Publications | Prof. Dr. Simon Blatt

Preprints

2023

A fractional Willmore-type energy functional – subcritical observations

Simon Blatt, Giovanni Giacomin, Julian Scheuer, and Armin Schikorra

Jun 2023

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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.
Existence of optimal flat ribbons

Simon Blatt, and Matteo Raffaelli

Jul 2023

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We apply the direct method of the calculus of variations to show that any nonplanar Frenet curve in \mathbbR^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.

2020

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

Simon Blatt

Jul 2020

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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.
On the analyticity of solutions to non-linear elliptic partial differential systems

Simon Blatt

Jul 2020

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2009

Note on continuously differentiable isotopies

Simon Blatt

Jul 2009

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2008

A Lower Bound for the Gromov Distortion of Knotted Submanifolds

Simon Blatt

Jul 2008

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Journal Articles

2022

Analyticity for solution of fractional integro-differential equations

Simon Blatt

Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods, Nov 2022

Id/No 113071

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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 results in (Albanese et al. 2015), where it was shown that such solutions belong to the Gevrey classes \mathcal G_2$ which is larger than the class of analytic functions.
A regularized gradient flow for the $p$-elastic energy

Simon Blatt, Christopher Hopper, and Nicole Vorderobermeier

Advances in Nonlinear Analysis, Jan 2022

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We prove long-time existence for the negative L^2-gradient flow of the p-elastic energy, p≥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 κ. 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.
A minimising movement scheme for the $p$-elastic energy of curves

Simon Blatt, Christopher P. Hopper, and Nicole Vorderobermeier

Journal of Evolution Equations, Jan 2022

Id/No 41

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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.
A Möbius invariant discretization of O’Hara’s Möbius energy

Simon Blatt, Aya Ishizeki, and Takeyuki Nagasawa

Journal of Knot Theory and its Ramifications, Jan 2022

Id/No 2250016

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We introduce a new discretization of O’Hara’s Möbius energy. In contrast to the known discretizations of Simon and Kim and Kusner it is invariant under Möbius transformations of the surrounding space. The starting point for this new discretization is the cosine formula of Doyle and Schramm. We then show Γ-convergence of our discretized energies to the Möbius energy under very natural assumptions.
On O’Hara knot energies. I: Regularity for critical knots

Simon Blatt, Philipp Reiter, and Armin Schikorra

Journal of Differential Geometry, Jan 2022

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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.

2021

Scale-invariant tangent-point energies for knots

Simon Blatt, Philipp Reiter, Armin Schikorra, and Nicole Vorderobermeier

Apr 2021

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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 \mathcalE^p,q and proving that the derivative γ’ of a parametrization of a \rm TP^p,q-critical curve γ induces a critical map with respect to \mathcalE^p,q acting on torus-to-sphere maps.

2020

The gradient flow of the Möbius energy: \varepsilon-regularity and consequences

Simon Blatt

Analysis & PDE, Apr 2020

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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.

2019

A note on singularities in finite time for the \protectL^2 gradient flow of the Helfrich functional

Simon Blatt

Journal of Evolution Equations, Feb 2019

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This work investigates the formation of singularities under the steepest descent L^2-gradient flow of the functional \mathrm W_\lambda_1, \lambda_2, the sum of the Willmore energy, \lambda_1 times the area, and \lambda_2 times the signed volume of an immersed closed surface without boundary in \mathbb R^3. We show that in the case that \lambda_1>1 and \lambda_2=0 any immersion develops singularities in finite time under this flow. If \lambda_1 >0 and \lambda_2 > 0, embedded closed surfaces with energy less than $8\pi+\min{(16 \pi \lambda_1^3)/(3\lambda_2^2), 8\pi} and positive volume evolve singularities in finite time. If in this case the initial surface is a topological sphere and the initial energy is less than 8 \pi, the flow shrinks to a round point in finite time. We furthermore discuss similar results for the case that \lambda_2$ is negative.

