Isoperimetric Problem and Willmore Problem
by
Antonio Ros
Universidad de Granada, Spain

Plenary Lecture

Given a 3-dimensional ambient space, the isoperimetric problem consists of studying the surfaces of least area among those which enclose a region of prescribed volume. In many important cases, these isoperimetric surfaces exist and we would like, for instance, to control their topology (we shall do it that in some cases) and, if the ambient space is simply enough, we want to know explicitely the isoperimetric solutions (that can be solved only in very few situations).

Another interesting question in classical Differential Geometry is the study of the functional \int_M H² , where M is a compact surface in R³ and H is its mean curvature. We want to know the infimum of the functional when M has some prescribed topology or lies in a fixed isotopy class. This infimum is unknown even in the case that M is a torus (the Willmore conjecture asserts that, for tori, that value is 2 ^2).

Both problems have received attention of geometers. We shall explain some ofthe ideas used in the study of these questions putting the emphasis in theconnections between them. We will give some results on the isoperimetric problem in the positive curvature case and as consequence we shall prove the Willmore conjecture for tori in R³ admitting a central symmetry.

Date received: May 29, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-77.