We study a certain improved fractional Sobolev–Poincaré inequality on domains, which can be considered as a fractional counterpart of the classical Sobolev–Poincaré inequality. We prove the equivalence of the corresponding weak and strong type inequalities; this leads to a simple proof of a strong type inequality on John domains. We also give necessary conditions for the validity of an improved fractional Sobolev–Poincaré inequality, in particular, we show that a domain of finite measure, satisfying this inequality and a ‘separation property’, is a John domain.

We study the variation of the convergence Newton polygon of a differential equation along a smooth Berkovich curve over a non-archimedean complete valued field of characteristic zero. Relying on work of the second author who investigated its properties on affinoid domains of the affine line, we prove that its slopes give rise to continuous functions that factorise by the retraction through a locally finite subgraph of the curve.

We consider the quintic generalized Korteweg–de Vries equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ which is a canonical mass critical problem, for initial data in$H$^{1}close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18].

In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function $${\varphi}$$ with an isolated singularity at 0 in an open subset of $${\mathbb{C}^n}$$ . This threshold is defined as the supremum of constants$c$> 0 such that $${e^{-2c\varphi}}$$ is integrable on a neighborhood of 0. We relate $${c(\varphi)}$$ to the intermediate multiplicity numbers $${e_j(\varphi)}$$ , defined as the Lelong numbers of $${(dd^c\varphi)^j}$$ at 0 (so that in particular $${e_0(\varphi)=1}$$ ). Our main result is that $${c(\varphi)\geqslant\sum_{j=0}^{n-1} e_j(\varphi)/e_{j+1}(\varphi)}$$ . This inequality is shown to be sharp; it simultaneously improves the classical result $${c(\varphi)\geqslant 1/e_1(\varphi)}$$ due to Skoda, as well as the lower estimate $${c(\varphi)\geqslant n/e_n(\varphi)^{1/n}}$$ which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals.

We show that any embedded minimal torus in$S$^{3}is congruent to the Clifford torus. This answers a question posed by H. B. Lawson, Jr., in 1970.

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In a recent important breakthrough D. Christodoulou has solved a long standing problem of General Relativity of evolutionary formation of trapped surfaces in the Einstein-vacuum space-times. He has identified an open set of regular initial conditions on an outgoing null hypersurface (both finite and at past null infinity) leading to a formation a trapped surface in the corresponding vacuum space-time to the future of the initial outgoing hypersurface and another incoming null hypersurface with the prescribed Minkowskian data. In this paper we give a simpler proof for a finite problem by enlarging the admissible set of initial conditions and, consistent with this, relaxing the corresponding propagation estimates just enough that a trapped surface still forms. We also reduce the number of derivatives needed in the argument from two derivatives of the curvature to just one. More importantly, the proof, which can be easily localized with respect to angular sectors, has the potential for further developments.

This paper resolves a number of problems in the perturbation theory of linear operators, linked with the 45-year-old conjecure of M. G. Kreĭn. In particular, we prove that every Lipschitz function is operator-Lipschitz in the Schatten–von Neumann ideals$S$^{$α$}, 1 <$α$< ∞. Alternatively, for every 1 <$α$< ∞, there is a constant$c$_{$α$}> 0 such that $$ {\left\| {f(a) - f(b)} \right\|_{\alpha }} \leqslant {c_{\alpha }}{\left\| f \right\|_{{{\text{Lip}}\,{1}}}}{\left\| {a - b} \right\|_{\alpha }}, $$ where$f$is a Lipschitz function with $$ {\left\| f \right\|_{{{\text{Lip}}\,{1}}}}: = \mathop{{\sup }}\limits_{{_{{\lambda \ne \mu }}^{{\lambda, \mu \in \mathbb{R}}}}} \left| {\frac{{f\left( \lambda \right) - f\left( \mu \right)}}{{\lambda - \mu }}} \right| < \infty, $$ $$ {\left\| \cdot \right\|_{\alpha }} $$ is the norm is$S$^{$α$}, and$a$and$b$are self-adjoint linear operators such that $$ a - b \in {S^{\alpha }} $$ .

We prove a quantitative bi-Lipschitz non-embedding theorem for the Heisenberg group with its Carnot–Carathéodory metric and apply it to give a lower bound on the integrality gap of the Goemans–Linial semidefinite relaxation of the sparsest cut problem.

An analogue of the well-known $$ \frac{3}{{16}} $$ lower bound for the first eigenvalue of the Laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non-elementary subgroup$L$of SL(2,$Z$). The proof in the case that the Hausdorff of the limit set of$L$is bigger than $$ \frac{1}{2} $$ is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting. In the case that delta is less than $$ \frac{1}{2} $$ we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function. These “spectral gaps” are then applied to sieving problems on orbits of such groups.