A strict inequality for the minimisation of the Willmore functional under isoperimetric constraint

Inspired by previous work of Kusner and Bauer-Kuwert, we prove a strict inequality between the Willmore energies of two surfaces and their connected sum in the context of isoperimetric constraints. Building on previous work by Keller-Mondino-Rivi\`ere, our strict inequality leads to existence of minimisers for the isoperimetric constrained Willmore problem in every genus, provided the minimal energy lies strictly below $8\pi$. Besides the geometric interest, such a minimisation problem has been studied in the literature as a simplified model in the theory of lipid bilayer cell membranes.Comment: 16 pages. Final version to appear in Advances in Calculus of Variation

On the minimisation of bending energies related to the Willmore functional under constraints on area and volume

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 2-sphere into the 3-space. Such a family of weak immersions can be parametrised by a single continuous map on the 2-sphere. They showed pre-compactness and continuity of the area functional on the class of bubble trees under weak convergence. In this thesis, we prove lower semi-continuity 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 bi-harmonic 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

Partielle Differentialgleichungen

The workshop dealt with partial differential equations in geometry and technical applications. The main topics were the combination of nonlinear partial differential equations and geometric measure theory, conformal invariance and the Willmore functional, and regularity of free boundaries

Coupled Evolution Equations for Immersions of Closed Manifolds and Vector Fields

A possible description of the elastic energy of a biological membrane is given by the L 2 -distance of its mean curvature from a spontaneous curvature plus a topological constant. Such energy functional is often referred to as Helfrich energy. We study a generalization of this model, where the spontaneous curvature arises as the divergence of a vector field along the surface. An abstract formulation for immersions of manifolds of arbitrary dimension is derived. For plane curves we prove for this energy functional the existence of global minimizers and regularity of stationary points subject to different constraints. The constraints we considered are the length and enclosed signed area of the curve and the range of the vector field. Furthermore, we derive a gradient-flow equation in the general situation of immersions of manifolds of arbitrary space dimension which leads to a coupled system of partial differential equations. For this coupled system we show local well-posedness even for a constraint preserving adaption of the flow. Moreover, we show a Łojasiewicz-Simon gradient inequality for the unrestricted functional as well as in the presence of constraints. From this we draw conclusions about the asymptotic behavior of the flow close to a local minimizer. For curves and vector fields in the euclidean plane we introduce a geometric quantity whose smallness guarantees smoothness of the flow. By a rescaling argument we achieve that even finiteness of this quantity suffices to exclude the formation of singularities

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described