Ricci flow from metrics with isolated conical singularities
Geometric Flows in Riemannian and Complex Geometry May 02, 2016  May 06, 2016
Location: SLMath: Eisenbud Auditorium
complex geometry
Riemannian geometry
geometric analysis
geometric flow
Ricci flow
singularities of flows
conical singularities
rescaling of solutions
positive curvature
51D25  Lattices of subspaces and geometric closure systems [See also 05B35]
51D30  Continuous geometries, geometric closure systems and related topics [See also 06Cxx]
51A40  Translation planes and spreads in linear incidence geometry
14495
Let $(M,g_0)$ be a compact ndimensional Riemannian manifold with a finite number of singular points, where at each singular point the metric is asymptotic to a cone over a compact (n1)dimensional manifold with curvature operator greater or equal to one. We show that there exists a smooth Ricci flow starting from such a metric with curvature decaying like C/t. The initial metric is attained in GromovHausdorff distance and smoothly away from the singular points. To construct this solution, we desingularize the initial metric by glueing in expanding solitons with positive curvature operator, each asymptotic to the cone at the singular point, at a small scale s. Localizing a recent stability result of DeruelleLamm for such expanding solutions, we show that there exists a solution from the desingularized initial metric for a uniform time T>0, independent of the glueing scale s. The solution is then obtained by letting s>0. We also show that the so obtained limiting solution has the corresponding expanding soliton as a forward tangent flow at each initial singular point. This is joint work with P. Gianniotis
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