2018

Curves Between Lipschitz and \protectC^1 and Their Relation to Geometric Knot Theory

Simon Blatt

The Journal of Geometric Analysis, Nov 2018

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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
On the analyticity of critical points of the Möbius energy

Simon Blatt, and Nicole Vorderobermeier

Calculus of Variations and Partial Differential Equations, Dec 2018

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We prove that smooth critical points of the Möbius energy }}\backslashmathcal {E}}}Eparametrized by arc-length are analytic. Together with the main result in Blatt et al. (Trans Am Math Soc 368(9):6391–6438, 2016) this implies that critical points of the Möbius energy with merely bounded energy are not only }}C^\backslashinfty }}C∞but also analytic. Our proof is based on Cauchy’s method of majorants and a decomposition of the gradient which has already proved useful in the proof of the regularity results in Blatt and Reiter (Manuscr Math 140(1–2):29–50, 2013) and Blatt et al. (Trans Am Math Soc 368(9):6391–6438, 2016). To the best of the authors knowledge, this is the first analyticity result in the context of non-local differential equations.

2017

The gradient flow of O’Hara’s knot energies

Simon Blatt

Mathematische Annalen, Apr 2017

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Jun O’Hara invented a family of knot energies E^j,p, j,p ∈(0, ∞). We study the negative gradient flow of the sum of one of the energies E^α= E^α,1, α∈(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.
Monotonicity formulas for extrinsic triharmonic maps and the triharmonic Lane–Emden equation

Simon Blatt

Journal of Differential Equations, Apr 2017

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We derive monotonicity formulas and \varepsilon-regularity results for extrinsic triharmonic maps and triharmonic Lane-Emden equations. As an application, we prove partial regularity results for both types of equations in the supercritical regime.

2016

Well-posedness of the supercritical Lane–Emden heat flow in Morrey spaces

Simon Blatt, and Michael Struwe

ESAIM: Control, Optimisation and Calculus of Variations, Apr 2016

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For any smoothly bounded domain Ω⊂\mathbb R^n, n≥3, and any exponent p>2^*=2n/(n-2) we study the Lane-Emden heat flow u_t-∆u = |u|^p-2u on Ω\times]0,∞[ and establish local and global well-posedness results for the initial value problem with suitably small initial data u\big|_t=0=u_0 in the Morrey space L^2,λ(Ω), where λ=4/(p-2). We contrast our results with results on instantaneous complete blow-up of the flow for certain large data in this space, similar to ill-posedness results of Galaktionov-Vazquez for the Lane-Emden flow on \mathbb R^n.

2015

Regularity theory for tangent-point energies: the non-degenerate sub-critical case

Simon Blatt, and Philipp Reiter

Adv. Calc. Var., Apr 2015

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In this article we introduce and investigate a new two-parameter family of knot energies TP^(p,q) that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the integrand in the original definition of the tangent-point energies. We will first characterize the curves of finite energy in the sub-critical range p∈(q+2,2q+1) and see that those are all injective and regular curves in the Sobolev-Slobodeckiĭ space W^(p-1)/q,q. We derive a formula for the first variation that turns out to be a non-degenerate elliptic operator for the special case q=2 — a fact that seems not to be the case for the original tangent-point energies. This observation allows us to prove that stationary points of TP^(p,2) + λlength, p∈(4,5), λ> 0, are smooth — so especially all local minimizers are smooth.
Towards a regularity theory for integral Menger curvature

Simon Blatt, and Philipp Reiter

Ann. Acad. Sci. Fenn. Math., Apr 2015

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We generalize the notion of integral Menger curvature introduced by Gonzalez and Maddocks by decoupling the powers in the integrand. This leads to a new two-parameter family of knot energies intM^p,q. We classify finite-energy curves in terms of Sobolev-Slobodeckij spaces. Moreover, restricting to the range of parameters leading to a sub-critical Euler-Lagrange equation, we prove existence of minimizers within any knot class via a uniform bi-Lipschitz bound. Consequently, intM^p,q is a knot energy in the sense of O’Hara. Restricting to the non-degenerate sub-critical case, a suitable decomposition of the first variation allows to establish a bootstrapping argument that leads to C^∞-smoothness of critical points.
Harmonic analysis meets critical knots. Critical points of the Möbius energy are smooth

