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On the two-sphere Σ , we consider the problem of minimising among suitable immersions f : Σ → R 3 the weighted L ∞ norm of the mean curvature H, with weighting given by a prescribed ambient function ξ , subject to a fixed surface area constraint. We show that, under a low-energy assumption which prevents topological issues from arising, solutions of this problem and also a more general set of "pseudo-minimiser" surfaces must satisfy a second-order PDE system obtained as the limit as p → ∞ of the Euler–Lagrange equations for the approximating L p problems. This system gives some information about the geometric behaviour of the surfaces, and in particular implies that their mean curvature takes on at most three values: H ∈ { ± ‖ ξ H ‖ L ∞ } away from the nodal set of the PDE system, and H = 0 on the nodal set (if it is non-empty). [ABSTRACT FROM AUTHOR]

Over forty years ago, Paneitz, and independently Fradkin and Tseytlin, discovered a fourth-order conformally invariant differential operator, intrinsically defined on a conformal manifold, mapping scalars to scalars. This operator is a special case of the so-termed extrinsic Paneitz operator defined in the case when the conformal manifold is itself a conformally embedded hypersurface. In particular, this encodes the obstruction to smoothly solving the five-dimensional scalar Laplace equation, and suitable higher dimensional analogs, on conformally compact structures with constant scalar curvature. Moreover, the extrinsic Paneitz operator can act on tensors of general type by dint of being defined on tractor bundles. Motivated by a host of applications, we explicitly compute the extrinsic Paneitz operator. We apply this formula to obtain: an extrinsically-coupled  Q -curvature for embedded four-manifolds, the anomaly in renormalized volumes for conformally compact five-manifolds with negative constant scalar curvature, Willmore energies for embedded four-manifolds, the local obstruction to smoothly solving the five-dimensional singular Yamabe problem, and new extrinsically-coupled fourth- and sixth-order operators for embedded surfaces and four-manifolds, respectively. [ABSTRACT FROM AUTHOR]

Owing to a variety in nanoscale material applications, the conjunction of two nanostructures is frequently researched for potential new applications. Numerous methods are used to model this conjunction process. One such method, the minimization of elastic energy, only considers the axial curvature when modeling conjoined structures. Another method minimizes the Willmore energy, which depends on both the axial and rotational curvatures. In particular, because the catenoid is an absolute minimizer of Willomre energy, a catenoid section can be utilized to conjoin nanostrucrures. Owing to the similarities among carbon nanostructures, we expanded the use of two different energies to join a boron nitride nanotube with a boron nitride nanotorus. The primary objective of this study was to formulate a basic underlying structure from which any small perturbations can be viewed as departures from an ideal model. Accordingly, elastic energy was used to determine the conjunction region for two-dimensional structures, whereas Willmore energy was used to determine the conjunction region for three-dimensional structures. This approach may be extended to produce other hybrid nanoscale structures. [ABSTRACT FROM AUTHOR]

The Riesz z-energy of a manifold X is the integration of the distance between two points to the power z over the product space X × X . Considered as a function of a complex variable z, it can be generalized to a meromorphic function by analytic continuation, which we will call the meromorphic energy function of X. It has only simple poles at some negative integers. The residues of a manifold X are the residues of the meromorphic energy function. For example, the volume and the Willmore energy for surfaces in R 3 can be obtained as residues. [ABSTRACT FROM AUTHOR]

