Geometric curvature energies: facts, trends, and open problems

This survey focuses on geometric curvature functionals, that is, geometrically defined self-avoidance energies for curves, surfaces, or more general kdimensional sets in Rd. Previous investigations of the authors and collaborators concentrated on the regularising effects of such energies, with a priori estimates in the regime above scale-invariance that allowed for compactness and variational applications for knotted curves and surfaces under topological restrictions. We briefly describe the impact of geometric curvature energies on geometric knot theory. Currently, various attempts are beingmade to obtain a deeper understanding of the energy landscape of these highly singular and nonlinear nonlocal interaction energies. Moreover, a regularity theory for critical points is being developed in the setting of fractional Sobolev spaces. We describe some of these current trends and present a list of open problems.

and also integral Menger 2.2 curvature [115] Mp( ) := ∫︁ ∫︁ ∫︁ 1 R p (x, y, z) dH 1 (x)dH 1 (y)dH 1 (z), p ≥ 3. (2.1.5) On a fixed loop of unit length, these energies are ordered as (2.1.6) where the last term is the ropelength of . Moreover, the p-th root of Mp , Ip, and of Up tends to ropelength as p → ∞ both on fixed conformations of knots and in the sense of Γ-convergence. Besides averaging and maximising over the multi-point interactions in the circumradius we investigated tangent-point interactions such as [121,57] Ep( ) := ∫︁ ∫︁ 1 r tp (x, y) p dH 1 (x)dH 1 (y), p ≥ 2, (2.1.7) as well, where r tp (x, y) is the radius of the unique circle through two given curve points x and y that is additionally tangent to in x.
(2.1.9) 2.2 Karl Menger considered in the 1930's the circumradius R(x, y, z) of three curve points x, y, z ∈ knowing that the coalescent limit of R(x, y, z) as x and y tend to z coincides with the local radius of curvature if the curve is sufficiently smooth. Menger was also aware of the fact that there is an elementary formula for the circumradius solely in terms of the mutual distances of the points x, y, and z. By means of multipoint functions such as R(·, ·, ·) Menger indeed intended to develop a purely metric geometry in contrast to classic differential geometry. The idea of using Menger curvature as a tool -both in harmonic analysis and in modeling -has been re-discovered in the last 20 years.

2.3
The most obvious choice to take as integrand in (2.1.8) a negative power of the circumsphere radius of a tetrahedron does not serve our purposes since there are smooth embedded surfaces for which such an integrand would not be bounded; see our detailed discussion on various integrands in [120,Appendix B].
Regularising effects. Summarising the essential results of this systematic research (which is well documented in a number of publications [119,114,115,121,117,118,120,123,61,62], we can say the following: we have a pretty clear understanding of the topological and regularising effects of each of these energies, with sharp regularity statements and uniform a priori estimates. For example, a rectifiable curve with finite integral Menger curvature Mp( ) for some p > 3 (i.e., above the scale-invariant case p = 3) is homeomorphic to the unit-circle or unit-interval, and the arclength parametrisation of that curve satisfies the uniform a priori estimate value Mp(Σ), such that for each x ∈ Σ the intersection B R (x) ∩ Σ equals the graph of a C 1,1−(8/p) -function with uniform estimates on this function solely depending on p and Mp(Σ). Again, this is a geometric variant of the Morrey-Sobolev embedding theorem with optimal Hölder exponent 1 − (8/p) for the oscillation of tangent planes. Similar results hold for k-dimensional admissible sets Σ k ⊂ R d with finite integral Menger curvature Mp(Σ) for p > k(k + 2), or finite tangent point energy Ep(Σ) for p > 2k, both in the regime above scale-invariance; see [61,62,123]. A thorough discussion of the respective admissibility class of sets can be found, e.g., in [64]. At this point, we may roughly describe our mild requirements on the set Σ k as a certain degree of local flatness around many (but not all) points, together with an amount of connectivity to allow for some degree-theoretic arguments.
