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ISSN 1088-6850(e) ISSN 0002-9947(p)

Local geometry of singular real analytic surfaces

Author(s): Daniel Grieser
Journal: Trans. Amer. Math. Soc. 355 (2003), 1559-1577.
MSC (2000): Primary 14P15; Secondary 32B20, 53B20, 58J99
Posted: 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 is equipped with a Riemannian metric that is induced from some analytic Riemannian metric on $\mathbb{R} ^N$ . We prove:

1.: Each point of 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 is the coefficient of the linear term in the expansion of (2).
4.: The Stokes Theorem, selfadjointness and discreteness of the Laplace-Beltrami operator on , an estimate on the heat kernel, and a Gauss-Bonnet Theorem for the Euler characteristic.

As a central tool we use resolution of singularities.

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