Local geometry of singular real analytic surfaces

Grieser, Daniel

doi:10.1090/S0002-9947-02-03168-9

Local geometry of singular real analytic surfaces
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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:

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