Weak immersions of surfaces with L 2 - bounded second fundamental form Lecture 2 PCMI Graduate Summer School 2013 Tristan Rivière

Weak immersions of surfaces with L 2 - bounded second fundamental form Lecture 3 PCMI Graduate Summer School 2013

Immersed Spheres of Finite Total Curvature into Manifolds

We prove that a sequence of, possibly branched, weak immersions of the two-sphere S into an arbitrary compact riemannian manifold (M, h) with uniformly bounded area and uniformly bounded L−norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a subsequence, to a Lipschitz mapping of S and whose image is made of a connected union of f...

Analysis Aspects of Willmore Surfaces

Abstract : We found a new formulation to the Euler-Lagrange equation of the Willmore functional for immersed surfaces in R. This new formulation of Willmore equation appears to be of divergence form, moreover, the nonlinearities are made of jacobians. Additionally to that, if ~ H denotes the mean curvature vector of the surface, this new form writes L ~ H = 0 where L is a well defined locally i...

Local Palais-Smale Sequences for the Willmore Functional

Using the reformulation in divergence form of the EulerLagrange equation for the Willmore functional as it was developed in the second author’s paper [Riv2], we study the limit of a local Palais-Smale sequence of weak Willmore immersions with locally square-integrable second fundamental form. We show that the limit immersion is smooth and that it satisfies the conformal Willmore equation: it is...

Triangles, Ellipses, and Cubic Polynomials

1. INTRODUCTION. Discussions that led to this paper began during an electronic version of the Secondary School Teachers Program, a part of the Park City Mathematics Institute (PCMI), offered at the University of Cincinnati in the summer of 2006 for local high school mathematics teachers. The authors were working with the teachers on daily problem sets provided by PCMI; complex numbers and polyn...