For smooth mappings of the unit disc into the oriented Grassmannian manifold G n , 2 , Hélein (Harmonic Maps Conservation Laws and Moving Frames, Cambridge University Press, Cambridge, 2002) conjectured the global existence of Coulomb frames with bounded conformal factor provided the integral of | A | 2 , the squared-length of the second fundamental form, is less than γ n = 8 π . It has since been shown that the optimal bounds that guarantee this result are: γ 3 = 8 π and γ n = 4 π for n ≥ 4 . For isothermal immersions in R 3 , this hypothesis is equivalent to saying the integral of the sum of the squares of the principal curvatures is less than γ 3 . The goal here is to prove that when n = 3 the same conclusion holds under weaker hypotheses. In particular, it holds for isothermal immersions when | A | 2 is integrable and the integral of | K | , where K is the Gauss curvature, is less than 4 π . Since 2 | K | ≤ | A | 2 this implies the known result for isothermal immersions, but | K | may be small when | A | 2 is large. The method, which is purely analytic, is then developed to examine the case n = 3 when | A | is only square-integrable. The possibility of extending that result in the language of Grassmannian manifolds to the case n > 3 is outlined in an Appendix. [ABSTRACT FROM AUTHOR]

We prove that a certain discrete energy for triangulated surfaces, defined in the spirit of discrete differential geometry, converges to the Willmore energy in the sense of Γ-convergence. Variants of this discrete energy have been discussed before in the computer graphics literature. [ABSTRACT FROM AUTHOR]

In this paper we show that locally there exists a Willmore deformation between minimal surfaces in S n + 2 and minimal surfaces in H n + 2 , i.e., there exists a smooth family of Willmore surfaces { y t : U ~ → S n + 2 , t ∈ [ 0 , 2 π) } such that (y t) | t = 0 is conformally equivalent to a minimal surface in S n + 2 and (y t) | t = π / 2 is conformally equivalent to a minimal surface in H n + 2 . Here U ~ is a simply connected open subset of the surface M. For some cases the deformations are global. By the Willmore deformations of the Veronese two-sphere and its generalizations in S 4 , for any positive number W 0 ∈ R + , we construct complete minimal surfaces in H 4 with Willmore energy being equal to W 0 . An example of complete minimal Möbius strip in H 4 with Willmore energy 6 5 π 5 ≈ 10.733 π is also presented. We also show that all isotropic minimal surfaces in S 4 admit Jacobi fields different from Killing fields, i.e., they are not “isolated”. [ABSTRACT FROM AUTHOR]

Technologies for 3D data acquisition and 3D printing have enormously developed in the past few years, and, consequently, the demand for 3D virtual twins of the original scanned objects has increased. In this context, feature-aware denoising, hole filling and context-aware completion are three essential (but far from trivial) tasks. In this work, they are integrated within a geometric framework and realized through a unified variational model aiming at recovering triangulated surfaces from scanned, damaged and possibly incomplete noisy observations. The underlying non-convex optimization problem incorporates two regularisation terms: a discrete approximation of the Willmore energy forcing local sphericity and suited for the recovery of rounded features, and an approximation of the ℓ 0 pseudo-norm penalty favouring sparsity in the normal variation. The proposed numerical method solving the model is parameterization-free, avoids expensive implicit volume-based computations and based on the efficient use of the Alternating Direction Method of Multipliers. Experiments show how the proposed framework can provide a robust and elegant solution suited for accurate restorations even in the presence of severe random noise and large damaged areas. [ABSTRACT FROM AUTHOR]

This article investigates stationary surfaces with boundaries, which arise as the critical points of functionals dependent on curvature. Precisely, a generalized "bending energy" functional W is considered which involves a Lagrangian that is symmetric in the principal curvatures. The first variation of W is computed, and a stress tensor is extracted whose divergence quantifies deviation from W -criticality. Boundary-value problems are then examined, and a characterization of free-boundary W -surfaces with rotational symmetry is given for scaling-invariant W -functionals. In case the functional is not scaling-invariant, certain boundary-to-interior consequences are discussed. Finally, some applications to the conformal Willmore energy and the p-Willmore energy of surfaces are presented. [ABSTRACT FROM AUTHOR]

