Tangent cones and regularity of real hypersurfaces

Details Tangent cones and regularity of real hypersurfaces Tangent cones and regularity of real hypersurfaces

2010-05-16

arxiv.org/abs/1005.2761v2

Mathematics

Algebraic Geometry

26 pages, 13 figures, and 1 appendix

Scientific paper

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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