A Sharp Characterization of the Willmore Invariant

2021/12/29 by Blitz, Samuel · 2 citations
#53A55 #53C18 #53C21 #58J32 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th)

paper · doi:10.48550/arxiv.2112.14378

Abstract

First introduced to describe surfaces embedded in ℝ³, the Willmore invariant is a conformally-invariant extrinsic scalar curvature of a surface that vanishes when the surface minimizes bending and stretching. Both this invariant and its higher dimensional analogs appear frequently in the study of conformal geometric systems. To that end, we provide a characterization of the Willmore invariant in general dimensions. In particular, we provide a sharp sufficient condition for the vanishing of the Willmore invariant and show that in even dimensions it can be described fully using conformal fundamental forms and one additional tensor.

Cited by

• Non-existence of higher-order conformal fundamental forms in odd dimensions
• Asymptotia of Kerr-de Sitter Black Holes

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