That these regularity estimates are indeed sharp, can be seen either by explicit examples constructed in [66,57], or by the complete characterisation of energy spaces for all these energies in the work of S. Blatt and Kolasiński [17,13,11]: based on our results that finite energy implies that the admissible sets are already C 1 -submanifolds in R d , they use the explicit structure of the energies to estimate locally the seminorms of fractional Sobolev spaces to find that Σ has finite energy if and only if Σ is embedded and has local graph representations of exactly that Sobolev regularity. Recall, e.g., from [128, Section 2.2.2], that a function u ∈ L p (R k ) belongs to the Sobolev-Slobodeckiȋ space W m+s,p (R k ) for some m ∈ N, s ∈ (0, 1), and p ∈ [1, ∞) if u belongs to the classic Sobolev space W m,p and satisfies, in addition, (2.1.14) As an example, let us mention Blatt were treated in cooperation with Kolasiński in [64] leading to the theorem that finite energy characterises embedded submanifolds of classic Sobolev regularity W 2,p if p > k; see [64,Theorem 1.4]. This result may be compared to Allard's famous C 1,αregularity theorem [3,37] for k-dimensional varifolds whose generalised mean curvature is p-integrable, where, again, p > k.
Connections to geometric knot theory. The uniform estimates obtained in [119,114,115] for finite energy curves (like the one in (2.1.13)) together with a uniform geometric rigidity of these curves (replacing the excluded volume constraint of thickness) was used to connect the respective energies to geometric knot theory, as described in detail in [116,Section 4]; see also the recent surveys [122,124]. This geometric rigidity 2.4 means, roughly speaking, that the curve may be equipped with a necklace of consecutive double-cones whose size and opening angle are determined purely in terms of the respective energy [116,Proposition 4.7]. The circular cross-sections of each piece of this necklace, i.e., of each such double cone (with its two tips located on the curve ), are intersected by transversally and exactly in one point; see Figure 2.1. Once this necklace is established one can fairly easily construct an ambient isotopy from to the inscribed polygon made of the consecutive double cones' axes.
Thus, any one of the geometric curvature energies for curves (2.1.3)-(2.1.5), (2.1.7), bounds the stick number, which is the minimal number of straight segments you need to build a polygonal representative of the same knot type. Since stick number is a knot invariant, any such energy bounds the number of knot types: given any constant E ≥ 0 there is a nonnegative integer N(E) depending only on E, such that at most N(E) knot types can be represented by curves of geometric curvature energy below the energy threshold E.
On the other hand, the double-cone property described above also serves as a substitute of the excluded volume constraint given by finite ropelength. This allows us to control the average crossing number acn( ), where you count the number of selfintersections of every planar projection of the given curve and then average over all directions of projections. Indeed, Freedman et al. derived in [41, Section 3] a double integral formula for acn( ), acn( ) := 1 4π where × denotes the usual cross-product in R 3 . While the local interaction terms in that formula may be estimated by the local smoothness properties of a finite energy curve , one can follow the strategy of G. Buck and J. Simon [24] for curves of finite ropelength, to estimate the global interaction terms by estimating the volume of a spatial region necessary to fit in a maximally compactified curve . Only here, one 2.4 Referred to as diamond property in [116].
Bereitgestellt von | Universitätsbibliothek der RWTH Aachen Angemeldet Heruntergeladen am | 06.03.19 15:35 has to replace the excluded volume constraint by the double-cone condition, so that our constants are far from being optimal; see [116,Proposition 4.13]. Since the average crossing number bounds the classic knot invariant crossing number, we thus have established another means to control the number of knot types below given energy thresholds.
