Related papers: Heat Flows with Prescribed Singularities from 3-di…

In this work we study the existence of singular flows satisfying shadowing-like properties. More precisely, we prove that if C1 -vector field on a closed manifold induces a chain-recurrent flow containing an attached hyperbolic singularity…

We develop estimates for the solutions and derive existence and uniqueness results of various local boundary value problems for Dirac equations that improve all relevant results known in the literature. With these estimates at hand, we…

We present a class of singularity free exact cosmological solutions of Einstein's equations describing a perfect fluid with heat flow. It is obtained as generalization of the Senovilla class [1] corresponding to incoherent radiation field.…

A {\em singular hyperbolic attractor} for flows is a partially hyperbolic attractor with singularities (hyperbolic ones) and volume expanding central direction \cite{mpp1}. The geometric Lorenz attractor \cite{gw} is an example of a…

We define and study the harmonic heat flow for almost complex structures which are compatible with a Riemannian structure $(M, g)$. This is a tensor-valued version of harmonic map heat flow. We prove that if the initial almost complex…

Consider a Brody hyperbolic foliation with non-degenerate singularities on a compact complex manifold. We show that the leafwise heat diffusions and the abstract heat diffusions coincide. In particular, this will imply that the abstract…

We first prove stochastic representation formulae for space-time harmonic mappings defined on manifolds with evolving Riemannian metric. We then apply these formulae to derive Liouville type theorems under appropriate curvature conditions.…

In this paper, we study the intrinsic mean curvature flow on certain closed spacelike manifolds, and prove the existence of hyperbolic structures on them.

We examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension. It is shown that, generically, singular data can give rise to two distinct solutions which are both…

In this paper, we study the global existence of classical solutions to the three dimensional ideal invicid compressible and heat conductive fluids with radial symmetrical data in $H^s(\mathbb{R}^3)$. Our proof is based on the symmetric…

We consider the Whitham equations for deformations of hyperelliptic spectral curves, which preserve all periods of a meromorphic differential. If the meromorphic differential has a root at a fixed point of the hyperelliptic involution, then…

We consider the heat flow of Yang-Mills-Higgs functional where the base manifold is a Riemannian surface and the fiber is a compact symplectic manifold. We show that the corresponding Cauchy problem admits a global weak solution for any…

We show that solutions to certain higher-order intrinsic geometric flows on a compact manifold, including some flows generated by the ambient obstruction tensor, are unique. With the goal of providing a complete self-contained proof,…

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 study the spectral theory and the resolvent of the vector field generating the frame flow of closed hyperbolic 3-dimensional manifolds on some family of anisotropic Sobolev spaces. We show the existence of a spectral gap and prove…

The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of…

Let $M$ be a Riemannian manifold and $\Omega$ a smooth domain of $M$. We study the following heat diffusion problem: assume that the initial temperature is equal to $1$, uniformly on $\Omega$, and is $0$ on its complement. Heat will then…

Motivated by questions in detecting minimal surfaces in hyperbolic manifolds, we study the behavior of geometric flows in complete hyperbolic three-manifolds. In most cases the flows develop singularities in finite time. In this paper, we…

We show that the evolution of isoparametric hypersurfaces of Riemannian manifolds by the mean curvature flow is given by a reparametrization of the parallel family in short time, as long as the uniqueness of the mean curvature flow holds…

Let $\Omega$ be a compact Riemannian manifold with smooth boundary and let $u_t$ be the solution of the heat equation on $\Omega$, having constant unit initial data $u_0=1$ and Dirichlet boundary conditions ($u_t=0$ on the boundary, at all…