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International Journal of Mathematics and Mathematical Sciences - Volume 2003 2003, Issue 7, Pages 405-450

Lincoln College, Oxford OX1 3DR, UK

Received 13 August 2001; Revised 12 March 2002

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The Yamabe problem proved in 1984 guarantees the existence of a metric of constant scalar curvature in eachconformal class of Riemannian metrics on a compact manifold ofdimension n≥3, which minimizes the total scalar curvature onthis conformal class. Let M′,g′ and M″,g″ be compact Riemannian n-manifolds. We form their connected sumM′#M″ by removing small balls of radius ϵ fromM′, M″ and gluing together the 𝒮n−1 boundaries, and make a metric g on M′#M″ by joiningtogether g′, g″ with a partition of unity. In this paper, weuse analysis to study metrics with constant scalar curvature onM′#M″ in the conformal class of g. By the Yamabe problem,we may rescale g′ and g″ to have constant scalar curvature1,0, or −1. Thus, there are 9 cases, which we handleseparately. We show that the constant scalar curvature metricseither develop small “necks” separating M′ and M″, or oneof M′, M″ is crushed small by the conformal factor. Whenboth sides have positive scalar curvature, we find three metricswith scalar curvature 1 in the same conformal class.

Author: Dominic Joyce



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