Local Geometry of Singular Real Analytic Surfaces

Abstract

Let V be a compact real analytic surface with isolated singularities embedded in $R^N$, and assume its smooth part is equipped with a Riemannian metric that is induced from some analytic Riemannian metric on $R^N$. 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 $V\cap\{q:\dist(q,p)=r\}$ as r tends to zero. 3. A Gauss-Bonnet Theorem, saying that horns do not contribute an extra term, while cones contribute the leading coefficient in the length expansion of 2. 4. The $L^2$ Stokes Theorem, self-adjointness and discreteness of the Laplace-Beltrami operator on the smooth part, and a Gauss-Bonnet Theorem for the $L^2$ Euler characteristic. As a central tool we use resolution of singularities.