Willmore functional and Lagrangian surfaces

Let \(\Sigma\) be a surface and \(f\) an immersion from \(\Sigma\) to \({\mathbb R}^n\). The Willmore functional is defined as 

\({\mathcal W}(f)=\frac{1}{4}\int_\Sigma\mid H\mid^2d\mu_f\),

where \(H\) is the mean curvature vector of \(f\) defined by \(H= tr A\), and \(d\mu_f\) is the area element on \(\Sigma\) induced by \(f\).

In this project, we will consider the Willmore functional for surfaces in \({\mathbb C}^2={\mathbb R}^4\) under a Lagrange constraint. This is initiated by Minicozzi (1993) who proved the existence of smooth minimzers of the Willmore functional among closed Lagrangian tori in \({\mathbb C}^2\) and posed the conjecture that the Clifford torus, which is a Lagrangian torus in \({\mathbb C}^2\), minimizes the Willmore energy among Lagrangian tori. Clearly, the n=4 Willmore conjecture implies the Lagrangian Willmore conjecture, which is also open. The objective of this project is to establish analytical tools to study the Willmore functional along Lagrangian surfaces.


In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the (n+1)-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for n=2 we obtain a Minkowski-type inequality and for n=3 we obtain an optimal Willmore-type inequality. To prove these estimates, we employ a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.


Related project(s):
22Willmore functional and Lagrangian surfaces

In this paper we prove that any smooth surfaces can be locally isometrically embedded into as Lagrangian surfaces. As a byproduct we obtain that any smooth surfaces are Hessian surfaces.


JournalAnnales de l'Institut Henri Poincare (C) Non Linear Analysis
Link to preprint version
Link to published version

Related project(s):
22Willmore functional and Lagrangian surfaces

  • 1

Team Members

Prof. Dr. Ernst Kuwert
Project leader
Albert-Ludwigs-Universität Freiburg im Breisgau

Prof. Dr. Guofang Wang
Project leader
Albert-Ludwigs-Universität Freiburg im Breisgau

This website uses cookies

By using this page, browser cookies are set. Read more