Li-Yau inequalities for the Helfrich functional and applications

Area of Science:

• Mathematical physics
• Differential geometry
• Materials science

Background:

• The Helfrich functional is crucial for understanding membrane mechanics.
• Spontaneous curvature introduces complexities in energy minimization.
• Previous existence results for minimizers were limited.

Purpose of the Study:

• To generalize the Li-Yau inequality for the Helfrich functional.
• To establish conditions for embeddedness based on spontaneous curvature.
• To prove the existence of smoothly embedded minimizers for the Canham-Helfrich model.

Main Methods:

• Derivation of a general Li-Yau inequality.
• Analysis of singular volume integrals related to spontaneous curvature.
• Application to the spherical Canham-Helfrich model.

Main Results:

• A generalized Li-Yau inequality for the Helfrich functional is proven.
• An energy threshold for embeddedness is derived from spontaneous curvature.
• Existence of smoothly embedded minimizers is shown for the Canham-Helfrich model when infimum energy is sufficiently small.

Conclusions:

• The study provides a rigorous mathematical framework for membrane mechanics.
• The results advance the understanding of shape determination in lipid bilayers.
• This work offers a new perspective on the existence of physically relevant membrane configurations.