Willmore energy

The Willmore energy is in the differential geometry of a size which measures the bending energy of embedded surfaces in the room.

It is named after Thomas Willmore.

Definition

A smooth, embedded, compact, medium -oriented surface curvature to define the energy Willmore

Motivation

Minimal surfaces in are by definition of areas whose mean curvature vanishes.

It follows from the maximum principle that there is in no compact minimal surfaces without boundary. Instead, you look for closed surfaces which minimize the Willmore energy.

Variant

Occasionally, the Willmore energy is also

Defined with the Gaussian curvature.

Because according to the Gauss -Bonnet

Applies, the two definitions differ only by a ( abhängende of the topology of the area) constant.

Spheres

A round sphere of arbitrary radius Willmore energy. An elementary application of the inequality between arithmetic and geometric means ( together with the set of Gauss -Bonnet ) shows that for every other sphere of the Willmore energy than is greater.

Tori

Clifford tori have Willmore energy.

Thomas Willmore conjectured in 1965 that for every surface of genus inequality

Applies. A proof of this conjecture was announced in February 2012 by Fernando Coda Marques and André Neves.

Immersions

The Willmore energy can also be defined for immersions. Li and Yau proved that the Willmore energy is for each non - embedded always ized surface at least. In particular, the minimum of the Willmore energy is actually realized by always ized spheres and tori by embedded surfaces.

Forever ized projective planes, the Willmore energy is a minimum, the minimum is realized by the Bryant - Kusner parametrization of the Boy surface.