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Annals of Global Analysis and Geometry

, Volume 56, Issue 1, pp 147–165 | Cite as

On the variation of curvature functionals in a space form with application to a generalized Willmore energy

  • Anthony GruberEmail author
  • Magdalena Toda
  • Hung Tran
Article
  • 43 Downloads

Abstract

Functionals involving surface curvature are important across a range of scientific disciplines, and their extrema are representative of physically meaningful objects such as atomic lattices and biomembranes. Inspired in particular by the relationship of the Willmore energy to lipid bilayers, we consider a general functional depending on a surface and a symmetric combination of its principal curvatures, and provided the surface is immersed in a 3-D space form of constant sectional curvature. We calculate the first and second variations of this functional, extending known results and providing computationally accessible expressions given entirely in terms of the basic geometric information found in the surface fundamental forms. Further, we motivate and introduce the p-Willmore energy functional, applying the stability criteria afforded by our calculations to prove a result about the p-Willmore energy of spheres.

Keywords

Curvature functionals Variational problems Surface immersions Willmore energy 

Mathematics Subject Classification

58E12 53C42 53A05 53A35 

Notes

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsTexas Tech UniversityLubbockUSA

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