The Marques-Neves theorem asserts that among all the torodial (i.e. genus 1) closed surfaces, the Clifford torus has the minimal Willmore energy \int H^2 \, dA. Since the Willmore energy is invariant under Möbius transformations, it can be shown that there is a one-parameter family, up to homotheties, of genus 1 Willmore minimizers. It is then a natural conjecture that such a minimizer is unique if one prescribes its isoperimetric ratio. In this article, we show that this conjecture can be reduced to the positivity question of a polynomial recurrence. A proof of the positivity can be found in the companion article by Melczer and Mezzarobba [submitted to J. Comb. Theory (2020)]. This establishes a first uniqueness result for the Canham model of biomembranes. [ABSTRACT FROM AUTHOR]

We consider the minimization of an average-distance functional defined on a two-dimensional domain Ω with an Euler elastica penalization associated with ∂Ω, the boundary of Ω. The average distance is given by ... where p ≥ 1 is a given parameter and dist(x, ∂Ω) is the Hausdorff distance between {x} and ∂Ω. The penalty term is a multiple of the Euler elastica (i.e., the Helfrich bending energy or the Willmore energy) of the boundary curve ∂Ω, which is proportional to the integrated squared curvature defined on ∂Ω, as given by ... where κ∂Ω denotes the (signed) curvature of ∂Ω and λ > 0 denotes a penalty constant. The domainΩ is allowed to vary among compact, convex sets of ℝ² with Hausdorff dimension equal to two. Under no a priori assumptions on the regularity of the boundary ∂Ω, we prove the existence of minimizers of Ep,λ. Moreover, we establish the C1,1-regularity of its minimizers. An original construction of a suitable family of competitors plays a decisive role in proving the regularity. [ABSTRACT FROM AUTHOR]

For a given family of smooth closed curves $γ^1,...,γ^\alpha⊂\mathbb{R}^3$ we consider the problem of finding an elastic connected compact surface $M$ with boundary $γ=γ^1\cup...\cupγ^\alpha$. This is realized by minimizing the Willmore energy $\mathcal{W}$ on a suitable class of competitors. While the direct minimization of the Area functional may lead to limits that are disconnected, we prove that, if the infimum of the problem is $

For a smooth closed embedded planar curve Γ, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus 픤 ≥ 1 having the curve Γ as boundary, without any prescription on the conormal. In case Γ is a circle we prove that do not exist minimizers and that the infimum of the problem equals β픤 − 4π, where β픤 is the energy of the closed minimizing surface of genus 픤. We also prove that the same result also holds if Γ is a straight line for the suitable analogously defined minimization problem on asymptotically flat surfaces. Then we study the case in which Γ is compact, 픤 = 1 and the competitors are restricted to a suitable class 풞 of varifolds that includes embedded surfaces. We prove that under suitable assumptions minimizers exists in this class of generalized surfaces. [ABSTRACT FROM AUTHOR]

In this paper we prove some geometric inequalities for closed surfaces in Euclidean three-space. Motivated by Gage's inequality for convex curves, we first verify that for convex surfaces the Willmore energy is bounded below by some scale-invariant quantities. In particular, we obtain an optimal scaling law between the Willmore energy and the isoperimetric ratio under convexity. In addition, we address Topping's conjecture relating diameter and mean curvature for connected closed surfaces. We prove this conjecture in the class of simply-connected axisymmetric surfaces, and moreover obtain a sharp remainder term which ensures the first evidence that optimal shapes are necessarily straight even without convexity. [ABSTRACT FROM AUTHOR]

Nanoscale materials have gained considerable interest because of their special properties and wide range of applications. Many types of boron nitride at the nanoscale have been realized, including nanotubes, nanocones, fullerenes, tori, and graphene sheets. The connection of these structures at the nanoscale leads to merged structures that have enhanced features and applications. Modeling the joining between nanostructures has been adopted by different methods. Namely, carbon nanostructures have been joined by minimizing the elastic energy in symmetric configurations. In other words, the only considerable curvature in the elastic energy is the axial curvature. Accordingly, because it has nanoscale structures similar to those in carbon, BN can also be joined and connected by using this method. On the other hand, different methods have been proposed to consider the rotational curvature because it has a similar size. Based on that argument, the Willmore energy, which depends on both curvatures, has been minimized to join carbon nanostructures. This energy is used to identify the joining region, especially for a three-dimensional structure. In this paper, we expand the use of Willmore energy to cover the joining of boron nitride nanostructures. Therefore, because catenoids are absolute minimizers of this energy, pieces of catenoids can be used to connect nanostructures. In particular, we joined boron nitride fullerene to three other BN nanostructures: nanotube, fullerene, and torus. For now, there are no experimental or simulation data for comparison with the theoretical connecting structures predicted by this study, which is some justification for the suggested simple model shown in this research. Ultimately, various nanoscale BN structures might be connected by considering the same method, which may be considered in future work. [ABSTRACT FROM AUTHOR]

