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Tangent cones and regularity of real hypersurfaces

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Published/Copyright: November 13, 2013
Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2014 Issue 697

Abstract

We characterize 𝒞1 embedded hypersurfaces of 𝐑n 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 𝒞1. 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⊂𝐑n is 𝒞1. Furthermore, if X is real algebraic, strictly convex, and unbounded, then its projective closure is a 𝒞1 hypersurface as well, which shows that X is the graph of a function defined over an entire hyperplane. Finally we show that the last property is a special feature of real algebraic sets, in the sense that it does not hold in the real analytic category.

Funding source: NSF

Award Identifier / Grant number: DMS-0336455

We thank Matt Baker, Saugata Basu, Igor Belegradek, Eduardo Casas Alvero, Joe Fu, Frank Morgan, Bernd Sturmfels, Serge Tabachnikov, and Brian White, for useful communications. Thanks also to the anonymous referee for informing us about Alexander Lytchak's work, and its connection to the last claim in Theorem 1.2.

Received: 2012-6-22
Revised: 2013-7-31
Published Online: 2013-11-13
Published in Print: 2014-12-1

© 2014 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Holomorphic one-forms, integral and rational points on complex hyperbolic surfaces
  3. Mordell–Weil groups of elliptic threefolds and the Alexander module of plane curves
  4. Reflexive differential forms on singular spaces. Geometry and cohomology
  5. Special cycles on unitary Shimura varieties II: Global theory
  6. Batalin–Vilkovisky structures on Ext and Tor
  7. Tangent cones and regularity of real hypersurfaces
https://doi.org/10.1515/crelle-2013-0091

Articles in the same Issue

  1. Frontmatter
  2. Holomorphic one-forms, integral and rational points on complex hyperbolic surfaces
  3. Mordell–Weil groups of elliptic threefolds and the Alexander module of plane curves
  4. Reflexive differential forms on singular spaces. Geometry and cohomology
  5. Special cycles on unitary Shimura varieties II: Global theory
  6. Batalin–Vilkovisky structures on Ext and Tor
  7. Tangent cones and regularity of real hypersurfaces
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