In addition, we could show that all these energies are charge and tight, which means that they blow up along sequences that converge to curves with selfintersections and also along sequences where one small knotted subarc pulls tight, i.e., vanishes in the limit. Being tight distinguishes these geometric curvature energies from the Möbius energy (2.1.1): O'Hara showed in [83, Theorem 3.1] that the Möbius energy does not prevent the pull-tight phenomenon. Moreover, Up and Ip could be shown to distinguish the knot from the unknot: there is a gap between the infimum over unknots and the infimum over non-trivially knotted curves. The infima of all these energies (in contrast, e.g., to the Möbius 2.5 energy) are attained on each given knot class. Freedman et al. [41] showed that the Möbius energy E Möb is uniquely minimised by the round circle, a knot energy with that property is called basic. A. Abrams et al. [1] extended that uniqueness result to a larger family of energies. Also ropelength is basic, and more generally, Up as well; see [119,Lemma 7]. A more recent monotonicity formula 2.6 for compact free boundary surfaces by A. Volkmann [129,Section 5] implies the same for the tangent-point energy Ep (see (2.1.7)), and hence also Ip by the arguments in [116, Proof of Cor. 3.7]. All these results resolve some of the open problems formulated in geometric knot theory, e.g. in [125,Section 2], or in [86,Chapter 8], and give some first insights into the presumably complicated energy landscape of these energies on knot space. Almost nothing is known about the actual shape of knotted energy minimisers, apart from the explicit continuous family of ideal links (minimising ropelength in fixed link classes) presented by J. Cantarella, R. B. Kusner, and J. M. Sullivan in [29], and necessary criticality conditions for ropelength minimisers [110,26,27]. Moreover, studying ideal knots in R 4 lead to the discovery of unique explicit solution families of longest (thick) ropes on the two-sphere by H. Gerlach and the second author; see [45] and the popular account in [44].

Trends and open problems
Regularity. Higher regularity of local minimisers or critical points is only known in a few cases. Freedman et al. [41] used the Möbius-invariance of the Möbius energy E Möb defined in (2.1.1) to apply reflection arguments to show that local minimisers are of class C 1,1 , and they derived the Euler-Lagrange equation for injective and regular curves and perturbations h both of class C 1,1 . Later Zh.-X. He [50] used this Euler-Lagrange equation to improve the regularity of local E Möbminimisers to C ∞ -smoothness; see also [98]. Quite recently, there has been considerable progress through the work of Blatt to first gain some additional regularity, i.e., a slightly higher integrability of the tangent [22,Theorem III], before a bootstrapping process can be started. One should point out that, similarly to other geometric equations like the variational equation for the Willmore functional, the Euler-Lagrange equation for E Möb is in a sense critical, which requires some very intricate techniques that were developed in the context of fractional harmonic mappings [34,33,102,101].
Somewhat less involved is the regularity proof for other members of O'Hara's families of repulsive potentials [82,84,85], namely for the energies E α , where a power α ∈ (2, 3) replaces the quadratic power in the denominators of (2.1.1). Blatt and Reiter established the Fréchet-differentiability of E α on the space of regular curves of finite energy, and proved C ∞ -smoothness for arclength parametrised critical points To carry over this regularity program to critical points of geometric curvature energies such as the tangent-point energy (2.1.7) or integral Menger curvature (2.1.5), Blatt and Reiter embedded those energies into larger two-parameter families of energies by decoupling the integrability exponent p into different powers for numerator and denominator of the integrands. In this way they obtain, for instance, modified tangentpoint energies by replacing the p-th power of the inverse tangent-point radius (Here, Tx denotes the tangent-line to through the point x ∈ .) In the parameter regime q > 1 and p ∈ (q + 2, 2q + 1) the modified tangent-point energies turn out to be well-behaved knot energies that are minimisable in every knot class. The fractional Sobolev regularity W (p−1)/q,q characterises finite energy (see [20, 16), and also for the one-dimensional prototype (2.1.3), non-smooth analysis tools such as Clarke gradients would have to be applied to derive the variational differential inclusion, similar to the analysis performed for the ropelength functional for curves involving the non-smooth expression (2.1.2) for thickness; see [109,110,26,27]. dimensional sets E ⊂ C with integrability exponent p = 2 (well below the scaleinvariant exponent p = 3) has played a fundamental rôle in harmonic analysis, e.g., in the proof of the famous Vitushkin conjecture on the removability of compact subsets of the complex plane for complex analytic functions; see, for instance, X. Tolsa's quite recent excellent monograph [126]. Motivated by some of G. David's methods [35] for his final proof of this conjecture, J.-C. Léger [70]

can be covered (up to sets of H k -measure zero) by a countable union of Lipschitz images of R k .