This is a survey of accumulated spectral analysis observations spanning more than a century, referring to the double layer potential integral equation, also known as Neumann-Poincaré operator. The very notion of spectral analysis has evolved along this path. Indeed, the quest for solving this specific singular integral equation, originally aimed at elucidating classical potential theory problems, has inspired and shaped the development of theoretical spectral analysis of linear transforms in XX-th century. We briefly touch some marking discoveries into the subject, with ample bibliographical references to both old, sometimes forgotten, texts and new contributions. It is remarkable that applications of the spectral analysis of the Neumann-Poincaré operator are still uncovered nowadays, with spectacular impacts on applied science. A few modern ramifications along these lines are depicted in our survey. [ABSTRACT FROM AUTHOR]

The Canham-Helfrich-Evans models of biomembranes consist of a family of geometric constrained variational problems. In this article, we compare two classes of numerical methods for these variational problems based on piecewise linear (PL) and subdivision surfaces (SS). Since SS methods are based on spline approximation and can be viewed as higher order versions of PL methods, one may expect that the only difference between the two methods is in the accuracy order. In this paper, we prove that a numerical method based on minimizing any one of the 'PL Willmore energies' proposed in the literature would fail to converge to a solution of the continuous problem, whereas a method based on minimization of the bona fide Willmore energy, well-defined for SS but not PL surfaces, succeeds. Motivated by this analysis, we propose also a regularization method for the PL method based on techniques from conformal geometry. We address a number of implementation issues crucial for the efficiency of our solver. A software package called W MINCON accompanies this article, and provides parallel implementations of all the relevant geometric functionals. When combined with a standard constrained optimization solver, the geometric variational problems can then be solved numerically. To this end, we realize that some of the available optimization algorithms/solvers are capable of preserving symmetry, while others manage to break symmetry; we explore the consequences of this observation. [ABSTRACT FROM AUTHOR]

Due to a variety of applications of nanoscaled materials, several researchers further investigate a joining between two nanostructures as a candidate for new potential applications. Here, the vertically joining between the nanotube with the nano-torus is investigated. Variational calculus is used to predict the joining curve between two nanostructures based on minimizing the elastic energy. Moreover, Willmore energy is also utilized to determine the join region especially for a three-dimensional structure. Since the surface of a catenoid is a minimizer obtained by the Willmore energy function, it is used to join two symmetric nanostructures. We find that these two approaches have less than 10% difference in the positions of the joining curves. These two methods might be used to design other hybrid nano-scaled structures. [ABSTRACT FROM AUTHOR]

We describe the implementation of a topological constraint in finite-element simulations of phase-field models, which ensures path-connectedness of preimages of intervals in the phase-field variable. The constraint takes the form of an energetic penalty for a suitable geodesic distance between all pairs of points in the domain. The main application of our method presented here is a discrete steepest descent of a phase-field version of a bending energy with spontaneous curvature and additional surface area penalty. This leads to disconnected surfaces without our topological constraint but connected surfaces with the constraint. Numerically, our constraint is treated by first transforming the double integral over all pairs of points in the domain to a weighted graph structure and then using Dijkstra's algorithm to calculate the distance between discrete connected components. [ABSTRACT FROM AUTHOR]

