Reiner Schätzle

The Willmore functional

For an immersed closed surface f: ∑ → ℝ n the Willmore functional is defined by

Almost Transversal Intersections of Convex Surfaces and Translative Integral Formulae

W\left( f \right) = \frac{1}{4}\int\limits_\Sigma {{{\left| {\overrightarrow H } \right|}^2}} d{\mu _g}.

Gradient flow for the Willmore functional

Let K,L IR n be two convex bodies with non-empty interiors and with boundaries @K, @L, and let denote the Euler characteristic as defined in singular homology theory. We prove two translative integral formulas involving boundaries of convex bodies. It is shown that the integrals of the functions t 7! (@K \ (@L + t)) and t 7! (@K \ (L + t)), t 2 IR n , with respect to an n- dimensional Haar measure of IR n can be expressed in terms of certain mixed volumes of K and L. In the particular case where K and L are outer parallel bodies of convex bodies at distance r > 0, the result will be deduced from a recent (local) translative integral formula for sets with positive reach. The general case follows from this and from the following (global) topological result. Let Kr,Lr denote the outer parallel bodies of K,L at distance r 0. Establishing a conjecture of Firey (1978), we show that the homotopy type of @Kr \ @Lr and @Kr \ Lr, respectively, is independent of r 0 if K \ L 6 ; and if @K and @L intersect almost transversally. As an immediate consequence of our translative integral formulas, we obtain a proof for two kinematic formulas which have also been conjectured by Firey.

Lower semicontinuity of the Willmore functional for currents

This work consists of three parts, the first and second of which are concerned with generalized forms of two conjectures by Wm. J. Firey(1978). Let K, L ⊂ R n be compact convex sets which have common interior points and intersect almost transversally. Let Kr ,L r, r ≥ 0, denote the outer parallel bodies of K, L at distance r and set Sr := ∂Kr ∩ ∂Lr, Hr := ∂Kr ∩ Lr .I t has been conjectured by Firey that the Euler characteristic of these sets is independent of r> 0. More generally, it has been shown in (10) that Sr and S0 as well as Hr and H0 are homotopy equivalent for all r ≥ 0. In the present work, we prove that, for all r ≥ 0, Sr and S0 as well as Hr and H0 are in fact bi-lipschitz homeomorphic lipschitz submanifolds of R n . In the second part, we establish translative integral geometric formulae involving such intersections for arbitrary pairs of compact convex sets. Formulae of this type have recently been proved in (10) for compact convex sets with interior points, while very special cases of these have been conjectured by Firey. Finally, the last part is devoted to a theoretical study of general convex surfaces in stochastic geometry.