Asymptotic Estimates for the Willmore Flow With Small Energy

Kuwert and Schätzle showed in 2001 that the Willmore flow converges to a standard round sphere, if the initial energy is small. In this situation, we prove stability estimates for the barycenter and the quadratic moment of the surface. Moreover, in codimension one, we obtain stability bounds for the enclosed volume and averaged mean curvature. As direct applications, we recover a quasi-rigidity estimate due to De Lellis and Müller (2006) and an estimate for the isoperimetric deficit by Röger and Schätzle (2012), whose original proofs used different methods.

Quantitative estimates for bending energies and applications to non-local variational problems

We discuss a variational model, given by a weighted sum of perimeter, bending and Riesz interaction energies, that could be considered as a toy model for charged elastic drops. The different contributions have competing preferences for strongly localized and maximally dispersed structures. We investigate the energy landscape in dependence of the size of the ‘charge’, that is, the weight of the Riesz interaction energy.In the two-dimensional case, we first prove that for simply connected sets of small elastica energy, the elastica deficit controls the isoperimetric deficit. Building on this result, we show that for small charge the only minimizers of the full variational model are either balls or centred annuli. We complement these statements by a non-existence result for large charge. In three dimensions, we prove area and diameter bounds for configurations with small Willmore energy and show that balls are the unique minimizers of our variational model for sufficiently small charge.

Reverse Bonnesen-style inequalities on surfaces of constant curvature

This paper deals with the isoperimetric deficit upper bound for the convex domain in a surface [Formula: see text] of constant curvature [Formula: see text] by the containment measure of a convex domain to contain another convex domain in integral geometry. Some reverse Bonnesen-style inequalities are obtained. In particular, two of them strengthen Zhou’s result in [Formula: see text] and Bottema’s result in the Euclidean plane [Formula: see text].