This paper considers an energy approach to assessing the state of a cerebral aneurysm as a hydroelastic system consisting of an elastic vessel wall and incoming blood flow. Assuming that the elastic energy of a vessel with an aneurysm, combined with the bending and kinetic energies, is spent only in viscous flow dissipation in the structure, we performed a series of numerical calculations for fusiform aneurysm configuration models with and without a diverticulum of different sizes relative to the size of the aneurysm body. It is shown that pressure–velocity diagrams are in good agreement with clinical data. It is shown by numerical simulation that a small diverticulum has a significant effect on hemodynamics inside the aneurysm body, and at a large diverticulum size, the vortex induced inside the diverticulum is almost completely localized in it. [ABSTRACT FROM AUTHOR]

We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact set in any dimension, subject to attractive Robin boundary conditions. As an improvement upon our previous work (Krejčiřík and Lotoreichik J. Convex Anal. 25, 319–337, 2018), we show that under either a constraint of fixed perimeter or area, the maximiser within the class of exteriors of simply connected planar sets is always the exterior of a disk, without the need of convexity assumption. Moreover, we generalise the result to disconnected compact planar sets. Namely, we prove that under a constraint of fixed average value of the perimeter over all the connected components, the maximiser within the class of disconnected compact planar sets, consisting of finitely many simply connected components, is again a disk. In higher dimensions, we prove a completely new result that the lowest point in the spectrum is maximised by the exterior of a ball among all sets exterior to bounded convex sets satisfying a constraint on the integral of a dimensional power of the mean curvature of their boundaries. Furthermore, it follows that the critical coupling at which the lowest point in the spectrum becomes a discrete eigenvalue emerging from the essential spectrum is minimised under the same constraint by the critical coupling for the exterior of a ball. [ABSTRACT FROM AUTHOR]

The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold. Recently it has been shown how, given a conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem [21]. This enables a route to proliferating conformal hypersurface invariants. The aim of this work is to give a self contained and explicit treatment of the calculus and identities required to use this machinery in practice. In addition we show how to compute the solution's asymptotics. We also develop the calculus for explicitly constructing the conformal hypersurface invariant differential operators discovered in [21] and in particular how to compute extrinsically coupled analogues of conformal Laplacian powers. Our methods also enable the study of integrated conformal hypersurface invariants and their functional variations. As a main application we prove that a class of energy functions proposed in a recent work have the right properties to be deemed higher-dimensional analogues of the Willmore energy. This complements recent progress on the existence and construction of different functionals in [22] and [20]. [ABSTRACT FROM AUTHOR]

We discuss the infinitesimal bending of curves and knots in ℛ 3 . A brief overview of the results on the infinitesimal bending of curves is outlined. Change of the Willmore energy, as well as of the Möbius energy under infinitesimal bending of knots is considered. Our visualization tool devoted to visual representation of infinitesimal bending of knots is presented. [ABSTRACT FROM AUTHOR]

We demonstrate that Radon measures which arise as the limit of the Modica-Mortola measures associated with phase-fields with uniformly bounded diffuse area and Willmore energy may be singular at the boundary of a domain and discuss implications for practical applications. We furthermore give partial regularity results for the phase-fields u ε at the boundary in terms of boundary conditions and counterexamples without boundary conditions. [ABSTRACT FROM AUTHOR]

Functionals involving surface curvature are important across a range of scientific disciplines, and their extrema are representative of physically meaningful objects such as atomic lattices and biomembranes. Inspired in particular by the relationship of the Willmore energy to lipid bilayers, we consider a general functional depending on a surface and a symmetric combination of its principal curvatures, and provided the surface is immersed in a 3-D space form of constant sectional curvature. We calculate the first and second variations of this functional, extending known results and providing computationally accessible expressions given entirely in terms of the basic geometric information found in the surface fundamental forms. Further, we motivate and introduce the p-Willmore energy functional, applying the stability criteria afforded by our calculations to prove a result about the p-Willmore energy of spheres. [ABSTRACT FROM AUTHOR]

