Related papers: On Poletsky type inequality for mappings of Rieman…

We establish a local model for the moduli space of holomorphic symplectic structures with logarithmic poles, near the locus of structures whose polar divisor is normal crossings. In contrast to the case without poles, the moduli space is…

We state a new Calderon-Zygmund decomposition for Sobolev spaces on a doubling Riemannian manifold. Our hypotheses are weaker than those of the already known decomposition which used classical Poincare inequalities.

For $4 \nmid L$ and $g$ large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level $L$ structures. In particular, we determine the divisibility properties of the…

We extend the validity of a Gromov's dimension comparison estimate for topological hypersurfaces to sufficiently large classes of rectifiable sets, arising from Sobolev mappings. Our tools are a suitably weak exterior differentiation for…

We study compactness and boundedness of embeddings from Sobolev type spaces on metric spaces into $L^q$ spaces with respect to another measure. The considered Sobolev spaces can be of fractional order and some statements allow also…

In this paper we give characterizations of mappings generate embeddings of Sobolev spaces in the terms of ring capacity inequalities. In addition we prove that such mappings are Lipschitz mappings in the sub-hyperbolic type capacitory…

For mappings in metric spaces satisfying one inequality with respect to modulus of families of curves, there is proved a lightness of preimage under the mapping. It is proved that, the mappings, satisfying estimate mentioned above, are…

We explore the phenomenon of cavitation in higher-dimensional elasticity, defining it as the mapping of a punctured ball onto a non-degenerate ring domain. Crucially, for the class of locally quasiconformal mappings (or more general…

We provide explicit bounds on the difference of heights of the $j$-invariants of isogenous elliptic curves defined over $\overline{\mathbb{Q}}$. The first one is reminiscent of a classical estimate for the Faltings height of isogenous…

The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class…

We show that on almost complex surfaces plurisubharmonic functions can be locally approximated by smooth plurisubharmonic functions. The main tool is the Poletsky type theorem due to U. Kuzman.

Interpolation inequalities play an important role in the study of PDEs and their applications. There are still some interesting open questions and problems that related to integral estimates and regularity of solutions to the elliptic…

We prove non local Hardy inequalities on Carnot groups and Riemannian manifolds, relying on integral representations of fractional Sobolev norms.

We prove non local Hardy inequalities on Carnot groups and Riemannian manifolds, relying on integral representations of fractional Sobolev norms.

The paper deals with planar polynomial vector fields. We aim to estimate the number of orbital topological equivalence classes for the fields of degree n. An evident obstacle for this is the second part of Hilbert's 16th problem. To…

We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky's global Torelli theorem for the…

We consider a large class of 2D area-preserving diffeomorphisms that are not uniformly hyperbolic but have strong hyperbolicity properties on large regions of their phase spaces. A prime example is the Standard map. Lower bounds for…

The modular spaces are a family of polarizations of the Hilbert space that are based on Aharonov's modular variables and carry a rich geometric structure. We construct here, step by step, a Feynman path integral for the quantum harmonic…

We establish boundary regularity estimates for elliptic systems in divergence form with VMO coefficients. Additionally, we obtain nondegeneracy estimates of the Hopf-Oleinik type lemma for elliptic equations. In both cases, the moduli of…

Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities. Sobolev…