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Thurston's triangulation conjecture asserts that every hyperbolic 3-manifold admits a geometric decomposition into ideal hyperbolic tetrahedra, a result proven only for certain special 3-manifolds. This paper presents combinatorial Ricci…

The paper is devoted to the study of topological properties, structure and classification of Morse flows with fixed points on the boundary of three-dimensional manifolds. We construct a complete topological invariant of a Morse flow,…

We prove the existence of unique regular solutions of steady-state buoyancy-driven flows of viscous incompressible heat-conducting fluids in 3D open channels under mixed boundary conditions. The model takes into account the phenomena of the…

Let $M$ be a closed, negatively curved Riemannian manifold of dimension $n \neq 4, 8$ with strictly $1/4$-pinched sectional curvature. We prove, that if the frame flow is ergodic and the sum of its unstable and stable bundles together with…

We show that there exist closed three-dimensional Riemannian manifolds where the incompressible Euler equations exhibit smooth steady solutions that are isolated in the $C^1$-topology. The proof of this fact combines ideas from dynamical…

The inclination or $\lambda$-Lemma is a fundamental tool in finite dimensional hyperbolic dynamics. In contrast to finite dimension, we consider the forward semi-flow on the loop space of a closed Riemannian manifold $M$ provided by the…

Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature, provided M is asymptotically harmonic of constant h > 0.

We investigate the motion of a family of closed curves evolving according to the geometric evolution law on a given two dimensional manifold which is embedded or immersed in the three-dimensional Euclidean space. We derive a system of…

In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a…

Given a closed hyperbolic 3-manifold M with a quasigeodesic flow we construct a \pi_1-equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal P to the lifted flow on H^3 has…

In this paper we prove existence, uniqueness and regularity of certain perturbed (subsonic--supersonic) transonic potential flows in a two-dimensional Riemannian manifold with "convergent-divergent" metric, which is an approximate model of…

We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness and regularity. Of particular interest are properties which characterize the underlying…

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all…

Over the last 10 years or so, advanced statistical properties, including exponential decay of correlations, have been established for certain classes of singular hyperbolic flows in three dimensions. The results apply in particular to the…

We study the spectrum of phase transitions with prescribed mean curvature in Riemannian manifolds. These phase transitions are solutions to an inhomogeneous semilinear elliptic PDE that give rise to diffuse objects (varifolds) that limit to…

We study the asymptotic behavior of the heat content on a compact Riemannian manifold with boundary and with singular specific heat and singular initial temperature distributions imposing Robin boundary conditions. Assuming the existence of…

We first prove that, unlike the biharmonic case, there exist triharmonic curves with nonconstant curvature in a suitable Riemannian manifold of arbitrary dimension. We then give the complete classification of triharmonic curves in surfaces…

We study a class of fourth order curvature flows on a compact Riemannian manifold, which includes the gradient flows of a number of quadratic geometric functionals, as for instance the L2 norm of the curvature. Such flows can develop a…

We prove a conjecture of Bernstein that the superconvexity of the heat kernel on hyperbolic space holds in all dimensions and, hence, there is an analog of Huisken's monotonicity formula for mean curvature flow in hyperbolic space of all…

The main objective of this article is to study the mean curvature flow into an ambient compact smooth manifold M with boundary and with a Riemannian metric that evolves by a self-similar solution of the Ricci flow coupled with the harmonic…