Contents Online

# 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

DOI: http://dx.doi.org/10.4310/CAG.2015.v23.n1.a6

#### Authors

#### Abstract

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.