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We study the fine geometric structure of bifurcation currents in the parameter space of cubic polynomials viewed as dynamical systems. In particular we prove that these currents have some laminar structure in a large region of parameter…

We deal with locally free $\mathcal{O}_X$-modules with connection over a Berkovich curve $X$. As a main result we prove local and global decomposition theorems of such objects by the radii of convergence of their solutions. We also derive a…

Minkowski's concept of a four-dimensional physical space is a central paradigm of modern physics. The three-dimensional Maxwellian electrodynamics is uniquely generalized to the covariant four-dimensional form. Is the (1+3) decomposition of…

Conics in the Euclidean space have been known for their geometrical beauty and also for their power to model several phenomena in real life. It usually happens that when thinking about the conics in a semi-Riemannian manifold, the equations…

We give streamlined proofs of theorems of S.\ Smirnov about the decomposition of vector fields of measures into curves.

We study an axisymmetric metric satisfying the Petrov type D property with some additional ansatze, but without assuming the vacuum condition. We find that our metric in turn becomes conformal to the Kerr metric deformed by one function of…

We prove that a Sobolev map from a Riemannian manifold into a complete metric space pushes forward almost every compactly supported integral current to an Ambrosio--Kirchheim integral current in the metric target, where "almost every" is…

We analyze the moduli space of spontaneously broken N=8 supergravity theories in 4 dimensions with classical Minkowski vacua. We find that all the known classical vacua, as well as the several new ones we construct here, can be connected by…

We present some results concerning currents of integration on finite-dimensional analytic spaces in Hilbert spaces, using the setting of metric currents. In particular, we obtain the characterization of such currents as positive closed…

We use Fra\" iss\'e theoretic methods to construct several universal and ultrahomogeneous Polish metric structures. Namely, universal and ultrahomogeneous Polish metric space equipped with countably many closed subsets of its powers,…

In this paper we prove an isoperimetric inequality of euclidean type for complete metric spaces admitting a cone-type inequality. These include all Banach spaces and all complete, simply-connected metric spaces of non-positive curvature in…

In a separably connected space any two points are contained in a separable connected subset. We show a mechanism that takes a connected bounded metric space and produces a complete connected metric space whose separablewise components form…

In this article we prove completeness results for Sobolev metrics with nonconstant coefficients on the space of immersed curves and on the space of unparametrized curves. We provide necessary as well as sufficient conditions for the…

A method is presented which allows the exact construction of conserved (i.e. divergence-free) current vectors from appropriate sets of multipole moments. Physically, such objects may be taken to represent the flux of particles or electric…

We introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally decomposed as a product manifold…

We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of…

We investigate trend to equilibrium for the damped wave equation with a confining potential in the Euclidean space. We provide with necessary and sufficient geometric conditions for the energy to decay exponentially uniformly. The proofs…

This paper is concerned with rational curves on real classical groups. Our contributions are three-fold: (i) We determine the structure of quadratic rational curves on real classical groups. As a consequence, we completely classify…

A basic problem in quantizing a field in curved space is the decomposition of the classical modes in positive and negative frequency. The decomposition is equivalent to a choice of a complex structure in the space of classical solutions. In…

We prove that if an orientable 3-manifold $M$ admits a complete Riemannian metric whose scalar curvature is positive and has at most $C$-quadratic decay at infinity for some $C > \frac{2}{3}$, then it decomposes as a (possibly infinite)…