Measure Concentration around Minimal Hypersurfaces

The classical measure concentration phenomenon on the standard sphere ${\mathbb{S}}^n$ asserts that the volume of any tubular neighborhood of a totally geodesic equator ${\mathbb{S}}^{n-1}$ concentrates almost the entire volume of ${\mathbb{S}}^n$ as the dimension $n$ grows to infinity. In this paper, we extend this \lq\lq Fat Equator Effect’’ to arbitrary closed minimal hypersurfaces ${\mathrm{\Sigma }}^{n-1}\subset {\mathbb{S}}^n$. By combining the analytic framework of Brownian motion and its Mean Exit Time (MET) function with the geometric tool of Schwarz symmetrization, we establish that minimality alone is sufficient to guarantee measure concentration around ${\mathrm{\Sigma }}^{n-1}$ in high dimensions.