Meurer's class of admissible integrands includes also the discrete curvatures used by G. Lerman and J. T. Whitehouse in [71,72] to give a characterisation of David's and S. Semmes' concept of uniform rectifiability; cf [36, Theorem 1.57]. J. Azzam and Tolsa [7] recently established a new rectifiability criterion in terms of P. Jones's β-numbers [56] which are fundamentally related to integral Menger curvature as shown in [70,75]. Interestingly, however, and somewhat surprising is the fact, that the integrands (2.1.9) and (2.1.12) we studied in the integrability regime above scale-invariance, are In the definition of rectifiability one covers the set (up to a set of measure zero) by Lipschitz images, and one might think about improving the regularity of the covering images. The step from Lipschitz to C 1 -images is immediate by Whitney's extension theorem; see, e.g. [112, Section 3, Lemma 11.1], but improving that to C 1,α (as in the regime above scale-invariance) is highly non-trivial. This was recently accomplished by Kolasiński [63] also for a large class of discrete curvatures with a certain overlap with Meurer's class including (2.2.4), so that, e.g., the following higher order rectifiability result holds true and can be deduced from [63, Theorem 1.1].
Theorem 2.2.7 (C 1,α -rectifiability). Any Borel set E ⊂ R n with 0 < H k (E) < ∞ and a.e. positive lower density, satisfying for some p > 2 is k-rectifiable of class C 1,α for some positive Hölder exponent α = α(p), i.e., the set E can be covered (up to sets of H k -measure zero) by a countable union of k-dimensional C 1,α -submanifolds of R n .  [127,7]?

Open questions 2.2.8. Can one extend Meurer's rectifiability result to the integrands
Not much is known about geometric curvature energies in the scale-invariant regime, but simple scaling arguments reveal the fact that cone-type singularities do lead to infinite geometric curvature energies; see Figure 2.2. S. Scholtes could indeed demonstrate that embedded polygons have finite integral Menger curvature Mp if and only if p ∈ (0, 3); see [103]. Recall that p = 3 is the scale-invariant exponent for integral Menger curvature for curves. In addition, Scholtes established certain weak tangential properties of arbitrary (a priori fairly wild) sets at every point if the one-dimensional set E ⊂ R n has finite integral Menger curvature M 3 (E) [104]. So, one can indeed hope for mild regularising effects, like for the energy U p for curves for p = 1, where we proved in [119, Theorem 1] that finite U 1 -energy implies that the curve is embedded and in the Sobolev class W 2,1 . However, not every embedded W 2,1 -curve has finite U 1 -energy; see [119,. Finiteness of the tangent-point energy Ep in the scale-invariant case p = 2 (see definition (2.1.7)) yields at least a topological one-dimensional manifold -possibly with boundary; see [121,Theorem 1.1]. Only for the Möbius energy (2.1.1), whose Möbius-invariance implies scale-invariance, one has Blatt's [11] characterisation of the appropriate energy spaces  as the fractional Sobolev space W 3/2,2 (assuming injective arclength parametrised curves), and already earlier Blatt and Reiter used an idea of He to construct a closed bi-Lipschitz curve with finite Möbius energy that is not differentiable [18,Corollary 4.2]. But very recently, Blatt has established a nice approximation result on convolutions of curves whose tangents have vanishing mean oscillations which in particular implies that arclength parametrised curves of finite Möbius energy can be approximated in the W 3/2,2 -norm and in energy 2.7 by smooth curves; see [14,Theorem 1.3]. At present there are a few suggestions how to generalise the Möbius energy to higher-dimensional submanifolds -we are aware of Kusner and Sullivan [67,68] and D. Auckly and L. Sadun [5] (see also the very recent contribution by O'Hara and G. Solanes [88], [87]) -but no satisfactory analysis regarding regularity or variational