ICTP Mathematics Seminar - Tuesday 23, January, at 14:00 (Karen CORRALES ESCALONA)

[ICTP Math Section]

M A T H E M A T I C S   S E M I N A R S  2018


Tuesday 23, January, at 14.00 hrs.


Karen Corrales Escalona (ICTP)


Title: Singularities of mean curvature flow and surgery


Abstract: Let γ : M × [0.T ) → Rn+1 be a family of smooth immersions of 
an n−dimensional manifold M evolving by mean curvature flow. When M is 
compact, it is well known that the mean curvature flow is defined up to 
a finite singular time T at which the curvature of the hypersurface 
becomes unbounded. For instance, if the initial surface is convex, the 
evolving surfaces become spherical and contract to a point. On the other 
hand, if the initial surface is not convex, the evolution may become 
singular without shrinking. The study of singularities for the mean 
curvature flow is of great interest but, also in the last decades 
several methods have been developed to continue the flow beyond the 
first singular time. Huisken and Sinestrari considered a new approach 
based on a surgery procedure to extend the mean curvature flow after 
singularities. This construction was inspired by a procedure originally 
introduced by Hamilton for the Ricci flow which played a fundamental 
role in the work by Perelman to prove the Poincaré and Geometrization 
conjectures. The results about singularities and surgery for the mean 
curvature flow were obtained for manifolds of codimension 1. In this 
case the comparison principle is fundamental which, among others things, 
ensures that the embeddedness is preserved. In contrast, the comparison 
principle cannot be applied for higher codimension manifolds making more 
difficult the study of such evolution. In this talk I will introduce the 
types of singularities as well as the procedure of surgery for mean 
curvature flow. We are going to discuss some new results concerning the 
simplest case in higher codimension, space curves.


VENUE:  Stasi Seminar Room (ICTP Leonardo da Vinci Building, first floor)

http://indico.ictp.it/event/8482/