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Transactions of the American Mathematical Society
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
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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 $V\cap\{q:{\rm dist\,}(q,p)=r\}$ as $r\to0$.
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.


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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
Email: grieser@mathematik.hu-berlin.de

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03168-9
PII: S 0002-9947(02)03168-9
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