Metric--Measure Geometry and Geometric Analogues of Holographic Extremal Surfaces
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.