Math @ Duke

Publications [#287079] of Hubert Bray
Papers Published
 Bray, HL; Neves, A, Classification of Prime 3Manifolds with Yamabe Invariant Greater than RP^3,
Annals of Mathematics, vol. 159 no. 1
(January, 2004),
pp. 407424 [p09]
(last updated on 2018/04/23)
Abstract: In this paper we compute the σinvariants (sometimes also called the smooth Yamabe invariants) of RP3 and RP2×S1 (which are equal) and show that the only prime 3manifolds with larger σinvariants are S3, S2×S1, and S2×~S1 (the nonorientable S2 bundle over S1). More generally, we show that any 3manifold with σinvariant greater than RP3 is either S3, a connect sum with an S2 bundle over S1, or has more than one nonorientable prime component. A corollary is the Poincaré conjecture for 3manifolds with σinvariant greater than RP3.
Surprisingly these results follow from the same inverse mean curvature flow techniques which were used by Huisken and Ilmanen in [7] to prove the Riemannian Penrose Inequality for a black hole in a spacetime. Richard Schoen made the observation [18] that since the constant curvature metric (which is extremal for the Yamabe problem) on RP3 is in the same conformal class as the Schwarzschild metric (which is extremal for the Penrose inequality) on RP3 minus a point, there might be a connection between the two problems. The authors found a strong connection via inverse mean curvature flow.


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