arXiv subjects

Fabian Rupp

Publications and source records attributed to Fabian Rupp.

At least 19 recordsLinked to original sources

arXiv2026

Calibration energy and mean curvature flow

We introduce the calibration energy for oriented immersions into Euclidean space, quantifying the deviation from calibrated geometry. A key property is that this energy may remain finite for infinite-volume immersions, while a null-Lagrangian structure ensures that it has the same first variation as the volume functional. We establish an exact dissipation identity for the calibration energy along proper oriented mean curvature flows in arbitrary dimensions and codimensions, under a mild local-volume bound. In fact, our result covers a class of singular calibrations and singular initial data. Even in the smooth setting, this provides a new finite variational framework for mean curvature flow beyond the finite-volume regime. As a main application, we establish a general dynamical rigidity theorem for calibrated cones in arbitrary codimension: no singular calibrated cone can be desingularized by a proper oriented mean curvature flow. Our framework further yields novel rigidity theorems for solitons and convergence for two-dimensional immortal flows.

math.DG↗

arXiv2026

Stability of the free boundary Willmore problem

We study the Willmore problem with free boundary by means of a new {\L}ojasiewicz-Simon gradient inequality for functionals on infinite dimensional manifolds. In contrast to previous works, we do not rely on a gradient-like representation of the Fr\'echet derivative, but merely on an inequality. For the free boundary Willmore flow, we prove that solutions starting sufficiently close to a local minimizer exist for all times and converge. In the static setting, we prove quantitative stability of free boundary Willmore immersions and a local rigidity result in a neighborhood of free boundary minimal surfaces.

math.AP↗

arXiv2025

Embeddedness and graphicality of the elastic flow for complete curves

We study positivity-preserving properties for the elastic flow of non-compact, complete curves in Euclidean space. Despite the fact that the canonical elastic energy is infinite in this context, we extend our recent work based on the adapted elastic energy to derive nontrivial optimal thresholds for maintaining planar embeddedness and graphicality, respectively. We also obtain a new Li--Yau type inequality for complete planar curves.

math.AP↗

arXiv2025

Elastic bilayer membranes under confinement -- Existence, regularity, and rigidity

Motivated by applications to cell biology, we study the constrained minimization of the Helfrich energy among closed surfaces confined to a container. We show existence of minimizers in the class of bubble trees of spherical weak branched immersions and derive the Euler--Lagrange equations which involve a measure-valued Lagrange multiplier that is concentrated on the coincidence set with the container boundary. We provide a careful analysis of this elliptic system and prove optimal regularity for solutions throughout the branch points. For surfaces confined in the unit ball we show that the minimization problem behaves rather rigid and identify a parameter range for which minimizers are always round spheres.

math.AP↗

arXiv2025

A new energy method for shortening and straightening complete curves

We introduce a novel energy method that reinterprets ``curve shortening'' as ``tangent aligning''. This conceptual shift enables the variational study of infinite-length curves evolving by the curve shortening flow, as well as higher order flows such as the elastic flow, which involves not only the curve shortening but also the curve straightening effect. For the curve shortening flow, we prove convergence to a straight line under mild assumptions on the ends of the initial curve. For the elastic flow, we establish a global well-posedness theory, and investigate the precise long-time behavior of solutions. In fact, our method applies to a more general class of geometric evolution equations including the surface diffusion flow, Chen's flow, and the free elastic flow.

math.AP↗

arXiv2025

Dimension reduction for Willmore flows of tori: fixed conformal class and analysis of singularities

This work studies Willmore flows of tori and their singularities via a dimension reduction approach. We introduce a Willmore flow that preserves the degenerate constraint of prescribed conformal class and, for rotationally symmetric initial data, we establish a strong relation with the length-preserving elastic flow in the hyperbolic plane. We provide a necessary condition for singularities and a criterion for the initial datum that allows to exclude them. Our results allow for initial data with arbitrarily large energy, in particular exceeding the usual Li-Yau threshold of $8π$. As an application, we obtain existence of a new class of conformally constrained Willmore tori. Moreover, we investigate singularities of the classical Willmore flow. For a class of tori, we identify a non-smooth object, the inverted catenoid, as the limit shape and we show that the flow can be restarted at this singular surface and converges to a round sphere.

