Li-Yau inequalities for the Helfrich functional and applications

TL;DR

This paper establishes a Li-Yau inequality for the Helfrich functional, linking spontaneous curvature to energy thresholds that ensure embeddedness, and demonstrates the existence of smooth minimizers in the spherical Canham-Helfrich model.

Contribution

It introduces a new Li-Yau inequality for the Helfrich functional with applications to existence of embedded minimizers in the spherical case.

Findings

Li-Yau inequality for Helfrich functional proved

Explicit energy thresholds for embeddedness derived

Existence of smooth embedded minimizers shown under certain energy conditions

Abstract

We prove a general Li-Yau inequality for the Helfrich functional where the spontaneous curvature enters with a singular volume type integral. In the physically relevant cases, this term can be converted into an explicit energy threshold that guarantees embeddedness. We then apply our result to the spherical case of the variational Canham-Helfrich model. If the infimum energy is not too large, we show existence of smoothly embedded minimizers. Previously, existence of minimizers was only known in the classes of immersed bubble trees or curvature varifolds.