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Atlanta, GA | Posted:
February 19, 2004 *

Professor Gerhard Huisken of the Albert-Einstein-Institut für Gravitationsphysik of the Max-Planck Society, will deliver this year's Stelson Lecture on February 19th. Professor Huisken, a master expositor and expert on curvature driven flows, will describe in two lectures the striking recent developments in curvature flow and the situations in which they apply, including relations to Perelman's recent proof of the Poincaré conjecture. Huisken's first lecture, Classifying manifolds and hypersurfaces by geometric evolution equations, is intended for a general audience. If every curve on a surface can shrink to a point, without leaving the surface or breaking, then the surface itself can be deformed, without tearing, into a round sphere. It has been known, at least since the 1970's, that surfaces can be deformed in interesting ways by moving each point on the surface in a way that depends on the curvature at that point. Until very recently, no general theory relating curvature driven deformations and deformations into special surfaces was known. A central problem arising in making the connection involves the development of singularities (kinds of tears or breaks). By understanding which kinds of singularities can occur, and how they occur, deep and surprising results emerge. The singularities actually give important information about how the surfaces fit together. Gerhard Huisken, recent recipient of the Leibniz Prize, works in the interface between differential geometry and general relativity. He is presently a director at the Albert-Einstein-Institut for Gravitational Physics.

Parabolic partial differential equations and systems can be used to deform a given geometrical object into a "nicer" or even canonical representative inside a certain class. Examples for this phenomenon are Hamilton's flow of Riemannian metrics by their Ricci curvature and the mean curvature flow of hypersurfaces. Recently the reach of these deformations has been greatly enhanced by controlling and extending them beyond singularities. The lecture explains recent work by Huisken and Sinestrari on surgery in mean curvature flow and explains the relation to the work of Perelman on the Ricci flow.

The lecture explains how the geometry of necks in a hypersurface can be controlled by a priori estimates for the curvature and then gives an explicit surgery construction. Finally it is shown how the surgery can be used to extend mean curvature flow beyond singularities for hypersurfaces with the sum of the two lowest principal curvature positive everywhere.