Tangent cones and regularity of real hypersurfaces

TL;DR

This paper characterizes when real hypersurfaces are smooth ($ ext{C}^1$) based on tangent cone properties, showing that flat tangent cones imply smoothness under certain measure and regularity conditions.

Contribution

It provides new characterizations of $ ext{C}^1$ hypersurfaces using tangent cones and measure-theoretic multiplicity, extending classical regularity results to broader classes of sets.

Findings

Embedded $ ext{C}^1$ hypersurfaces are uniquely characterized by flat tangent cones and measure conditions.

Any topological hypersurface with flat tangent cones supported by uniform balls is $ ext{C}^1$.

Convex real analytic hypersurfaces are necessarily $ ext{C}^1$.

Abstract

We characterize embedded $\C^1$ hypersurfaces of $\R^n$ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most $m<3/2$. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is $\C^1$. In the real analytic case the same conclusion holds under the weakened hypothesis that each tangent cone be a hypersurface. In particular, any convex real analytic hypersurface $X\subset\R^n$ is $\C^1$. Furthermore, if $X$ is real algebraic, strictly convex, and unbounded then its projective closure is a $\C^1$ hypersurface as well, which shows that $X$ is the graph of a function defined over an entire hyperplane.