Classification of prime 3-manifolds with Yamabe invariant greater than RP3

• 成果类型: Article
• 署名作者: Bray, HL; Neves, A
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2004.159.407
• 发表日期: 2004
• 页码: 407-424
• 关键词: curvature

摘要

In this paper we compute the a-invariants (sometimes also called the smooth Yamabe invariants) of RP3 and RP2 X S-1 (which are equal) and show that the only prime 3-manifolds with larger a-invariants are S-3, S-2 x S-1, and S-2 x S-1 (the nonorientable S-2 bundle over S-1). More generally, we show that any 3-manifold with sigma-invariant greater than RP3 is either S-3, a connect sum with an S-2 bundle over S-1, or has more than one nonorientable prime component. A corollary is the Poincare conjecture for 3-manifolds with sigma-invariant greater than RP3. Surprisingly these results follow from the same inverse mean curvature flow techniques which were used by Huisken and Emanen 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.