Tangent cones and regularity of real hypersurfaces

Mohammad Ghomi; Ralph Howard

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arXiv · 1005.2761

Tangent cones and regularity of real hypersurfaces

Mohammad Ghomi·Ralph Howard

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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.

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BibTeX RIS

Mohammad Ghomi, Ralph Howard. 2010-05-16. Tangent cones and regularity of real hypersurfaces. https://arxiv.org/abs/1005.2761

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