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.

## Publications

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):****22**Willmore 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.

Journal | Annales de l'Institut Henri Poincare (C) Non Linear Analysis |

Link to preprint version | |

Link to published version |

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

## Team Members

**Prof. Dr. Ernst Kuwert**

Project leader

Albert-Ludwigs-Universität Freiburg im Breisgau

ernst.kuwert(at)math.uni-freiburg.de

**Prof. Dr. Guofang Wang**

Project leader

Albert-Ludwigs-Universität Freiburg im Breisgau

guofang.wang(at)math.uni-freiburg.de