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On the minimisation of bending energies related to the Willmore functional under constraints on area and volume
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Scharrer, Christian (2021) On the minimisation of bending energies related to the Willmore functional under constraints on area and volume. PhD thesis, University of Warwick.

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Official URL: http://webcat.warwick.ac.uk/record=b3763873
Abstract
The Willmore energy of a closed surface is defined as the integrated squared mean curvature. It appears in many areas of science and technology in current research. A slight variation, known as Canham–Helfrich functional, is obtained as a linear combination of the Willmore functional, the total mean curvature and the area. The Canham–Helfrich energy is the associated bending energy of a lipid bilayer cell membrane. Its minimisation amongst closed spherical surfaces with given fixed area and volume is referred to as the Helfrich problem. Minimising purely the Willmore functional in the class of surfaces with given fixed genus, while keeping the constraints on area and volume will be referred to as the Canham problem. By the scaling invariance of the Willmore functional, the two constraints on area and volume reduce to a single constraint on the scaling invariant isoperimetric ratio.
This thesis presents results that substantially contributed in fully solving existence of minimisers for both the Helfrich and Canham problems. Previously, Mondino–Rivière developed the notion of bubble tree which is a finite family of weak possibly branched immersions of the 2sphere into the 3space. Such a family of weak immersions can be parametrised by a single continuous map on the 2sphere. They showed precompactness and continuity of the area functional on the class of bubble trees under weak convergence. In this thesis, we prove lower semicontinuity of the Canham–Helfrich functional as well as continuity of the volume under weak convergence of bubble trees, leading to existence of minimisers. Moreover, we show that critical bubble trees are smooth outside of finitely many branch points. In fact, the regularity result holds true for critical surfaces of any genus.
In the early 2000s, Bauer–Kuwert proved a strict inequality between the Willmore energies of two surfaces and their connected sum leading to existence of minimisers for the Willmore functional with any prescribed genus. In the proof they use biharmonic interpolation in order to patch an inverted surface into a huge copy of a second surface. Using their connected sum construction, we show that the same inequality remains valid in the context of isoperimetric constraints. In a first step, we determine the precise order of convergence for the isoperimetric ratio of the connected sum under scaling (up) of the second surface. Then, inspired by Huisken’s volume preserving mean curvature flow, we show that the small isoperimetric deficit can be adjusted using a variational vector field supported away from the patching region. By a previous result of Keller–Mondino–Rivière, our strict inequality leads to existence of minimisers for the Canham problem, provided the minimal energy lies strictly below 8_.
In order to complete the existence part for the Canham problem in the genus one case, we construct rotationally symmetric tori consisting of two opposite signed constant mean curvature surfaces. The tori converge as varifolds to a double round sphere. Using complete elliptic integrals, we show that the resulting family can be used to obtain comparison tori of any isoperimetric ratio with Willmore energy strictly below 8_.
Additionally, we prove a general Li–Yau inequality for varifolds on Riemannian manifolds by testing the first variation identity against vector fields which are proportional to the gradient of the distance function.
Item Type:  Thesis (PhD)  

Subjects:  Q Science > QA Mathematics  
Library of Congress Subject Headings (LCSH):  Curvature, Curves on surfaces, Convergence, Varifolds, Riemannian manifolds  
Official Date:  18 October 2021  
Dates: 


Institution:  University of Warwick  
Theses Department:  Mathematics Institute  
Thesis Type:  PhD  
Publication Status:  Unpublished  
Supervisor(s)/Advisor:  Mondino. Andrea  
Sponsors:  Engineering and Physical Sciences Research Council  
Format of File:  
Extent:  [vi], 127 leaves : illustrations  
Language:  eng 
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