Simon Blatt, Philipp Reiter, and Armin Schikorra

Transactions of the American Mathematical Society, Apr 2015

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Motivated by the Coulomb potential of an equidistributed charge on a curve, Jun O’Hara introduced and investigated the first geometric knot energy, the MÃ¶bius energy. We prove that every critical curve of this MÃ¶bius energy is of class and thus extend the corresponding result due to Freedman, He, and Wang for minimizers of the MÃ¶bius energy.
An Analytic Framework for the Supercritical Lane–Emden Equation and its Gradient Flow

Simon Blatt, and Michael Struwe

Int. Math. Res. Not. IMRN, Apr 2015

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The natural setting for the Lane-Emden equation -∆u = |u|^p-2u on a domain Ω⊂\R^n, n\ge 3, for supercritical exponents p>2^*=2n/(n-2) is identified as the space of functions u∈H^1_0∩L^p(Ω) with finite scale-invariant Morrey norms. We show that this Morrey regularity is propagated by the heat flow associated with this equation, and we study the blow-up profiles.
Boundary regularity for the supercritical Lane-Emden heat flow

Simon Blatt, and Michael Struwe

Calculus of Variations and Partial Differential Equations, Apr 2015

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We establish partial regularity up to the boundary for the Lane-Emden heat flow u_t-∆u = |u|^p-2u on a domain Ω⊂\R^n, n\ge 3, for exponents p>2^*=2n/(n-2). We obtain full regularity near the boundary at blow-up time in the mildly supercritical regime when 2^*<p<2^+=2(n-1)/(n-3).
Erratum to: Boundary regularity for the supercritical Lane-Emden heat flow

Simon Blatt, and Michael Struwe

Calculus of Variations and Partial Differential Equations, Apr 2015

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2014

How nice are critical knots? Regularity theory for knot energies

Simon Blatt, and Philipp Reiter

Journal of Physics: Conference Series, Apr 2014

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In this note we report on some recent developments in geometric knot theory which aims at finding links between geometric properties of a given knotted curve and its knot type. The central object of this field are so-called knot energies which are defined on closed embedded curves. First we present three important examples of two-parameter knot energy families, namely O’Hara’s energies, the (generalized) integral Menger curvature, and the (generalized) tangent- point energies. Subsequently we outline the main steps that lead to C inf -regularity of stationary points- especially minimizers-in the non-degenerate sub-critical range of parameters. Particular attention is devoted to the appearing parallels between these energies which, surprisingly at first glance, are quite similar from an analyst’s perspective.
Modeling repulsive forces on fibres via knot energies

Simon Blatt, and Philipp Reiter

Molecular Based Mathematical Biology, Apr 2014

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2013

A note on integral Menger curvature for curves

Simon Blatt

Math. Nachr., Apr 2013

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For continuously differentiable embedded curves γ, we will give a necessary and sufficient condition for the boundedness of integral versions of the Menger curvature. We will show that for p>3 the integral Menger curvature \mathfrak M_p(γ) is finite if and only if γbelongs to the Sobolev Slobodeckij space W^2-\frac 2 p,p(\mathbb R / \mathbb Z, \mathbb R^n). The quantity \mathfrak J_p (γ) - defined by taking the supremum of the p-th power of Menger curvature with respect to one variable and then integrating over the remaining two - is finite for p>2, if and only if γbelongs to the space W^2-\frac 1 p,p.
The energy spaces of the tangent point energies

Simon Blatt

J. Topol. Anal., Apr 2013

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In this small note, we will give a necessary and sufficient condition under which the tangent point energies introduced by Heiko von der Mosel and Pawel Strzelecki are bounded. We show that an admissible submanifold has bounded E_q-energy if and only if it is injective and locally agrees with the graph of functions that belong to Sobolev-Slobodeckij space W^2- \frac m q, q. The known Morrey embedding theorems of Heiko von der Mosel and Pawel Strzelecki can then be interpreted as standard Morrey embedding theorem for these spaces. Especially, this shows that the Hölder exponents for the embeddings by Heiko von der Mosel and Pasel Stzelecki are sharp.
The gradient flow of the Möbius energy

