Weyl energy and connected sums of four-manifolds

TL;DR

This paper proves that for certain four-manifolds, one can construct a connected sum with a metric whose Weyl energy is less than the sum of the original manifolds' Weyl energies, revealing new geometric properties.

Contribution

It demonstrates the existence of metrics on connected sums of four-manifolds with reduced Weyl energy, advancing understanding of conformal geometry in four dimensions.

Findings

Existence of metrics with lower Weyl energy on connected sums

Reduction of Weyl energy compared to sum of individual energies

Applicable to non-conformally flat, non-self-dual four-manifolds

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 $g_Z$.