Director of Institute for Mathematical Research, ETH Zürich
The study of the variations of curvature functionals takes its origins in the works of Euler and Bernouilli from the XVIIIth century on the Elastica. Since these very early times, special curves and surfaces such as geodesics, minimal surfaces, elastica, Willmore surfaces...etc have become central objects in mathematics much beyond the field of geometry stricto sensu with applications in analysis, in applied mathematics, in theoretical physics and natural sciences in general.Despite its venerable age the calculus of variations of length, area or curvature functionals for curves and surfaces is still a very active field of research with important developments that took place in the last decades.In this talk we will first make a brief tour of the various main strategies for the minmax constructions of these critical curves and surfaces in euclidian space or closed manifolds.We will start by recalling the origins of minmax methods for the length functional and present in particular the "curve shortening process" of Birkhoff. We will mention the generalization of Birkhoff`s approach to surfaces and the "harmonic map replacement" method by Colding and Minicozzi.We will comment on the contribution of the Geometric Measure Theory to the minmax theory of minimal surfaces with the seminal works of Almgren and Pitts and the recent specular applications by Marques and Neves.
In the last part of the talk we will present a new method based on smoothing arguments combined with Palais Smale deformation theory for performing successful minmax procedures for surfaces. We will present various applications of this so called "viscosity method".
Tristan Rivière 教授是瑞士联邦理工学院数学研究所 (Institute for Mathematical Research, ETH Zürich) 现任所长，1996 年获得法国 National Centre for Scientific Research (CNRS) Bronze Medal，2003年获得意大利数学学会 Gold Medal “Guido Stampacchia”，是2002年北京国际数学家大会 (ICM) 和 2016年柏林欧洲数学家大会 (ECM) 特邀报告人。