arXiv research

Researchers identify only two types of tori with specific energy constraints.

Energy quantization for surfaces with area, volume, and mean curvature constraints.

Constrained Willmore surfaces are critical points of the Willmore functional under conformal variations. As shown in [5] one can associate to any conformally immersed constrained Willmore torus f a compact Riemann surface Σ, such that f can be reconstructed in terms of algebraic data on Σ. Particularly interesting exam…

Study of large area-constrained Willmore surfaces in Schwarzschild-like manifolds.

Study fourth-order geometric problems on Willmore surfaces.

The paper finds new constrained Willmore minimizers for non-rectangular tori.

Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy $W=\int H^2$ under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the crit…

We give an overview of the constrained Willmore problem and address some conjectures arising from partial results and numerical experiments. Ramifications of these conjectures would lead to a deeper understanding of the Willmore functional over conformal immersions from compact surfaces.

In this paper we study equivariant constrained Willmore tori in the 3-sphere. These tori admit a 1-parameter group of Möbius symmetries and are critical points of the Willmore energy under conformal variations. We show that the associated spectral curve of an equivariant torus is given by a double covering of $\mathbb …

The paper shows how to foliate a manifold near a critical point of scalar curvature with Willmore spheres.

New proof of Willmore conjecture using tori minimizers.

We use the dressing method to construct transformations of constrained Willmore surfaces in arbitrary codimension. An adaptation of the Terng--Uhlenbeck theory of dressing by simple factors to this context leads us to define Bäcklund transforms of these surfaces for which we prove Bianchi permutability. Specialising to…

Study on area-constrained Willmore spheres in asymptotic Schwarzschild manifolds.

The Willmore flow preserves surface volume, leading to convergence to a sphere.

This work is dedicated to the study of the Moebius invariant class of constrained Willmore surfaces and its symmetries. We define a spectral deformation by the action of a loop of flat metric connections; Baecklund transformations, by applying a dressing action; and, in 4-space, Darboux transformations, based on the so…

In this paper we consider two special classes of constrained Willmore tori in the 3-sphere. The first class is given by the rotation of closed elastic curves in the upper half plane - viewed as the hyperbolic plane - around the x-axis. The second is given as the preimage of closed constrained elastic curves, i.e., elas…

The tori $T_r = r S^1 \times s S^1 \subset S^3$, where $r^2 + s^2 = 1$, are constrained Willmore surfaces, i.e. critical points of the Willmore functional among tori of the same conformal type. We compute which of the $T_r$ are stable critical points.

We prove that a constrained Willmore immersion of a 2-torus into the conformal 4-sphere is either of "finite type", that is, has a spectral curve of finite genus, or is of "holomorphic type" which means that it is super conformal or Euclidean minimal with planar ends. This implies that all constrained Willmore tori in …

Researchers define and prove existence of minimizers for generalized Willmore functionals.

This paper studies the regularity of constrained Willmore immersions into $\R^{m\ge3}$ locally around both "regular" points and around branch points, where the immersive nature of the map degenerates. We develop local asymptotic expansions for the immersion, its first, and its second derivatives, given in terms of resi…

We prove that the critical points of various energies such as the area, the Willmore energy, the frame energy for tori...etc among possibly branched immersions constrained to evolve within a smooth sub-manifold of the Teichmüller space satisfy the corresponding constrained Euler Lagrange equation. We deduce that critic…

Researchers create families of tori minimizing Willmore energy.

Ejiri's torus in $S^5$ is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any space forms. Li and Vrancken classified all Willmore surfaces of tensor product in $S^{n}$ by reducing them into elastic curves in $S^3$, and the Ejiri torus appeared as a special example. I…

This study reduces Willmore flows of tori to simpler problems and finds new conformally constrained Willmore tori.

Unified view of integrable systems linking CMC, isothermic, and Willmore surfaces.

Delaunay tori minimize Willmore energy under isoperimetric constraints.

Study of tori of revolution under Willmore flow converges to Clifford Torus.

Study solutions of elliptic sinh-Gordon equation, focusing on spectral genus two.

Stable 2-lobed Delaunay tori found in 3-sphere.

Refines geometric center of mass analysis for Einstein field equations.

We consider a free boundary problem for the Willmore functional. Given a smooth domain $Ω$ in ${\mathbb R}^3$, we construct Willmore disks wich are critical in the class of surfaces meeting $\partial Ω$ orthogonally along their boundary and having small prescribed area. Using rescaling we first obtain constrained solut…

Study finds small surfaces in space times with new functionals.

Flow preserves isoperimetric ratio for immersed surfaces.

Refined estimates for surfaces in curved spaces based on Willmore functional.

Study of immersions with Willmore energy leading to spherical and catenoid bubbles.

Proves strict inequality for minimizers of Willmore energy under isoperimetric constraints.

We construct embedded Willmore tori with small area constraint in Riemannian three-manifolds under some curvature condition used to prevent Möbius degeneration. The construction relies on a Lyapunov-Schmidt reduction; to this aim we establish new geometric expansions of exponentiated small symmetric Clifford tori and a…

We define a hierarchy of special classes of constrained Willmore surfaces by means of the existence of a polynomial conserved quantity of some type, filtered by an integer. Type 1 with parallel top term characterises parallel mean curvature surfaces and, in codimension 1, type 1 characterises constant mean curvature su…

Given a 3-dimensional Riemannian manifold $(M,g)$, we prove that if $(Φ_k)$ is a sequence of Willmore spheres (or more generally area-constrained Willmore spheres), having Willmore energy bounded above uniformly strictly by $8 π$, and Hausdorff converging to a point $\bar{p}\in M$, then $Scal(\bar{p})=0$ and $\nabla Sc…

Develops adiabatic theory for ACW flow on surfaces.

The article explores Helfrich flow with spontaneous curvature, finding singularities and convergence behaviors.

This is the second of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three-manifolds. In both papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty being the Möbius degeneration of the tori. In the first paper the construction was perfo…

A proof of the Willmore conjecture is presented. With the help of the global Weierstrass representation the variational problem of the Willmore functional is transformed into a constrained variational problem on the moduli space of all spectral curves corresponding to periodic solutions of the Davey-Stewartson equation…

For every $g\in\mathbb{N}_0$ and $ε>0$, we construct a smooth genus $g$ surface embedded into the unit ball with area $8π$ and Willmore energy smaller than $8π+ ε$. From this we deduce that a minimising sequence for Willmore's energy in the class of genus $g$ surfaces embedded in the unit ball with area $8π$ converges …

We consider closed immersed hypersurfaces in $\R^{3}$ and $\R^4$ evolving by a class of constrained surface diffusion flows. Our result, similar to earlier results for the Willmore flow, gives both a positive lower bound on the time for which a smooth solution exists, and a small upper bound on a power of the total cur…

The paper is devoted to the variational analysis of the Willmore, and other L^2 curvature functionals, among immersions of 2-dimensional surfaces into a compact riemannian m-manifold (M^m,h) with m>2. The goal of the paper is twofold, on one hand, we give the right setting for doing the calculus of variations (includin…

New foliations found for critical surfaces of Hawking energy, resolving discrepancies.

Since the pioneering work of Canham and Helfrich, variational formulations involving curvature-dependent functionals, like the classical Willmore functional, have proven useful for shape analysis of biomembranes. We address minimizers of the Canham-Helfrich functional defined over closed surfaces enclosing a fixed volu…