No abstract uploaded!

We prove the existence of a smooth minimizer of the Willmore energy in the class of conformal immersions of a given closed Riemann surface into IRn, n = 3, 4, if there is one conformal immersion with Willmore energy smaller than a certain bound Wn,p depending on codimension and genus p of the Riemann surface. For tori in codimension 1, we know W3,1 = 8 .

No abstract uploaded!

We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov–Hausdorff distance to the nearest cone. The monotonicity formulas are related to the classical Bishop–Gromov volume comparison theorem and Perelman’s celebrated monotonicity formula for the Ricci flow. We will explain the connection between all of these.

We consider a 1-parameter family of strictly convex hypersurfaces in Rn+1 moving with speed −Kαν, where ν denotes the outward-pointing unit normal vector and α⩾1/(n+2). For α>1/(n+2), we show that the flow converges to a round sphere after rescaling. In the affine invariant case α=1/(n+2), our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling.