arXiv subjects

Christian Scharrer

Publications and source records attributed to Christian Scharrer.

At least 19 recordsLinked to original sources

arXiv2026

Gradient flow dynamics for cell membranes in the Canham-Helfrich model

The energetically most efficient way how a deformed red blood cell regains equilibrium is mathematically described by the gradient flow of the Canham-Helfrich functional, including a spontaneous curvature and the conservation of surface area and enclosed volume. Using a recently discovered multiplicity inequality, we prove global existence and convergence of smooth solutions for spheres and axisymmetric tori, provided the initial energy lies below explicit thresholds.

math.AP↗

arXiv2026

Delaunay surfaces with free boundary on the unit $2$-sphere

We characterize all symmetric, embedded surfaces of revolution with constant mean curvature that meet the unit sphere orthogonally. It is well known that there exists a unique free boundary catenoid inside of the unit ball. By analogy, we prove the existence of a unique free boundary nodoid inside of the unit ball with constant mean curvature equal to $-1$. Moreover, there exists a unique compact free boundary nodoid \emph{outside} of the unit ball with constant mean curvature equal to $-1$. However, there exists no symmetric surface of revolution with constant mean curvature $1$ that meets the unit sphere orthogonally.

math.DG↗

arXiv2026

Existence and convergence of the area-constrained elastic flow

We study the evolution of plane closed curves with fixed area moving by the negative $L^2$-gradient of their elastic energy. For smooth initial data, we establish local and global existence of the flow. By imposing a simplicity assumption and an initial energy bound, we show that the length of the evolving curve remains uniformly bounded. This yields subconvergence to a critical point, which is then improved to full convergence by utilizing a Łojasiewicz--Simon inequality. Conversely, an analysis of the energy profile curve, which maps a given length to the minimal energy among all curves with that length and fixed area, reveals that the length diverges to infinity for initial data satisfying specific length and energy criteria. We visualize our findings through numerical simulations.

math.AP↗

arXiv2026

Global regularity of integral 2-varifolds with square integrable mean curvature

We provide sharp sufficient criteria for an integral $2$-varifold to be induced by a $W^{2,2}$-conformal immersion of a smooth surface. Our approach is based on a fine analysis of the Hausdorff density for $2$-varifolds with critical integrability of the mean curvature and a recent local regularity result by Bi-Zhou. In codimension one, there are only three possible density values below $2$, each of which can be attained with equality in the Li--Yau inequality for the Willmore functional by the unit sphere, the double bubble, and the triple bubble. We show that below an optimal threshold for the Willmore energy, a varifold induced by a current without boundary is in fact a curvature varifold with a uniform bound on its second fundamental form. Consequently, the minimization of the Willmore functional in the class of curvature varifolds with prescribed even Euler characteristic provides smooth solutions for the Willmore problem. In particular, the "ambient" varifold approach and the "parametric" approach are equivalent for minimizing the Willmore energy.

math.DG↗

arXiv2026

Bubble classification of immersions at the boundary of the moduli space with $8π$ Willmore energy

We study the asymptotic bubbling behavior of sequences of weak genus-$p$ immersions with diverging conformal classes and limiting Willmore energy of $8π$. After applying suitable Möbius transformations, in a strong $W^{2,2}_{\mathrm{loc}}$-limit, we obtain two round spheres at the largest scale and $p+1$ catenoids at the smallest scales. Moreover, we apply this classification to sequences of isoperimetrically, conformally and normalized-total-mean-curvature constrained Willmore minimizers when the constraints approach the boundary of the domain where minimizers exist, respectively.

math.DG↗

arXiv2025

A priori bounds for geodesic diameter. Part II. Fine connectedness properties of varifolds

For varifolds whose first variation is representable by integration, we introduce the notion of indecomposability with respect to locally Lipschitzian real valued functions. Unlike indecomposability, this weaker connectedness property is inherited by varifolds associated with solutions to geometric variational problems phrased in terms of sets, $G$ chains, and immersions; yet it is strong enough for the subsequent deduction of substantial geometric consequences therefrom. Our present study is based on several further concepts for varifolds put forward in this paper: real valued functions of generalised bounded variation thereon, partitions thereof in general, partition thereof along a real valued generalised weakly differentiable function in particular, and local finiteness of decompositions.

math.DG↗

arXiv2025

Energy quantization for constrained Willmore surfaces

We establish an energy quantization for constrained Willmore surfaces, where the constraints are given by area, volume, and total mean curvature, assuming that the underlying conformal structures remain bounded. Furthermore, we show strong compactness of constrained Willmore surfaces under some energy threshold, proving in particular the strong compactness of minimizers of two previously studied problems.

math.DG↗

arXiv2025

Short closed geodesics and the Willmore energy

We prove a lower bound on the length of closed geodesics for spheres with Willmore energy below $6π$. The energy threshold is optimal and the inequality cannot be extended to surfaces of higher genus. Moreover, we discuss consequences for the injectivity radius.

math.DG↗

arXiv2024

Li-Yau inequalities for the Helfrich functional and applications

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.

math.DG↗

arXiv2024

A priori bounds for geodesic diameter. Part III. A Sobolev-Poincaré inequality and applications to a variety of geometric variational problems

