Abstract.
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.
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Oblatum 14-III-2000 & 8-II-2001¶Published online: 4 May 2001
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LeBrun, C. Ricci curvature, minimal volumes, and Seiberg-Witten theory. Invent. math. 145, 279–316 (2001). https://doi.org/10.1007/s002220100148
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DOI: https://doi.org/10.1007/s002220100148