Communications in Analysis and Geometry

Volume 23 (2015)

Number 1

On geometrically constrained variational problems of the Willmore functional I: The Lagrangian-Willmore problem

Pages: 191 – 223



Yong Luo (Mathematisches Institut, Albert-Ludwigs-Universität, Freiburg, Germany; and The Collaborative Innovation Center of Mathematics, School of Mathematics and Statistics, Wuhan University, Hubei, China)

Guofang Wang (Mathematisches Institut, Albert-Ludwigs-Universität, Freiburg, Germany)


In this paper, we study a kind of geometrically constrained variational problem of the Willmore functional. A surface $l : \Sigma \to \mathbb{C}^2$ is called a Lagrangian-Willmore surface (in short, an LW surface) or a Hamiltonian-Willmore surface (in short, a HW surface) if it is a critical point of the Willmore functional under Lagrangian deformations or Hamiltonian deformations, respectively. We extend the $L^\infty$ estimates of the second fundamental form of Willmore surfaces to both HW and LW surfaces and thus get a gap theorem for both HW and LW surfaces. To investigate the existence of HW surfaces we introduce a sixth-order flow which is called by us the Hamiltonian-Willmore flow (in short, the HW flow) decreasing the Willmore energy and we prove that this flow is well posed.

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