Maximum principle for tensors and Ricci flow invariant curvature conditions

Maximum principle for tensors and Ricci flow invariant curvature conditions

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Yury Ustinovskiy , Princeton University
Fine Hall 110

Maximum principle is an important feature of parabolic/elliptic PDEs having numerous applications. In particular, it allows to bound solutions to these equations by the initial/boundary values. The Ricci flow is an evolution equation for a Riemannian metric resembling parabolic heat equation and it is highly desirable to have analogues of maximum principle for various geometric objects (primarily curvature tensor) associated to a metric. In 1986 Richard Hamilton proved a general maximum principle for tensors, which allowed to find many conditions on curvature tensor preserved under the Ricci flow. We discuss Hamilton's approach and formulate the most general approach to Ricci flow invariant curvature conditions formulated in 2013 by Burkhard Wilking.