Renormalized Area and Properly Embedded Minimal Surfaces in Hyperbolic 3-Manifolds
Renormalized Area and Properly Embedded Minimal Surfaces in Hyperbolic 3-Manifolds
复制标题
双曲 3 流形中的重正化面积和正确嵌入的最小曲面
DOI：
10.1007/s00220-010-1054-3
复制
发表时间：
2008
影响因子：
2.4
通讯作者：
R. Mazzeo
中科院分区：
文献类型：
--
作者：
Spyridon Alexakis;R. Mazzeo
我们研究 AdS/CFT 对应关系中的重整化面积函数 $${\mathcal{A}}$$，特别是对于凸协紧双曲 3 流形（以及更广泛的庞加莱-爱因斯坦空间）中正确嵌入的最小曲面。我们的主要结果包括作为局部几何量积分的最小曲面 Y 的重整化面积的显式公式，以及 $${\mathcal{A}}$$ 的第一和第二变体的公式，这些公式由 Y 的渐近边界环 γ 上的全局量的积分给出。所有这些公式也针对更广泛的非最小曲面类别而获得。研究此泛函的正确设置（当环境空间为双曲时）需要了解具有平滑嵌入渐近边界的所有正确嵌入的最小曲面的模空间。我们证明这个模空间是一个光滑的 Banach 流形，并为自然映射采用最小曲面到其边界曲线开发了 $${\mathbb{Z}}$$ 值度理论。我们表征 $${\mathbb{H}^3}$$ 中最小曲面的 $${\mathcal{A}}$$ 的非退化临界点，最后讨论 $${\mathcal{A}}$$ 与 Willmore 泛函的关系。
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.