Henri Anciaux

On the three-dimensional Blaschke-Lebesgue problem

The width of a closed convex subset of n-dimensional Euclidean space is the distance between two parallel supporting hyperplanes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension n � 3. In this paper we describe a necessary condition that the minimizer of the Blaschke-Lebesgue must satisfy in dimension n = 3: we prove that the smooth components of the boundary of the minimizer have their smaller principal curvature constant, and therefore are either spherical caps or pieces of tubes (canal surfaces). 2000 MSC: 52A40, 52A15

Cyclic and ruled Lagrangian surfaces in Euclidean four space

We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve in the 3-sphere or a Legendrian curve in the anti-de Sitter 3-space. We describe ruled Lagrangian surfaces and characterize the cyclic and ruled Lagrangian surfaces which are solutions to the self-similar equation of the Mean Curvature Flow. Finally, we give a partial result in the case of Hamiltonian stationary cyclic surfaces.

Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Abstract Given an oriented Riemannian surface ( Σ , g ) , its tangent bundle T Σ enjoys a natural pseudo-Kahler structure, that is the combination of a complex structure J , a pseudo-metric G with neutral signature and a symplectic structure Ω . We give a local classification of those surfaces of T Σ which are both Lagrangian with respect to Ω and minimal with respect to G . We first show that if g is non-flat, the only such surfaces are affine normal bundles over geodesics. In the flat case there is, in contrast, a large set of Lagrangian minimal surfaces, which is described explicitly. As an application, we show that motions of surfaces in R 3 or R 1 3 induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in T S 2 or T H 2 respectively. We relate the area of the congruence to a second-order functional F = ∫ H 2 − K d A on the original surface.

MARGINALLY TRAPPED SUBMANIFOLDS IN SPACE FORMS WITH ARBITRARY SIGNATURE

We give explicit representation formulas for marginally trapped submanifolds of co-dimension two in pseudo-Riemannian spaces with arbitrary signature and constant sectional curvature. This paper is dedicated to the memory of Franki Dillen, 1963-2013.