ON THE CANHAM PROBLEM: BENDING ENERGY MINIMIZING SURFACES OF ANY GENUS AND ISOPERIMETRIC RATIO

Abstract: In 1970, the biophysicist Peter CANHAM proposed that the shapes of red blood cells could be described variationally, leading to the Canham problem: find the surfaces of genus $g$ embedded in $\R^3$ that minimize their Willmore bending energy $W=\frac14 \int H^2$ with given area and enclosed volume, or equivalently (since $W$ is scale invariant) with given isoperimetric ratio $v \in (0, 1)$. Building on very recent work of Andrea MONDINO & Christian SCHARRER, we solve the “existence” part of the problem; it suffices to find a comparison surface of genus $g$ with arbitrarily small isoperimetric ratio $v$ and $W < 8π$, which we construct by gluing $g+1$ small catenoidal bridges to the bigraph of a singular solution for the linearized Willmore equation $∆(∆+2)φ = 0$ on the $(g+1)$-punctured sphere. (If time permits, we may discuss our ongoing work to understand the “small $v$” limit, as well as “uniqueness” aspects of the Canham problem.)

— Rob KUSNER, UMassAmherst & CoronavirusU

[joint work with Peter MCGRATH, NorthCarolinaStateU]

condensed mattermathematical physicsanalysis of PDEsdifferential geometrybiophysics

Audience: researchers in the topic

( paper )

---

Online Seminar "Geometric Analysis"

Series comments: We discuss recent trends related to geometric analysis in a broad sense. The general idea is to solve geometric problems by means of advanced tools in analysis. We will include a wide range of topics such as geometric flows, curvature functionals, discrete differential geometry, and numerical simulation.

Registration and links to videos available at blatt.sbg.ac.at/onlineseminar.php

---