arXiv subjects

Manuel Schlierf

Publications and source records attributed to Manuel Schlierf.

9 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

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↗

arXiv2026

Stability of the free boundary Willmore problem

We study the Willmore problem with free boundary by means of a new Łojasiewicz-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 Fréchet 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

The length-preserving elastic flow with free boundary on hypersurfaces in $\mathbb{R}^n$

We study the length-preserving elastic flow of curves in arbitrary codimension with free boundary on hypersurfaces. This constrained gradient flow is given by a nonlocal evolution equation with nonlinear higher-order boundary conditions. We prove global existence and subconvergence to critical points. The proof strategy involves a careful treatment of short-time existence, uniqueness, and parabolic energy estimates.

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

On the convergence of the Willmore flow with Dirichlet boundary conditions

Very little is yet known regarding the Willmore flow of surfaces with Dirichlet boundary conditions. We consider surfaces with a rotational symmetry as initial data and prove a global existence and convergence result for solutions of the Willmore flow with initial data below an explicit, sharp energy threshold. Strikingly, this threshold depends on the prescribed boundary conditions - it can even be made to be $0$. We show sharpness for some critical boundary data by constructing surfaces above this energy threshold so that the corresponding Willmore flow develops a singularity. Finally, a Li-Yau inequality for open curves in $\mathbb{H}^2$ is proved.

math.AP↗

arXiv2024

Singularities of the hyperbolic elastic flow: Convergence, quantization and blow-ups

We study the elastic flow of closed curves and of open curves with clamped boundary conditions in the hyperbolic plane. While global existence and convergence toward critical points for initial data with sufficiently small energy is already known, this study pioneers an investigation into the flow's singular behavior. We prove a convergence theorem without assuming smallness of the initial energy, coupled with a quantification of potential singularities: Each singularity carries an energy cost of at least 8. Moreover, the blow-ups of the singularities are explicitly classified. A further contribution is an explicit understanding of the singular limit of the elastic flow of $λ$-figure-eights, a class of curves that previously served in showing sharpness of the energy threshold 16 for the smooth convergence of the elastic flow of closed curves.

math.AP↗

arXiv2024

Introducing spontaneous curvature to the Helfrich flow: Singularities and convergence

While there are various results on the long-time behavior of the Willmore flow, the Helfrich flow with non-zero spontaneous curvature as its natural generalization is not yet well-understood. Past results for the gradient flow of a locally area- and volume-constrained Willmore flow indicate the existence of finite-time singularities which corresponds to the scaling-behavior of the underlying energy. However, for a non-vanishing spontaneous curvature, the scaling behavior is not quite as conclusive. Indeed, in this article, we find that a negative spontaneous curvature corresponds to finite-time singularities of the locally constrained Helfrich flow if the initial surface is energetically close to a round sphere. Conversely however, in the case of a positive spontaneous curvature, we find a positive result in terms of the convergence behavior: The locally area-constrained Helfrich flow starting in a spherical immersion with suitably small Helfrich energy exists globally and converges to a Helfrich immersion after reparametrization. Moreover, this energetic smallness assumption is given by an explicit energy threshold depending on the spontaneous curvature and the local area constraint of the energy.

math.AP↗

arXiv2024

Global existence for the Willmore flow with boundary via Simon's Li-Yau inequality

It is well-known that the Willmore flow of closed spherical immersions exists globally in time and converges if the initial datum has Willmore energy below $8π$ - exactly the Li-Yau energy threshold below which all closed immersions are embedded. Extending the Li-Yau inequality for closed surfaces via Simon's monotonicity formula also for surfaces with boundary, given Dirichlet boundary conditions, one obtains an energy threshold $C_{\mathrm{LY}}$ below which surfaces with this boundary are embedded. By a slight modification, one obtains a threshold $C_{\mathrm{LY}}^{\mathrm{rot}}$ below which surfaces of revolution satisfying the boundary data have no self-intersections on the rotation axis. With a new argument, using this modified Li-Yau inequality and tools from geometric measure theory, we show that the Willmore flow with Dirichlet boundary data starting in cylindrical surfaces of revolution exists globally in time if the energy of the initial datum is below $C_{\mathrm{LY}}^{\mathrm{rot}}$. Moreover, given Dirichlet boundary data, we also obtain the existence of a Willmore minimizer in the class of cylindrical surfaces of revolution if the corresponding infimum lies below $C_{\mathrm{LY}}^{\mathrm{rot}}$ which improves previous results for the stationary problem.

math.AP↗