closed, connected, oriented manifolds, and let $Y:=Z\#M$ denote their connected sum. Assume that $g_M, g_Z$ are not locally conformally flat and that they are not both self-dual or both anti-self-dual. Then there exists a metric $g_Y$ on $Y$ such that

Theorems & Definitions (46)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 2.1
  • ...and 36 more