Simon Blatt

Oberwolfach Reports, Apr 2013

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Stationary points of O’Hara’s knot energies

Simon Blatt, and Philipp Reiter

Manuscripta Math., Apr 2013

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In this article we study the regularity of stationary points of the knot energies E^αintroduced by O’Hara in the range α∈(2,3). In a first step we prove that E^αis C^1 on the set of all regular embedded closed curves belonging to H^(α+1)/2,2 and calculate its derivative. After that we use the structure of the Euler-Lagrange equation to study the regularity of stationary points of E^αplus a positive multiple of the length. We show that stationary points of finite energy are of class C^∞- so especially all local minimizers of E^αamong curves with fixed length are smooth.

2012

The gradient flow of the Möbius energy near local minimizers

Simon Blatt

Calc. Var. Partial Differential Equations, Apr 2012

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In this article we show that for initial data close to local minimizers of the Möbius energy the gradient flow exists for all time and converges smoothly to a local minimizer after suitable reparametrizations. To prove this, we show that the heat flow of the Möbius energy possesses a quasilinear structure which allows us to derive new short-time existence results for this evolution equation and a Łojasiewicz-Simon gradient inequality for the Möbius energy.
Boundedness and regularizing effects of O’Hara’s knot energies

Simon Blatt

J. Knot Theory Ramifications, Apr 2012

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Sharp boundedness and regularizing effects of the integral Menger curvature for submanifolds

Simon Blatt, and Sławomir Kolasiński

Adv. Math., Apr 2012

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In this paper we show that embedded and compact C^1 manifolds have finite integral Menger curvature if and only if they are locally graphs of certain Sobolev-Slobodeckij spaces. Furthermore, we prove that for some intermediate energies of integral Menger type a similar characterization of objects with finite energy can be given.

2010

Loss of convexity and embeddedness for geometric evolution equations of higher order

Simon Blatt

Journal of Evolution Equations, Apr 2010

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We show that for a large class of geometric evolution equations of immersed surfaces in the Euclidean space, there are compact embedded surfaces that lose their embeddedness and compact strictly convex surfaces that lose their convexity under these evolution equations.

2009

On a generalization of Jentzsch’s theorem

Hans-Peter Blatt, Simon Blatt, and Wolfgang Luh

J. Approx. Theory, Apr 2009

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A singular example for the Willmore flow

Simon Blatt

Analysis, Apr 2009

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In this article it will be proven that there are surfaces that become singular under the Willmore flow. For this purpose, we investigate the Willmore flow of radially symmetric immersions of the sphere using a blow-up construction due to E. Kuwert and R. Schätzle. It will be shown that in this situation the blow-up limit is a surface of revolution as well and is either a round sphere or consists of planes and catenoids. Furthermore, we give an estimate for the number of these planes and catenoids in terms of the Willmore energy of the initial surface. This will enable us to show that there are immersions of the sphere with a Willmore energy arbitrarily close to 8π that do not converge to a Willmore immersion under the Willmore flow. Either a small quantum of the curvature concentrates or the diameter of the surface does not stay bounded under the Willmore flow.
Chord-arc constants for submanifolds of arbitrary codimension

Simon Blatt

Adv. Calc. Var., Apr 2009

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In this article we show that for k-dimensional submanifolds of which go through infinity in a smooth way, smallness of the Gromov distortion and some Ahlfors regularity is equivalent to smallness of the BMO norm of the unit normal and globally δ-Reifenberg flatness with small δ. This generalizes a result due to Semmes for hypersurfaces to surfaces of arbitrary codimension.

2008

Compactness results for the Ricci flow

Simon Blatt

Apr 2008

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Does finite knot energy lead to differentiability?

Simon Blatt, and Philipp Reiter

Journal of Knot Theory and Its Ramifications, Apr 2008

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