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Topology of three-manifolds with positive $ P$-scalar curvature

Author: Edward M. Fan
Journal: Proc. Amer. Math. Soc. 136 (2008), 3255-3261
MSC (2000): Primary 53C21; Secondary 58E12, 49Q05
Published electronically: May 6, 2008
MathSciNet review: 2407091
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Abstract: Consider an $ n$-dimensional smooth Riemannian manifold $ (M^n,g)$ with a given smooth measure $ m$ on it. We call such a triple $ (M^n,g,m)$ a Riemannian measure space. Perelman introduced a variant of scalar curvature in his recent work on solving Poincaré's conjecture $ P(g)=R^m_\infty(g) = R(g) - 2\Delta_g log\phi - \vert\nabla log\phi\vert^2_g$, where $ dm = \phi dvol(g)$ and $ R$ is the scalar curvature of $ (M^n,g)$. In this note, we study the topological obstruction for the $ \phi$-stable minimal submanifold with positive $ P$-scalar curvature in dimension three under the setting of manifolds with density.

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Additional Information

Edward M. Fan
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000

Keywords: Minimal submanifold, scalar curvature, Riemannian geometry
Received by editor(s): April 17, 2006
Received by editor(s) in revised form: November 30, 2006
Published electronically: May 6, 2008
Additional Notes: The author was partially supported by an NSF graduate research fellowship.
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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