issues has been performed yet. Existence of critical points. For all geometric curvature energies above scaleinvariance one can find (at least one) minimising knot in a given isotopy class. This even works for higher-dimensional geometric curvature energies such as integral Menger curvature or tangent-point energies for submanifolds as described above. But are there other critical points, and how can one prove their existence? One of the first attempts in that direction is the work of D. Kim and Kusner [59] on the Möbius energy. They applied R. S. Palais' principle of symmetric criticality [89] to obtain E Möb -critical torus knots by minimising the Möbius energy within the appropriate subclass of torus knots enjoying particular symmetries . In addition, together with G. Stengle [59, p. 4] they used classic residue calculus from complex analysis to calculate their energy values. Further numerical experiments lead them to conjecture that most of these E Möbcritical torus knots are not local minimisers. For the non-smooth ropelength functional Cantarella et al. [28] successfully modified Palais' symmetric criticality principle to find new critical points in several symmetry classes of knots and links, e.g. in the non-trivial (a, b)-torus knots. They used their numerical ropelength minimising algorithm ridge runner to compute their respective values for ropelength; see Figure 2.3. In ongoing cooperative work with A. Gilsbach we apply Palais' principle to O'Hara's repulsive energies, integral Menger curvature, and tangent-point energies to produce symmetric critical configurations in every prescribed knot class. Specifically, in nontrivial (a, b)-torus knot classes we even obtain two distinct symmetric critical knots with this method [46], [47]. Very helpful in that context is the knowledge of the respec-   [28]. (Images by courtesy of J. Cantarella.) tive correct energy spaces described in Section 2.1. Gilsbach also uses Γ-convergence arguments to show that her symmetric critical points of integral Menger curvature Mp do converge to ropelength-critical points as p → ∞. Recently, Gilsbach has modified T. Hermes' numerical code [52] to actually compute the energy values of the symmetric critical points of integral Menger curvature. Hermes had rigorously derived the first variation formula for integral Menger curvature in the suitable fractional Sobolev space, and could prove that the round circle is a critical point. He created a numerical tool to explore the presumably quite complicated energy landscape of integral Menger curvature. His numerical experiments exhibit among other things the ability of the Menger gradient flow to untangle complicated unknots to the round circle after fairly short time, as well as varying features as p approaches infinity. For p only slightly above the scale-invariant exponent one finds smoothing as the predominant feature (while keeping the curves embedded in contrast to, e.g., the classic mean curvature flow on space curves), whereas for large p, say p ≥ 50, the similarity to Cantarella's ridge runner (corresponding to the case p = ∞) is striking [4]: both flows try to embed the curves as nicely as possible.

2.7
A second variation formula has been derived and analysed in detail for the Möbius energy by A. Ishizeki and T. Nagasawa [54]. They used a very interesting decomposition theorem for the Möbius energy itself [53], and studied recently the Möbiusinvariance of the various parts of that decomposition [55]. Also quite recently J. Knapp-Bereitgestellt von | Universitätsbibliothek der RWTH Aachen Angemeldet Heruntergeladen am | 06.03. 19 15:35 The gradient flow for integral Menger curvature flows the initial configuration above to the trefoil on the top right for p = 3.5, and to the bottom trefoil on the right as p = 50. mann [60] succeeded in deriving rigorously a second variation formula for integral Menger curvature Mp on curves in the appropriate fractional Sobolev spaces.
