The Isoperimetric Inequality and Mean Curvature Flow Singularities in Riemannian Manifolds with Bounded Geometry

Mathematical Applications of Science Fiction We present a rigorous analysis of the Mean Curvature Flow (MCF) of hypersurfaces immersed in a Riemannian manifold of dimension n+1n+1n+1 with geometry bounded to infinite order. By establishing a localized Michael–Simon–Sobolev inequality adapted to the injectivity radius and curvature bounds of the ambient space, we derive a generalized Huisken monotonicity formula with error terms explicitly controlled by the background geometry. We prove that for any Type I or Type II singularity, the rescaled flow converges in the Cheeger–Gromov sense to a self-shrinker or a translating soliton in Euclidean space, provided the initial hypersurface satisfies an entropy bound derived from the isoperimetric profile. We introduce the Thorne–Entropic Capacity functional SΞS_{\Xi}SΞ and show that singularity formation is topologically obstructed by its lower bounds. This work extends the classical singularity analysis of Huisken and White to non-Euclidean settings with transfinite geometric precision.