Simon Blatt

Chord-arc constants for submanifolds of arbitrary codimension

Abstract 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.

THE ENERGY SPACES OF THE TANGENT POINT ENERGIES

In this note, we will give a necessary and sufficient condition under which the tangent point energies introduced by von der Mosel and Strzelecki in [J. Geom. Anal., pp. 1–55 (2011), J. Knot Theory Ramifications21 (2012) 1250044] are bounded. We show that an admissible submanifold has bounded 𝔈q-energy if and only if it is injective and locally agrees with the graph of functions that belong to Sobolev–Slobodeckij space

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

W^{2- \frac{m}{q}, \,q}

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

. The known Morrey embedding theorems of von der Mosel and Strzelecki can then be interpreted as standard Morrey embedding theorems for these spaces. Especially, this shows that the Holder exponents for the embeddings in [J. Geom. Anal., pp. 1–55 (2011)] are sharp.

A singular example for the Willmore flow

Abstract In this article we introduce and investigate a new two-parameter family of knot energies TP (p,q)

Modeling repulsive forces on fibres via knot energies

{\operatorname{TP}^{(p,\,q)}}

The gradient flow of O’Hara’s knot energies

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 TP (p,q)

Harmonic analysis meets critical knots. Critical points of the Möbius energy are smooth

{\operatorname{TP}^{(p,\,q)}}

A Lower Bound for the Gromov Distortion of Knotted Submanifolds

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 (ℝ/ℤ,ℝ n )

BOUNDEDNESS AND REGULARIZING EFFECTS OF O'HARA'S KNOT ENERGIES

{W^{\scriptstyle (p-1)/q,q}(\mathbb {R}/\mathbb {Z},\mathbb {R}^n)}