The decay of the Ricci curvature at a real hypersurface singularity
نویسنده
چکیده
The Ricci curvature at an isolated singularity of an immersed hypersurface exhibits a local behaviour echoing a global property of the Ricci curvature on a complete hypersurface in euclidean space (i.e., the Efimov theorem [12], that sup Ric ≥ 0). This local behaviour takes the form of a near universal bound on the decay of the Ricci curvature at a simple singularity (eg. a cone singularity) in a real hypersurface in R, m ≥ 3 (Theorem 1). The bound is dimension-free, sharp and may be universal. If the bound is not universal, such exceptions as occur must be even-dimensional hypersurfaces with profiles (intersections with small spheres centred at the singularity) which are topological spheres admitting — by a canonical construction (Remark 3, §4) — Riemannian metrics with positive curvature operator. Thus any exceptional singularity would be topologically trivial and if, as long suspected, there are no exotic spheres with positive curvature operator then it would also be differentiably trivial.
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