度量-测度几何与全息极值曲面的几何类比
Metric--Measure Geometry and Geometric Analogues of Holographic Extremal Surfaces
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中文总结 AI 辅助
该研究基于度量-测度空间构建几何框架，推导极值曲面方程的内在几何类比，分析广义泛函的变分性质，在史瓦西和反德西特几何中验证了相关特性，为极值曲面构造提供了最小几何框架。
中文摘要 AI 辅助
我们基于度量-测度空间$(M,g,f)$构建了一个几何框架，其中函数$f$定义了受Perelman里奇流公式启发的黎曼测度变形。在该框架内，我们引入了测度加权超曲面及相关几何泛函，并推导了余维数1和余维数2子流形的修正极值条件。这些条件提供了极值曲面方程的内在几何类比，仅源于度量-测度结构，独立于全息对偶或量子场论输入。我们进一步定义了结合测度加权几何项与有效体贡献的广义泛函，并分析其变分性质。所得欧拉-拉格朗日方程与半经典广义熵泛函具有结构对应性，同时保持了与Perelman的$W$-泛函意义下热力学熵不同的纯几何解释。对史瓦西和反德西特几何的应用表明，函数$f$诱导了优选几何尺度的出现和紫外标度行为的修正。这些结果表明，度量-测度几何提供了一个最小框架，极值曲面构造的关键结构特征可从内在几何原理中产生。
英文摘要
We develop a geometric framework based on metric--measure spaces $(M,g,f)$, where the function $f$ defines a deformation of the Riemannian measure motivated by Perelman's formulation of Ricci flow. Within this setting, we introduce measure-weighted hypersurfaces and associated geometric functionals, and derive modified extremality conditions for codimension-one and codimension-two submanifolds. These conditions provide intrinsic geometric analogues of extremal surface equations, arising solely from the metric--measure structure and independent of holographic duality or quantum field theoretic input. We further define a generalized functional combining a measure-weighted geometric term with an effective bulk contribution and analyze its variational properties. The resulting Euler--Lagrange equation exhibits a structural correspondence with semiclassical generalized entropy functionals, while maintaining a purely geometric interpretation distinct from thermodynamic entropy in the sense of Perelman's $W$-functional. Applications to Schwarzschild and Anti-de Sitter geometries illustrate the emergence of preferred geometric scales and the modification of ultraviolet scaling behavior induced by the function $f$. These results suggest that metric--measure geometry provides a minimal framework in which key structural features of extremal surface constructions can arise from intrinsic geometric principles.
