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Simon Blatt

Publications and source records attributed to Simon Blatt.

At least 19 recordsLinked to original sources
arXiv2025

Stability of knot equivalence at low regularity, and symmetric critical knots for the M\"obius energy

We present sufficient criteria for the equivalence of tame knots at low regularity. To this end, we introduce a localized version of Gromov's distortion for any closed path-connected subset of $\R^n$. If two such sets have local Gromov distortion below a universal dimension-dependent constant $g_n$ at some scale, and if their Hausdorff-distance is less than one quarter of that scale, we can show that the fundamental groups of their complements are isomorphic. In addition, we construct this isomorphism so that it restricts to the corresponding peripheral subgroups as an isomorphism as well. Applied to the images of one-dimensional knots it follows that two knots are equivalent if their Hausdorff-distance is bounded in terms of the scale under which their local Gromov distortion is controlled. From that we deduce novel stability results for knot equivalence in the Lipschitz category, and in the setting of fractional Sobolev regularity below $C^1$. Moreover, we prove a compactness theorem of knot equivalence classes with respect to weak $W^{3/2,2}$-convergence. As an application we show that the M\"obius energy introduced by O'Hara~\cite{ohara_1991a} can be minimized within arbitrary prime knot classes under a symmetry constraint, and that these minimizers are in fact critical points and therefore smooth and even real analytic. In particular, in every torus knot class there are at least two distinct critical knots for the M\"obius energy.

math.GT↗
arXiv2024

Mathematics of Family Planning in Talmud

Motivated by the commitments from the Talmud in Judaism, we consider the family planning rules which require a couple to get children till certain numbers of boys and girls are reached. For example, the rabbinical school of Beit Hillel says that one boy and one girl are necessary, whereas Beit Shammai urges for two boys. Surprisingly enough, although the corresponding average family sizes differ in both cases, the gender ratios remain constant. We show more that for any family planning rule the gender ratio is equal to the birth odds. The proof of this result is given by using different mathematical techniques, such as induction principle, Doob's optional-stopping theorem, and brute-force. We conclude that, despite possible asymmetries in the religiously motivated family planning rules, they discriminate neither boys nor girls.

math.HO↗
arXiv2023

Existence of optimal flat ribbons

We apply the direct method of the calculus of variations to show that any nonplanar Frenet curve in $\mathbb{R}^{3}$ can be extended to an infinitely narrow flat ribbon having minimal bending energy. We also show that, in general, minimizers are not free of planar points, yet such points must be isolated under the mild condition that the torsion does not vanish.

math.DG↗
arXiv2023

A fractional Willmore-type energy functional -- subcritical observations

We investigate surfaces with bounded L^p-norm of the fractional mean curvature, a quantity we shall refer to as fractional Willmore-type functional. In the subcritical case and under convexity assumptions we show how this Willmore-functional controls local parametrization, and conclude as consequences lower Ahlfors-regularity, a weak Michael-Simon type inequality, and an application to stability.

math.AP↗
arXiv2021

A regularized gradient flow for the $p$-elastic energy

We prove long-time existence for the negative $L^2$-gradient flow of the $p$-elastic energy, $p\geq 2$, with an additive positive multiple of the length of the curve. To achieve this result we regularize the energy by adding a small multiple of a higher order energy, namely the square of the $L^2$-norm of the normal gradient of the curvature $\kappa$. Long-time existence is proved for the gradient flow of these new energies together with the smooth sub-convergence of the evolution equation's solutions to critical points of the regularized energy in $W^{2,p}$. We then show that the solutions to the regularized evolution equations converge to a weak solution of the negative gradient flow of the $p$-elastic energies. These latter weak solutions also sub-converge to critical points of the $p$-elastic energy.