The only approach to deal with higher-dimensional critical points for geometric curvature energies is the ingenious paper by A. Nabutovsky [80] who combined complexity theory with real algebraic geometry to prove the existence of infinitely many critical unknotted hyperspheres in R n for n ≥ 6 for a higher-dimensional variant of ropelength.  Further implications on geometric knot theory. The round circle minimises many of the known geometric knot energies [41,Corollary 2.2], [1], [119,Lemma 7], [129,Corollary 5.12], and we expect the same for integral Menger curvature due to strong numerical evidence based on Hermes' numerical experiments with his gradient flow algorithm [52,Section 4.3]. In addition, we mentioned the explicit continuous families of ropelength-minimising links constructed by Cantarella et al. [29]. More recently, I. Agol, F. C. Marques, and A. Nèves applied their ingenious min-max-theory for minimal currents to resolve not only the famous Willmore conjecture [74] but also a conjecture by Freedman et al. by proving that the stereographic projection of the standard Hopf-link minimises the Möbius energy; see [2]. Apart from these results nothing is known analytically about the shape of non-trivially knotted minimising curves. For the ropelength-minimising trefoil, the so-called ideal trefoil one has presumably fairly accurate numerical solutions [8,30,9,4,91] and some local analytic information on the possible shape of general ideal knots [110,26,39,40,27] Elastic (2, b)-torus knots). For any odd integer |b| ≥ 3 the unique elastic (2, b)-torus knot is the doubly covered circle. In particular, the elastic trefoil is the doubly covered circle.
This result confirms mechanical and numerical experiments (see Figure 2.6), as well as the heuristics and Metropolis Monte Carlo simulations of R. Gallotti and O. Pierre-Louis [42,90], and the numerical gradient-descent results by S. Avvakumov and A. Sossinsky [6]. However, adding twist changes the geometry of the springy wire drastically; see bottom right of Figure 2.6. And there is no theory yet, describing these twist effects for knotted elastic wires. A higher-dimensional branch of geometric knot theory is naturally much less developed yet. The shape of possible minimising configurations for higher-dimensional geometric curvature energies is wide open and seems currently out of reach. There is one exception, however: We proved in [64,Theorem 1.5] with the isoperimetric inequality and a simple measure-theoretic argument the following uniqueness result for the global tangent-point-energy (2.1.16): Theorem 2.2.14 (Spheres are unique minimisers). The round sphere uniquely minimises the global tangent-point energy (2.1.16) among all compact embedded C 1hypersurfaces in R n .
Recently, we proved in [65] that many higher-dimensional geometric curvature energies including integral Menger curvature (2.1.8), (2.1.11) or tangent-point energies (2.1.10), or their more singular variants (2.1.15), (2.1.16), are valuable knot energies. All these energies are self-repulsive (on the scale above scale-invariance), lowersemicontinuous on sublevel sets with respect to Hausdorff-convergence, they enjoy nice compactness properties and can thus be minimised in given isotopy classes; see [65, Theorem 2, Corollary 1]. They also bound the number of isotopy types with explicit constants only depending on the energy level and the integrability exponent, on a diameter bound, and on the dimensions [ This result can be compared to a whole series of finiteness theorems of diffeomorphism types under given bounds on classic curvatures, beginning with the work of J. Cheeger [31], and extended by many others, see, e.g, Cheeger's exhaustive survey [32] and the references therein. The notable difference is here that we deal with embedded submanifolds of lower regularity, whose Riemannian metrics are just Hölder continuous so that the classic notion of curvature does not make sense. The geometric curvature energies in the regime above scale-invariance turn out to be valuable substitutes. The Bereitgestellt von | Universitätsbibliothek der RWTH Aachen Angemeldet Heruntergeladen am | 06.03. 19 15:35 only comparable result with this emphasis is the work of O. Durumeric [38] who, however, works in the context of C 1,1 -submanifolds with positive thickness.
Higher-dimensional variants of elastic knots have not been discussed explicitly yet, but L. Simon's pioneering work [113] solves the problem of minimising the Willmore energy ∫︁ Σ H 2 (x) dH 2 (x) (2.2.9) in the class of two-dimensional embedded surfaces with prescribed genus or under alternative constraints; see also [10,100,111,69,58,99,79]. But minimising the Willmore energy or related functionals such as the Helfrich functional [51,81] on given isotopy classes has to the best of our knowledge not been investigated yet -with the exception of recent work of P. Breuning