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• W2150964489 abstract "We study the renormalized area functional $${mathcal{A}}$$ in the AdS/CFT correspondence, specifically for properly embedded minimal surfaces in convex cocompact hyperbolic 3-manifolds (and somewhat more broadly, Poincaré-Einstein spaces). Our main results include an explicit formula for the renormalized area of such a minimal surface Y as an integral of local geometric quantities, as well as formulæ for the first and second variations of $${mathcal{A}}$$ which are given by integrals of global quantities over the asymptotic boundary loop γ of Y. All of these formulæ are also obtained for a broader class of nonminimal surfaces. The proper setting for the study of this functional (when the ambient space is hyperbolic) requires an understanding of the moduli space of all properly embedded minimal surfaces with smoothly embedded asymptotic boundary. We show that this moduli space is a smooth Banach manifold and develop a $${mathbb{Z}}$$ -valued degree theory for the natural map taking a minimal surface to its boundary curve. We characterize the nondegenerate critical points of $${mathcal{A}}$$ for minimal surfaces in $${mathbb{H}^3}$$ , and finally, discuss the relationship of $${mathcal{A}}$$ to the Willmore functional." @default.
• W2150964489 created "2016-06-24" @default.
• W2150964489 creator A5034417161 @default.
• W2150964489 creator A5087642309 @default.
• W2150964489 date "2010-05-06" @default.
• W2150964489 modified "2023-09-30" @default.
• W2150964489 title "Renormalized Area and Properly Embedded Minimal Surfaces in Hyperbolic 3-Manifolds" @default.
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• W2150964489 doi "https://doi.org/10.1007/s00220-010-1054-3" @default.
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