math.AP↗
arXiv2021

Scale-invariant tangent-point energies for knots

We investigate minimizers and critical points for scale-invariant tangent-point energies ${\rm TP}^{p,q}$ of closed curves. We show that a) minimizing sequences in ambient isotopy classes converge to locally critical embeddings in all but finitely many points and b) show regularity of locally critical embeddings. Technically, the convergence theory a) is based on a gap-estimate of a fractional Sobolev spaces in comparison to the tangent-point energy. The regularity theory b) is based on constructing a new energy $\mathcal{E}^{p,q}$ and proving that the derivative $\gamma'$ of a parametrization of a ${\rm TP}^{p,q}$-critical curve $\gamma$ induces a critical map with respect to $\mathcal{E}^{p,q}$ acting on torus-to-sphere maps.

math.AP↗
arXiv2021

A minimising movement scheme for the $p$-elastic energy of curves

We prove short-time existence for the negative $L^2$-gradient flow of the $p$-elastic energy of curves via a minimising movement scheme. In order to account for the degeneracy caused by the energy's invariance under curve reparametrisations, we write the evolving curves as approximate normal graphs over a fixed smooth curve. This enables us to establish short-time existence and give a lower bound on the solution's lifetime that depends only on the $W^{2,p}$-Sobolev norm of the initial data.

math.AP↗
arXiv2020

A Reverse Isoperimetric Inequality and its Application to the Gradient Flow of the Helfrich Functional

We prove a quantitative reverse isoperimetric inequality for embedded surfaces with Willmore energy bounded away from $8\pi$. We use this result to analyze the negative $L^2$ gradient flow of the Willmore energy plus a positive multiple of the inclosed volume. We show that initial surfaces of Willmore energy less than $8\pi$ with positive inclosed volume converge to a round point in finite or infinite time.

math.AP↗
arXiv2020

Analyticity for Solution of Integro-Differential Operators

We prove that for a certain class of kernels $K(y)$ that viscosity solutions of the integro-differential equation $$ \int_{\mathbb R^n} (u(x+y) - 2 u(x) + u(x-y)) K(y) dy = f(x,u(x)) $$ are locally analytic if $f$ is an analytic function. This extends the result of Albanese, Fiscella, Valdinoci that such solutions belong to certain Gevrey classes.

math.AP↗
arXiv2019

On O'hara knot energies I: Regularity for critical knots

We develop a regularity theory for extremal knots of scale invariant knot energies defined by J. O'hara in 1991. This class contains as a special case the Möbius energy. For the Möbius energy, due to the celebrated work of Freedman, He, and Wang, we have a relatively good understanding. Their approch is crucially based on the invariance of the Möbius energy under Möbius transforms, which fails for all the other O'hara energies. We overcome this difficulty by re-interpreting the scale invariant O'hara knot energies as a nonlinear, nonlocal $L^p$-energy acting on the unit tangent of the knot parametrization. This allows us to draw a connection to the theory of (fractional) harmonic maps into spheres. Using this connection we are able to adapt the regularity theory for degenerate fractional harmonic maps in the critical dimension to prove regularity for minimizers and critical knots of the scale-invariant O'hara knot energies.

math.AP↗
arXiv2019

A Möbius invariant discretization and decomposition of the Möbius energy

The Möbius energy, defined by O'Hara, is one of the knot energies, and named after the Möbius invariant property which was shown by Freedman-He-Wang. The energy can be decomposed into three parts, each of which is Möbius invariant, proved by Ishizeki-Nagasawa. Several discrete versions of Möbius energy, that is, corresponding energies for polygons, are known, and it showed that they converge to the continuum version as the number of vertices to infinity. However already-known discrete energies lost the property of Möbius invariance, nor the Möbius invariant decomposition. Here a new discretization of the Möbius energy is proposed. It has the Möbius invariant property, and can be decomposed into the Möbius invariant components which converge to the original components of decomposition in the continuum limit. Though the decomposed energies are Möbius invariant, their densities are not. As a by-product, it is shown that the decomposed energies have alternative representation with the Möbius invariant densities.

