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Ralph Howard : Tangent cones and regularity of real hypersurfaces (Apr 17, 2012 4:25 PM)

We characterize $C^1$ embedded 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 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 $X$ of $R^n$ 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. This is joint work with Mohammad Ghomi.

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