Abstract We develop an efficient calculus for varying hypersurface embeddings based on variations of hypersurface defining functions. This is used to show that the functional gradient of a new Willmore-like, conformal hypersurface energy agrees exactly with the obstruction to smoothly solving the singular Yamabe problem for conformally compact four-manifolds. We give explicit formulæ for both the energy functional and the obstruction. Vanishing of the latter is a necessary condition for solving the vacuum cosmological Einstein equations in four spacetime dimensions with data prescribed on a conformal infinity, while the energy functional generalizes the scheme-independent contribution to entanglement entropy across surfaces to hypersurfaces. [ABSTRACT FROM AUTHOR]

We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant (Q -curvature, transgression)-type pairs for hypersurfaces with boundary. For the special case of anomalies coming from the volume enclosed by a minimal hypersurface ending on the boundary of a Poincaré–Einstein structure, this result recovers Branson's Q -curvature and corresponding transgression. When the singular metric solves a boundary version of the constant scalar curvature Yamabe problem, the anomaly gives generalized Willmore energy functionals for hypersurfaces with boundary. Our approach yields computational algorithms for all the above quantities, and we give explicit results for surfaces embedded in 3-manifolds. [ABSTRACT FROM AUTHOR]

We show that local minimizers of the Canham-Helfrich energy are asymptotically stable with respect to a model for relaxational fluid vesicle dynamics that we already studied in previous papers ([13, 14]). The proof is based on a Łojasiewicz-Simon inequality. [ABSTRACT FROM AUTHOR]

Numerous types of carbon nanostructure have been found experimentally, including nanotubes, fullerenes and nanocones. These structures have applications in various nanoscale devices and the joining of these structures may lead to further new configurations with more remarkable properties and applications. The join profile between different carbon nanostructures in a symmetric configuration may be modelled using the calculus of variations. In previous studies, carbon nanostructures were assumed to deform according to perfect elasticity, thus the elastic energy, depending only on the axial curvature, was used to determine the join profile consisting of a finite number of discrete bonds. However, one could argue that the relevant energy should also involve the rotational curvature, especially when its size is comparable to the axial curvature. In this paper, we use the Willmore energy, a natural generalisation of the elastic energy that depends on both the axial and rotational curvatures. Catenoids are absolute minimisers of this energy and pieces of these may be used to join various nanostructures. We focus on the cases of joining a fullerene to a nanotube and joining two fullerenes along a common axis. By comparing our results with the earlier work, we find that both energies give similar joining profiles. Further work on other configurations may reveal which energy provides a better model. [ABSTRACT FROM AUTHOR]

We use the minimizing movement theory to study the gradient flow associated to a non-regular relaxation of a geometric functional derived from the Willmore energy. Thanks to the coarea formula, we can define a Willmore energy on regular functions of L1(Rd). This functional is extended to every L1 function by taking its lower semicontinuous envelope. We study the flow generated by this relaxed energy for radially non-increasing functions (functions with balls as superlevel sets). In the first part of the paper, we prove a coarea formula for the relaxed energy of such functions. Then, we show that the flow consists of an erosion of the initial data. The erosion speed is given by a first order ordinary equation. [ABSTRACT FROM AUTHOR]

This paper investigates the regularity of constrained Willmore immersions intoℝm≥3locally around both “regular” points and around branch points, where the immersive nature of the map degenerates. We develop local asymptotic expansions for the immersion and its first and second derivatives, given in terms of residues computed as circulation integrals. We deduce explicit “point removability” conditions ensuring that the immersion is smooth. Our results apply in particular to Willmore immersions and to parallel mean curvature immersions in any codimension. [ABSTRACT FROM AUTHOR]

The change of the Willmore energy, as a special case of so-called Helfrich energy, under second order infinitesimal bending of curves in a three-dimensional Euclidean space is studied. The first and the second variation of the Willmore energy are obtained. [ABSTRACT FROM AUTHOR]