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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.

Publications

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):
22Willmore functional and Lagrangian surfaces

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Team Members

Prof. Dr. Ernst Kuwert