The Mathematical Institute, University of Oxford, Eprints Archive

Constant Scalar Curvature Metrics on Connected Sums

Joyce, Dominic (2003) Constant Scalar Curvature Metrics on Connected Sums. International Journal of Mathematics and Mathematical Sciences, 7 . pp. 405-450.

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Abstract

The Yamabe problem (proved in 1984) guarantees the existence of a metric of constant scalar curvature in each conformal class of Riemannian metrics on a compact manifold of dimension $n \geq 3$, which minimizes the total scalar curvature of this conformal class.

Let $(M',g')$ and $(M'',g'')$ be compact Riemannian $n$-manifolds. We form their connected sum $M'\#M''$ by removing small balls of radius $\epsilon$ from $M'$, $M''$ and gluing together the $S^{n-1}$ boundaries, and make a metric $g$ on $M'\#M''$ by joining together $g'$,$g''$ with a partition of unity.

In this paper we use analysis to study metrics with constant scalar curvature on $M'\#M''$ in the conformal class of $g$. By the Yamabe problem, we may rescale $g'$ and $g''$ to have constant scalar curvature 1, 0 or -1. Thus there are 9 cases, which we handle separately.

We show that the constant scalar curvature metrics either develop small `necks' separating $M'$ and $M''$, or one of $M'$, $M''$ is crushed small by the conformal factor. When both sides have positive scalar curvature we find three metrics with scalar curvature 1 in the same conformal class.

Item Type:Article
Subjects:D - G > Differential geometry
Research Groups:Geometry Group
ID Code:78
Deposited By:Dominic Joyce
Deposited On:07 Jun 2004
Last Modified:20 Jul 2009 14:18

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