Based on a novel type of Sobolev-Poincaré inequality (for generalised weakly differentiable functions on varifolds), we establish a finite upper bound of the geodesic diameter of generalised compact connected surfaces-with-boundary of arbitrary dimension in Euclidean space in terms of the mean curvatures of the surface and its boundary. Our varifold setting includes smooth immersions, surfaces with finite Willmore energy, two-convex hypersurfaces in level-set mean curvature flow, integral currents with prescribed mean curvature vector, area minimising integral chains with coefficients in a complete normed commutative group, varifold solutions to Plateau's problem furnished by min-max methods or by Brakke flow, and compact sets solving Plateau problems based on Čech homology. Due to the generally inevitable presence of singularities, path-connectedness was previously known neither for the class of varifolds (even in the absence of boundary) nor for the solutions to the Plateau problems considered.

math.DG↗

arXiv2024

On the minimization of the Willmore energy under a constraint on total mean curvature and area

Motivated by a model for lipid bilayer cell membranes, we study the minimization of the Willmore functional in the class of oriented closed surfaces with prescribed total mean curvature, prescribed area, and prescribed genus. Adapting methods previously developed by Keller-Mondino-Rivière, Bauer-Kuwert, and Ndiaye-Schätzle, we prove existence of smooth minimizers for a large class of constraints. Moreover, we analyze the asymptotic behaviour of the energy profile close to the unit sphere and consider the total mean curvature of axisymmetric surfaces.

math.DG↗

arXiv2023

Properties of surfaces with spontaneous curvature

A model describing cell membranes as optimal shapes with regard to the $L^2$-deficit of their mean curvature to a given constant called spontaneous curvature is considered. It is shown that the corresponding energy functional is lower semi-continuous with respect to oriented varifold convergence on a space of surfaces whose second fundamental form is uniformly bounded in $L^2$. Elementary examples are presented showing that the latter condition is necessary. As a consequence, smoothly embedded minimisers among surfaces of higher genus are obtained. Moreover, it is shown that the diameter of a connected surface is controlled by the $L^1$-deficit of its mean curvature to the spontaneous curvature leading to an improved condition for the existence of minimisers. Finally, the diameter bound can be applied to obtain an isoperimetric inequality.

math.DG↗

arXiv2023

Diameter estimates for surfaces in conformally flat spaces

The aim of this paper is to give an upper bound for the intrinsic diameter of a surface with boundary immersed in a conformally flat three dimensional Riemannian manifold in terms of the integral of the mean curvature and of the length of its boundary. Of particular interest is the application of the inequality to minimal surfaces in the three-sphere and in the hyperbolic space. Here the result implies an a priori estimate for connected solutions of Plateau's problem, as well as a necessary condition on the boundary data for the existence of such solutions. The proof follows a construction of Miura and uses a diameter bound for closed surfaces obtained by Topping and Wu-Zheng.

math.DG↗

arXiv2022

A priori bounds for geodesic diameter. Part I. Integral chains with coefficients in a complete normed commutative group

As service to the community, we provide - for Euclidean space - a basic treatment of locally rectifiable chains and of the complex of locally integral chains. In this setting, we may beneficially develop the idea of a complete normed commutative group bundle over the Grassmann manifold whose fibre is the coefficient group of the chains. Our exposition also sheds new light on some algebraic aspects of the theory. Finally, we indicate an extension to a geometric approach to locally flat chains centring on locally rectifiable chains rather than completion procedures.

math.DG↗

arXiv2022

Embedded Delaunay tori and their Willmore energy

A family of embedded rotationally symmetric tori in the Euclidean 3-space consisting of two opposite signed constant mean curvature surfaces that converge as varifolds to a double round sphere is constructed. Using complete elliptic integrals, it is shown that their Willmore energy lies strictly below $8π$. Combining such a strict inequality with previous works by Keller-Mondino-Rivière and Mondino-Scharrer allows to conclude that for every isoperimetric ratio there exists a smoothly embedded torus minimising the Willmore functional under isoperimetric constraint, thus completing the solution of the isoperimetric-constrained Willmore problem for tori. Similarly, we deduce the existence of smoothly embedded tori minimising the Helfrich functional with small spontaneous curvature. Moreover, it is shown that the tori degenerate in the moduli space which gives an application also to the conformally-constrained Willmore problem. Finally, because of their symmetry, the Delaunay tori can be used to construct spheres of high isoperimetric ratio, leading to an alternative proof of the known result for the genus zero case.

math.DG↗

arXiv2022

Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities

Using Rauch's comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li-Yau inequality which is applicable in any Riemannian manifold whose sectional curvature is bounded above (possibly positive). We show that the monotonicity inequalities can also be used to obtain Simon type diameter bounds, Sobolev inequalities and corresponding isoperimetric inequalities for Riemannian submanifolds with small volume. Moreover, we infer lower diameter bounds for closed minimal submanifolds as corollaries. All the statements are intrinsic in the sense that no embedding of the ambient Riemannian manifold into Euclidean space is needed. Apart from Rauch's comparison theorem, the proofs mainly rely on the first variation formula, thus are valid for general varifolds.

math.DG↗

arXiv2021

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-Rivière, our strict inequality leads to existence of minimisers for the isoperimetric constrained Willmore problem in every genus, provided the minimal energy lies strictly below $8π$. 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.

math.DG↗

arXiv2020

Existence and Regularity of Spheres Minimising the Canham-Helfrich Energy

We prove existence and regularity of minimisers for the Canham-Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the $2$-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to prove the main results we establish the lower semicontinuity of the Canham-Helfrich energy under weak convergence of (possibly branched and bubbled) weak immersions.

math.DG↗