The squared-mean-curvature integral was introduced two centuries
ago by Sophie Germain to model the bending energy and vibration patterns of
thin elastic plates. By the 1920s the Hamburg geometry school realized
this energy is invariant under the Möbius group of conformal
transformations, and thus that minimal surfaces in R^3 or S^3 are
equilibria. In 1965 Willmore observed that the round sphere minimizes the
bending energy among all closed surfaces, and he conjectured that a certain
torus of revolution - the stereographic projection of the Clifford minimal
torus in S^3 - minimizer for surfaces of genus 1. This conjecture was
proven this spring by Fernando Coda and Andre Neves; they use the
Almgren-Pitts minimax construction, the Hersch-Li-Yau notion of conformal
area, and Urbano's characterization of the Clifford torus by its Morse
index. We will discuss what is known or conjectured for other topological
types of bending-energy minimizing or equilibrium surfaces, and how
bending-energy gradient flow might be applied to problems in
low-dimensional topology.

We study curvature estimates of Ricci flow on complete 3-dim manifolds
without bounded curvature assumptions. Especially, from a more general
curvature preservation condition, we derived that nonnegative Ricci
curvature is preserved for any complete solution of 3-dim Ricci flow. A local
version of Hamilton-Ivey estimates is also obtained. Using that the nonnegative
Ricci is preserved under any 3-dim Ricci flow complete solution, we can prove the strong uniqueness of the Ricci flow with bounded nonnegative Ricci curvature and uniform injective radius lower bound as initial assumptions. This is joint work with Bing-Long Chen and Zhuhong Zhang.

We consider non-infinitesimal deformations of G2-structures on 7-dimensional
manifolds and derive a closed expression for the torsion of the deformed
G2-structure. We then specialize to the case where the deformation lies in
the seven-dimensional representation of G2 and is hence defined by a vector
v. In this case, we explicitly derive the expressions for the different
torsion components of the new G2-structure in terms of the old torsion
components and derivatives of v. In particular this gives a set of
differential equations for the vector v which have to be satisfied for a
transition between G2-structures with particular torsions. For some specific
torsion classes we then explore the solutions of these equations.