math.DG↗
arXiv2018

A M\"obius invariant discretization of O'Hara's M\"obius energy

We introduce a new discretization of O'Hara's M\"obius energy. In contrast to the known discretizations of Simon and Kim and Kusner it is invariant under M\"obius transformations of the surrounding space. The starting point for this new discretization is the cosine formula of Doyle and Schramm. We then show $\Gamma$-convergence of our discretized energies to the M\"obius energy under very natural assumptions.

math.FA↗
arXiv2018

A note on singularities in finite time for the constrained Willmore flow

This work investigates the formation of singularities under the steepest descent $L^2$-gradient flow of the functional $\mathcal W_{λ_1, λ_2}$, the sum of the Willmore energy, $λ_1$ times the area, and $λ_2$ times the signed volume of an immersed closed surface without boundary in $\mathbb R^3$. We show that in the case that $λ_1>1$ and $λ_2=0$ any immersion develops singularities in finite time under this flow. If $λ_1 >0$ and $λ_2 > 0$, embedded closed surfaces with energy less than $$8π+\min\{(16 πλ_1^3)/(3λ_2^2), 8π\}$$ and positive volume evolve singularities in finite time. If in this case the initial surface is a topological sphere and the initial energy is less than $8 π$, the flow shrinks to a round point in finite time. We furthermore discuss similar results for the case that $λ_2$ is negative.

math.AP↗
arXiv2018

On the analyticity of critical points of the Möbius energy

We prove that smooth critical points of the Möbius energy parametrized by arc-length are analytic. Together with the main result in \cite{BRS16} this implies that critical points of the Möbius energy with merely bounded energy are not only $C^\infty$ but also analytic. Our proof is based on Cauchy's method of majorants and a decomposition of the gradient which already proved useful in the proof of the regularity results in \cite{BR13} and \cite{BRS16}. To best of the authors knowledge, this is the first analyticity result in the context of non-local differential equations.

math.AP↗
arXiv2016

Curves between Lipschitz and $C^1$ and their relation to geometric knot theory

In this article we investigate regular curves whose derivatives have vanishing mean oscillations. We show that smoothing these curves using a standard mollifier one gets regular curves again. We apply this result to solve a couple of open problems. We show that curves with finite Möbius energy can be approximated by smooth curves in the energy space $W^{\frac 32,2}$ such that the energy converges which answers a question of He. Furthermore, we extend the result of Scholtes on the $Γ$-convergence of the discrete Möbius energies towards the Möbius energy and prove conjectures of Ishizeki and Nagasawa on certain parts of a decomposition of the Möbius energy. Finally, we extend a theorem of Wu on inscribed polygons to curves with derivatives with vanishing mean oscillation

math.CA↗
arXiv2016

The Gradient Flow of the Möbius energy: $\varepsilon$-regularity and consequences

In this article we study the gradient flow of the Möbius energy introduced by O'Hara in 1991. We will show a fundamental $\varepsilon$-regularity result that allows us to bound the infinity norm of all derivatives for some time if the energy is small on a certain scale. This result enables us to characterize the formation of a singularity in terms of concentrations of energy and allows us to construct a blow-up profile at a possible singularity. This solves one of the open problems listed by Zheng-Xu He. Ruling out blow-ups for planar curves, we will prove that the flow transforms every planar curve into a round circle.

math.AP↗
arXiv2016

The Gradient Flow of O'Hara's Knot Energies

Jun O'Hara invented a family of knot energies $E^{j,p}$, $j,p \in (0, \infty)$. We study the negative gradient flow of the sum of one of the energies $E^α= E^{α,1}$, $α\in (2,3)$, and a positive multiple of the length. Showing that the gradients of these knot energies can be written as the normal part of a quasilinear operator, we derive short time existence results for these flows. We then prove long time existence and convergence to critical points.

math.AP↗