Related papers: On holomorphic flows on Stein surfaces: transversa…

Two flows on a finite-dimensional normed space $X$ are equivalent if some homeomorphism $h$ of $X$ preserves all orbits, i.e., $h$ maps each orbit onto an orbit. Under the assumption that $h$, $h^{-1}$ both are $\beta$-H\"{o}lder continuous…

Structurally stable (rough) flows on surfaces have only finitely many singularities and finitely many closed orbits, all of which are hyperbolic, and they have no trajectories joining saddle points. The violation of the last property leads…

We prove that every Stein manifold X of dimension n admits [(n+1)/2] holomorphic functions with pointwise independent differentials, and this number is maximal for every n. In particular, X admits a holomorphic function without critical…

We study joining rigidity in the class of von Neumann flows with one singularity. They are given by a smooth vector field $\mathcal{X}$ on $\mathbb T^2\setminus \{a\}$, where $\mathcal{X}$ is not defined at $a\in \mathbb T^2$. It follows…

Let $\AAutH (X)$ be the subgroup of the group $\AutH (X)$ of holomorphic automorphisms of a normal affine algebraic surface $X$ generated by elements of flows associated with complete algebraic vector fields. Our main result is a…

Let $Q$ be a smooth compact orientable 3--manifold with smooth boundary $\partial Q$. Let $\mathcal{B}$ be the set of exact 2--forms $B\in\Omega^2(Q)$ such that $j_{\partial Q}^*B=0$, where $j_{\partial Q}:{\partial Q}\to Q$ is the…

Flows on surfaces are one of the most fundamental and classical objects in dynamical systems, and are studied from various areas (e.g. integrable systems, differential equations, fluid mechanics). Though hyperbolic flows and recurrent flows…

Hamiltonian flows on compact surfaces are characterized, and the topological invariants of such flows with finitely many singular points are constructed from the viewpoints of integrable systems, fluid mechanics, and dynamical systems.…

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that, when the genus of the surface is two, almost every such locally Hamiltonian flow with…

We perform a systematic analysis of flow-like solutions in theories of Einstein gravity coupled to multiple scalar fields, which arise as holographic RG flows as well as in the context of cosmological solutions driven by scalars. We use the…

In this paper, we classify all noncollapsed singularities of the mean curvature flow in $\mathbb{R}^4$. Specifically, we prove that any ancient noncollapsed solution either is one of the classical historical examples (namely…

We prove that any Inoue surface admits a unique holomorphic connection. Using this result we show that two Inoue surfaces $S=H\times\mathbb{C}/G$, $S'=H\times\mathbb{C}/G'$ are biholomorphic if and only if $G$, $G'$ are conjugate in the…

In this paper we show that every complex hypersurface $A$ in a Stein manifold $X$ with $H^2(X;\mathbb Z)=0$ is the divisor of a holomorphic function $f$ on $X$ whose critical points are precisely the singular points of $A$. A similar result…

We solve the problem of topological classification for smooth structurally stable flows on closed four-dimensional manifolds, the non-wandering set of which contains exactly two saddle equilibria, and the wandering set contains isolated…

In this paper, we study transversely holomorphic partially hyperbolic flows, i.e. those whose holonomy pseudo-group is given by biholomorphic maps. We prove in the seven-dimensional case that under the assumption that the subcenter…

By studying the weak closure of multidimensional off-diagonal self-joinings we provide a criterion for non-isomorphism of a flow with its inverse, hence the non-reversibility of a flow. This is applied to special flows over rigid…

We investigate left-invariant Hitchin and hypo flows on $5$-, $6$- and $7$-dimensional Lie groups. They provide Riemannian cohomogeneity-one manifolds of one dimension higher with holonomy contained in $SU(3)$, $G_2$ and $Spin(7)$,…

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that almost every such locally Hamiltonian flow with only simple saddles has singular…

We study $2$-dimensional unit vector flows on graphs, that is, nowhere-zero flows that assign to each oriented edge a unit vector in $\mathbb R^{3}$. We give a new geometric characterization of $\mathbb S^{2}$-flows on cubic graphs. We also…

We study mean curvature flow of smooth, axially symmetric surfaces in $\mathbb{R}^3$ with Neumann boundary data. We show that all singularities at the first singular time must be of type I.