Beijing Institute of Mathematical Sciences and Applications

Helein's Convergence Theorem is a powerful tool for solving variational problems related to the Willmore function. The theorem asserts that a conformal immersion from a 2-disk in $R^n$ with small $L^2$-norm of the second fundamental form, will either converge to a conformal immersion or collapse to a point. When the collapse occurs, it is natural to hope that a rescaled sequence will converge to a conformal immersion. However, this is not generally true. In this talk, we will present sufficient conditions under which a rescaled sequence converges to a conformal immersion when the $L^2$-norm of the second fundamental form is small. Furthermore, we will establish an intrinsic version of Helein's Convergence Theorem.

In this talk we will show how the DPW method in integrable system can be used in the study of Willmore surfaces in spheres. Moreover, some geometric properties of Willmore surfaces from the DPW methods, including characterizations of minimal surfaces in space forms, Willmore surfaces with symmetries, etc. Some interesting new examples of Willmore surfaces can be derived in this way. This is based on joint works with Prof. Dorfmeister and Prof. Changping Wang.

We will recall the description of conformal maps and the description of harmonic maps in terms of Minkowski frames and loops of Minkowski frames respectively. Starting from a result of Burstall and Calderbank we will outline a generalized Weierstrass representation (= generalized loop group method) for constrained Willmore surfaces. This is joint work with Idrisse Khemar.

In this talk, we show the first and second variational formulas of the Willmore functional for closed surfaces in 4-dimensional conformal manifolds. As an application, the Clifford torus in CP^2 is proved to be strongly Willmore-stable. This provides a strong support to the conjecture of Montile and Urbano, which states that the Clifford torus in CP^2 minimizes the Willmore functional among all tori. In 4-dimensional locally symmetric spaces, by constructing some holomorphic differentials, we prove that among all minimal 2-spheres only those super-minimal ones can be Willmore. This is a joint work with Prof. Changping Wang.

Homogeneous and Delaunay tori are the only embedded tori in the 3-sphere of constant mean curvature. By a result of Schätzle and Ndiaye, the constrained Willmore minimizers of rectangular conformal classes near the square class are homogeneous. At discrete values, the family of homogeneous tori allows bifurcation into n-lobed Delaunay tori. In this talk, I show that the 2-lobed Delaunay tori are stable as constrained Willmore surfaces in 3-space and minimizes the Willmore energy in the class of isothermic surfaces.