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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Local geometry of singular real analytic surfaces
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by Daniel Grieser PDF
Trans. Amer. Math. Soc. 355 (2003), 1559-1577 Request permission

Abstract:

Let $V\subset \mathbb {R}^N$ be a compact real analytic surface with isolated singularities, and assume its smooth part $V_0$ is equipped with a Riemannian metric that is induced from some analytic Riemannian metric on $\mathbb {R}^N$. We prove:

  1. Each point of $V$ has a neighborhood which is quasi-isometric (naturally and “almost isometrically”) to a union of metric cones and horns, glued at their tips.

  2. A full asymptotic expansion, for any $p\in V$, of the length of $V\cap \{q:\operatorname {dist}(q,p)=r\}$ as $r\to 0$.

  3. A Gauss-Bonnet Theorem, saying that each singular point contributes $1-l/(2\pi )$, where $l$ is the coefficient of the linear term in the expansion of (2).

  4. The $L^2$ Stokes Theorem, selfadjointness and discreteness of the Laplace-Beltrami operator on $V_0$, an estimate on the heat kernel, and a Gauss-Bonnet Theorem for the $L^2$ Euler characteristic.

As a central tool we use resolution of singularities.

References
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Additional Information
  • Daniel Grieser
  • Affiliation: Institut für Mathematik, Humboldt Universität zu Berlin, Sitz: Rudower Chaussee 25, 10099 Berlin, Germany
  • Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139
  • MR Author ID: 308546
  • Email: grieser@mathematik.hu-berlin.de
  • Received by editor(s): July 9, 2002
  • Published electronically: November 18, 2002
  • Additional Notes: The author gratefully acknowledges support by the Deutsche Forschungsgemeinschaft (Gerhard-Hess-Programm) and the Erwin Schrödinger Institute
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 355 (2003), 1559-1577
  • MSC (2000): Primary 14P15; Secondary 32B20, 53B20, 58J99
  • DOI: https://doi.org/10.1090/S0002-9947-02-03168-9
  • MathSciNet review: 1946405