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.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.
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 -