Curvature functionals and p-Willmore energy

Date

2019-08

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Abstract

Functionals involving surface curvature are frequently encountered when modeling the behavior of important biological structures such as lipid membranes. To better understand these objects, we consider a general functional on surface immersions which is dependent on the surface mean and Gauss curvatures. Variations of this functional are presented, and stability criteria are given in terms of basic geometric invariants coming from the surface fundamental forms. These results are then applied to a particular curvature functional which generalizes the Willmore energy, and a nonexistence result is presented. A constrained minimization problem is then considered, leading to a stability result involving round spheres.

Further study is done on a generalization of the Willmore flow of surfaces in R3 -- a geometric tool known for its aesthetic beauty. In particular, two finite-element formulations of this problem are presented: one which is applicable to surfaces presented graphically, and the other which models closed immersed (possibly self-intersecting) surfaces and is amenable to constraints on surface area and enclosed volume. It is shown in both cases that the energy decreases along the flow. Moreover, stability and consistency results are obtained in the closed surface model, and examples of the implementation are discussed. Inspired by conformal geometry, a post-processing procedure is also presented, which ensures that a given surface mesh remains nearly conformal along the Willmore flow despite its initial regularity. This abolishes the mesh degeneration that usually accompanies position-based surface flows, and leads to a robust model that can accommodate variable time steps as well as surface genera.

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Keywords

Curvature functionals, Variational problems, Surface immersions, Willmore energy, Willmore flow, Mesh regularization

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