Some remarks on stable minimal surfaces in the critical point of the total scalar curvature
Commun. Korean Math. Soc. 2008 Vol. 23, No. 4, 587-595
Printed December 1, 2008
Seungsu Hwang
Chung-Ang University
Abstract : It is well known that critical points of the total scalar curvature functional ${\mathcal S}$ on the space of all smooth Riemannian structures of volume $1$ on a compact manifold $M$ are exactly the Einstein metrics. When the domain of ${\mathcal S}$ is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is isometric to a standard sphere. In this paper we investigate the relationship between the first Betti number and stable minimal surfaces, and study the analytic properties of stable minimal surfaces in $M$ for $n=3$.
Keywords : total scalar curvature, critical points, stable minimal surfaces
MSC numbers : 58E11, 53C25
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