Joyce, Dominic (2003) Constant Scalar Curvature Metrics on Connected Sums. International Journal of Mathematics and Mathematical Sciences, 7 . pp. 405-450.
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 , which minimizes the total scalar curvature of this conformal class.
Let and be compact Riemannian -manifolds. We form their connected sum by removing small balls of radius from , and gluing together the boundaries, and make a metric on by joining together , with a partition of unity.
In this paper we use analysis to study metrics with constant scalar curvature on in the conformal class of . By the Yamabe problem, we may rescale and 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 and , or one of , 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.
|Subjects:||D - G > Differential geometry|
|Research Groups:||Geometry Group|
|Deposited By:||Dominic Joyce|
|Deposited On:||07 Jun 2004|
|Last Modified:||20 Jul 2009 14:18|
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