Ricci curvature, minimal volumes, and Seiberg-Witten theory

• 成果类型: Article
• 署名作者: LeBrun, C
• 署名单位: State University of New York (SUNY) System; Stony Brook University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s002220100148
• 发表日期: 2001
• 页码: 279-316
• 关键词: simply connected manifolds; einstein-metrics; 4-manifolds; SURFACES; EXISTENCE; invariant; TOPOLOGY; genus

摘要

We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with non-trivial Seiberg-Witten invariants. These allow one, for example, to exactly compute the infimum of the L-2-norm of Ricci curvature for any complex surface of general type. We are also able to show that the standard metric on any complex-hyperbolic 4-manifold minimizes volume among all metrics satisfying a point-wise lower bound on sectional curvature plus suitable multiples of the scalar curvature. These estimates also imply new non-existence results for Einstein metrics.