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

## TL;DR

This paper investigates a generalized curvature functional in 3D space forms, deriving its variations and stability conditions, with a focus on spheres, inspired by applications in physics and biology.

## Contribution

It introduces a new generalized curvature functional, computes its first and second variations, and analyzes the stability of spheres within this framework.

## Key findings

- Derived explicit formulas for first and second variations of the functional.
- Proved stability criteria for spheres under the generalized functional.
- Established connections to physical models like lipid bilayers.

## 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, provided the surface is immersed in a 3-D space form. We compute the first and second variations of this functional, leading to expressions given entirely in terms of the surface fundamental forms. We then apply the stability criteria afforded by our calculations to a generalization of the Willmore functional, proving a result regarding the stability of spheres.