Local Geometry of Singular Real Analytic Surfaces

Let V ⊂ R be a compact real analytic surface with isolated singularities, and assume its smooth part V0 is equipped with a Riemannian metric that is induced from some analytic Riemannian metric on R . 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 ∈ V , of the length of V ∩{q : dist (q, p) = r} as r → 0. 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 Stokes Theorem, self-adjointness and discreteness of the Laplace-Beltrami operator on V0, and a Gauss-Bonnet Theorem for the L Euler characteristic. As a central tool we use resolution of singularities.