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Inverse Curvature Flows in Hyperbolic Space

Student-Run Geometry/Topology Seminar

Speaker: Brian Harvie, UC Davis
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Location: 2112 MSB
Start time: Thu, Dec 6 2018, 12:10PM

Geometric flows have proven themselves in recent decades to be powerful tools in geometry, topology, and mathematical physics. Geometric flows are nonlinear, (generally) parabolic PDEs which arise in two flavors: intrinsic flows, which describe the evolution of some Riemannian metric according to its curvature tensor, and extrinsic flows, which describe the motion of submanifolds, typically hypersurfaces, in a Riemannian manifold according to to their principal curvatures at each point. An example of the former is Hamilton's Ricci Flow, famous for its role in Perelman's proof of the Poincare Conjecture, and an example of the latter is Inverse Mean Curvature Flow (IMCF), which will be the main focus of this presentation.

IMCF has produced some purely geometric results, but it is probably best known for its application to mathematical problems in General Relativity, specifically in proving the Riemannian Penrose Inequality (RPI). Although IMCF was recently shown to imply the RPI over asymptotically flat Riemannian Manifolds, it is still an open question whether IMCF possesses the proper long-time existence and convergence properties to prove the RPI over a different family of spaces, namely asymptotically hyperbolic manifolds. Understanding the behavior of this flow over these manifolds begins with further study of it over hyperbolic space. This talk will include an introduction to extrinsic flows, then examine some of the history, motivation, and elementary properties of IMCF, and finally close with new work on IMCF over non-star-shaped surfaces in hyperbolic space.