math.AP↗

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

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↗

arXiv2024

Existence and convergence of the length-preserving elastic flow of clamped curves

We study the evolution of curves with fixed length and clamped boundary conditions moving by the negative $L^2$-gradient flow of the elastic energy. For any initial curve lying merely in the energy space we show existence and parabolic smoothing of the solution. Applying previous results on long time existence and proving a constrained Lojasiewicz-Simon gradient inequality we furthermore show convergence to a critical point as time tends to infinity.

math.AP↗

arXiv2024

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↗

arXiv2024

Conservation, convergence, and computation for evolving heterogeneous elastic wires

The elastic energy of a bending-resistant interface depends both on its geometry and its material composition. We consider such a heterogeneous interface in the plane, modeled by a curve equipped with an additional density function. The resulting energy captures the complex interplay between curvature and density effects, resembling the Canham-Helfrich functional. We describe the curve by its inclination angle, so that the equilibrium equations reduce to an elliptic system of second order. After a brief variational discussion, we investigate the associated nonlocal $L^2$-gradient flow evolution, a coupled quasilinear parabolic problem. We analyze the (non)preservation of quantities such as convexity, positivity, and symmetry, as well as the asymptotic behavior of the system. The results are illustrated by numerical experiments.

math.AP↗

arXiv2024

The Willmore flow with prescribed isoperimetric ratio

We introduce a non-local $L^2$-gradient flow for the Willmore energy of immersed surfaces which preserves the isoperimetric ratio. For spherical initial data with energy below an explicit threshold, we show long-time existence and convergence to a Helfrich immersion. This is in sharp contrast to the locally constrained flow, where finite time singularities occur.

math.AP↗

arXiv2023

A Li-Yau inequality for the 1-dimensional Willmore energy

By the classical Li-Yau inequality, an immersion of a closed surface in $\mathbb{R}^n$ with Willmore energy below $8π$ has to be embedded. We discuss analogous results for curves in $\mathbb{R}^2$, involving Euler's elastic energy and other possible curvature functionals. Additionally, we provide applications to associated gradient flows.

math.DG↗

arXiv2023

A dynamic approach to heterogeneous elastic wires

We consider closed planar curves with fixed length and arbitrary winding number whose elastic energy depends on an additional density variable and a spontaneous curvature. Working with the inclination angle, the associated $L^2$-gradient flow is a nonlocal quasilinear coupled parabolic system of second order. We show local well-posedness, global existence of solutions, and full convergence of the flow for initial data in a weak regularity class.

math.AP↗

arXiv2023

Curvature-dependent Eulerian interfaces in elastic solids

We propose a sharp-interface model for a hyperelastic material consisting of two phases. In this model, phase interfaces are treated in the deformed configuration, resulting in a fully Eulerian interfacial energy. In order to penalize large curvature of the interface, we include a geometric term featuring a curvature varifold. Equilibrium solutions are proved to exist via minimization. We then utilize this model in an Eulerian topology optimization problem that incorporates a curvature penalization.

math.AP↗

arXiv2023

Optimal thresholds for preserving embeddedness of elastic flows

We consider elastic flows of closed curves in Euclidean space. We obtain optimal energy thresholds below which elastic flows preserve embeddedness of initial curves for all time. The obtained thresholds take different values between codimension one and higher. The main novelty lies in the case of codimension one, where we obtain the variational characterization that the thresholding shape is a minimizer of the bending energy (normalized by length) among all nonembedded planar closed curves of unit rotation number. It turns out that a minimizer is uniquely given by a nonclassical shape, which we call ``elastic two-teardrop''.

math.AP↗

arXiv2023

Short closed geodesics and the Willmore energy

We prove a lower bound on the length of closed geodesics for spheres with Willmore energy below $6\pi$. 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↗

arXiv2023

The volume-preserving Willmore flow

We consider a closed surface in $\mathbb{R}^3$ evolving by the volume-preserving Willmore flow and prove a lower bound for the existence time of smooth solutions. For spherical initial surfaces with Willmore energy below $8π$ we show long time existence and convergence to a round sphere by performing a suitable blow-up and by proving a constrained Lojasiewicz-Simon inequality.

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