Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



Local geometry of singular real analytic surfaces

Author: Daniel Grieser
Journal: Trans. Amer. Math. Soc. 355 (2003), 1559-1577
MSC (2000): Primary 14P15; Secondary 32B20, 53B20, 58J99
Published electronically: November 18, 2002
MathSciNet review: 1946405
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


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 [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14P15, 32B20, 53B20, 58J99

Retrieve articles in all journals with MSC (2000): 14P15, 32B20, 53B20, 58J99

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

Keywords: Real analytic sets, quasi-isometry, Gauss-Bonnet theorem, $L^2$ Stokes theorem, resolution of singularities
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
Article copyright: © Copyright 2002 American Mathematical Society