Ricci Curvature, Minimal Volumes, and Seiberg-Witten Theory

Details Ricci Curvature, Minimal Volumes, and Seiberg-Witten Theory Ricci Curvature, Minimal Volumes, and Seiberg-Witten Theory

2000-03-13

arxiv.org/abs/math/0003068v1

Mathematics

Differential Geometry

41 pages, LaTeX2e

Scientific paper

10.1007/s002220100148

We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with a non-trivial Seiberg-Witten invariant. These allow one, for example, to exactly compute the infimum of the L2-norm of Ricci curvature for all complex surfaces 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.

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