Ricci curvature, minimal volumes, and Seiberg-Witten theory
By C. LeBrun. 2000
109 citations recorded by the source corpus.
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 L2-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.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/ab10379c011b1152e3354b23a45fbd6ca97a7dec
How researchers see this paper
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In Applied Mathematics
- Jeffrey Streets
- V. Kapovitch
- Anton Petrunin
- G. Tian
- M. Ishida
- V. Apostolov
- Tedi Drăghici
- Gabriel P. Paternain
In Geometry and Topology
- Jeffrey Streets
- Anton Petrunin
- G. Tian
- M. Ishida
- V. Apostolov
- Tedi Drăghici
- Gabriel P. Paternain
- J. Petean
Citations in the corpus
- Minimal entropy and collapsing with curvature bounded from below (2000, background): “M which admit F -structures and verify MinVol(M )> 0. Also C. LeBrun proved (see [26, 27 ]) that, for instance, an elliptic compact complex surface collapses with bounded Ricci curvature (i.e Vol|r| = 0) if and only if it is minimal.”
- Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces (2005, background): “⋄ If a closed manifold has negative Yamabe constant, then it cannot volume collapse with scalar curvature bounded from below (see [Sch89, LeB01 ]).”
- The curvature and the integrability of almost-Kahler manifolds: a survey (2003, background): “ some recent local classification results in dimension four, which can be used to obtain further integrability results. The last section is inspired by the recent works of Donaldson [32] and C. LeBrun [63], which naturally lead to the study of Einstein-like conditions with respect to the canonical He”
- Symplectic curvature flow (2010, background): “[14] LeBrun, C. Ricci curvature, minimal volumes, and Seiberg-Witten theory Invent.”
- Curvature, connected sums, and Seiberg-Witten theory (2001, methodology): “ar X iv :m at h/ 01 11 22 8v 2 [ m at”