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 mainly consider the relative isoperimetric inequalities for minimal submanifolds in R*n*+*m*. We first provide, following Cabré \cite{Cabre2008}, an ABP proof of the relative isoperimetric inequality proved in Choe-Ghomi-Ritoré \cite{CGR07}, by generalizing ideas of restricted normal cones given in \cite{CGR06}. Then we prove a relative isoperimetric inequalities for minimal submanifolds in R*n*+*m*, which is optimal when the codimension *m*≤2. In other words we obtain a relative version of isoperimetric inequalities for minimal submanifolds proved recently by Brendle \cite{Brendle2019}. When the codimension *m*≤2, our result gives an affirmative answer to an open problem proposed by Choe in \cite{Choe2005}, Open Problem 12.6. As another application we prove an optimal logarithmic Sobolev inequality for free boundary submanifolds in the Euclidean space following a trick of Brendle in \cite{Brendle2019b}.

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

Abstract: For an immersed Lagrangian submanifold, let \check{A} be the Lagrangian trace-free second fundamental form. In this note we consider the equation \nabla^*T=0 on Lagrangian surfaces immersed in \mathbb{C}^2, where T=-2\nabla^*(\check{A}\lrcornerω), and we prove a gap theorem for the Whitney sphere as a solution to this equation.

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

In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, provided the initial data is star-shaped.

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

In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. This solves completely a long-standing open problem. In the proof one of crucial ingredients is a new Minkowski type formula. We also prove a Heintze-Karcher-Ros type inequality for hypersurfaces in a ball, which, together with the new Minkowski formula, yields a new proof of Alexandrov's Theorem for embedded CMC hypersurfaces in a ball with free boundary.

Journal | Math. Ann. |

Volume | 374 |

Pages | 1845--1882 |

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

In this note, we first introduce a boundary problem for Lagrangian submanifolds, analogous to the free boundary hypersurfaces and capillary hypersurfaces. Then we present some interesting minimal Lagrangian submanifolds examples satisfying this boundary condition and we prove a Lagrangian version of Nitsche (or Hopf) type theorem. Some problems are proposed at the end of this note.

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

We study immersed surfaces in R3 which are critical points of the Willmore functional under boundary constraints. The two cases considered are when the surface meets a plane orthogonally along the boundary, and when the boundary is contained in a line. In both cases we derive weak forms of the resulting free boundary conditions and prove regularity by reflection.

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

For the Willmore flow of spheres in R^n with small energy, we prove stability estimates for the barycenter, the quadratic moment, and in case n=3 also for the enclosed volume and averaged mean curvature. As applications, we give a new proof for a quasi-rigidity estimate due to De Lellis and Müller, also for an inequality by Röger and Schätzle for the isoperimetric deficit.

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

Let M be a compact Riemannian manifold which does not admit any immersed surface which is totally geodesic. We prove that then any completely immersed surface in M has area bounded in terms of the L^2 norm of the second fundamental form.

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

Journal | SIAM J. Math. Anal. |

Volume | 50 |

Pages | 4407--4425 |

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**Related project(s):****22**Willmore functional and Lagrangian surfaces**25**The Willmore energy of degenerating surfaces and singularities of geometric flows

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 |

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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