TY - JOUR

T1 - On the variation of curvature functionals in a space form with application to a generalized Willmore energy

AU - Gruber, Anthony

AU - Toda, Magdalena

AU - Tran, Hung

N1 - Publisher Copyright:
© 2019, Springer Nature B.V.

PY - 2019/7/1

Y1 - 2019/7/1

N2 - 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.

AB - 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.

KW - Curvature functionals

KW - Surface immersions

KW - Variational problems

KW - Willmore energy

UR - http://www.scopus.com/inward/record.url?scp=85065769452&partnerID=8YFLogxK

U2 - 10.1007/s10455-019-09661-0

DO - 10.1007/s10455-019-09661-0

M3 - Article

AN - SCOPUS:85065769452

VL - 56

SP - 147

EP - 165

JO - Annals of Global Analysis and Geometry

JF - Annals of Global Analysis and Geometry

SN - 0232-704X

IS - 1

ER -