arXiv research

The Liouville theorem is proven for V T-harmonic map heat flow.

Paper studies heat flow for maps on manifolds, avoiding singularities.

Exponential rate of convergence for harmonic heat flow maps.

Study heat flow for half-harmonic maps and harmonic maps with free boundary.

Paper studies heat flow for VT harmonic maps on compact manifolds.

New approach to heat flow for half-harmonic maps, related to minimal surfaces.

Global existence and convergence of heat flow for p-harmonic maps.

Study proves short-term existence for harmonic maps under evolving metrics.

Paper studies Laplace operator estimates in harmonic map heat flows.

The heat flow for Dirac-harmonic maps on Riemannian spin manifolds is a modification of the classical heat flow for harmonic maps by coupling it to a spinor. It was introduced by Chen, Jost, Sun, and Zhu as a tool to get a general existence program for Dirac-harmonic maps. For source manifolds with boundary they obtain…

The paper studies harmonic map heat flow stability and decay rates.

Improved Liouville theorems for ancient solutions to V-harmonic map heat flows.

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 structure $J$ has small energy (depending on the norm $|\nabla J|$), then the flow ex…

We show an energy convexity along any harmonic map heat flow with small initial energy and fixed boundary data on the unit 2-disk. In particular, this gives an affirmative answer to a question raised by W. Minicozzi asking whether such harmonic map heat flow converges uniformly in time strongly in the W^{1,2}-topology,…

Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.

Researchers find stable solutions for heat map flow in higher dimensions.

Paper studies smoothness of bi-conformal heat flow on 4-manifolds.

The paper proves a Liouville theorem for heat flows on manifolds with specific curvature conditions.

Harmonic maps from complex plane to hyperbolic space constructed using heat flow.

Paper shows no finite time singularities for smooth conformal heat flow of harmonic maps.

New heat flow for harmonic maps avoids singularities but not bubbles.

Study Liouville theorems for harmonic maps along ancient super Ricci flows.

Continuous time analysis of bubble formation in harmonic maps.

Study shows Ricci flow's convergence and harmonic map heat flow's long-time existence.

The paper examines how parabolic frequency behaves under Ricci flow and Ricci-harmonic flow on manifolds.

In this paper, we first derive a Sobolev inequality along the harmonic-Ricci flow. We then prove a linear parabolic estimate based on the Sobolev inequality and Moser's iteration. As an application, we will obtain an upper bound estimate for the heat kernel under the flow.

We investigate the well-posedness of (i) the heat flow of harmonic maps from $R^n$ to a compact Riemannian manifold without boundary for initial data in BMO; and (ii) the hydrodynamic flow $(u,d)$ of nematic liquid crystals on $\mathbb R^n$ for initial data in $BMO^{-1}\times BMO$.

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. Space-time harmonic mappings which are defined globally in time correspond to an…

New rigidity result for maps between curved spaces.

The paper studies harmonic map heat flow to flat tori, proving ergodic behavior and convergence to hyperbolic measure.

The paper considers the Ricci flow, coupled with the harmonic map flow between two manifolds. We derive estimates for the fundamental solution of the corresponding conjugate heat equation and we prove an analog of Perelman's differential Harnack inequality. As an application, we find a connection between the entropy fu…

New method proves heat flow of harmonic maps into CAT(0) spaces.

In this paper, we establish the uniqueness of heat flow of harmonic maps into (N, h) that have sufficiently small renormalized energies, provided that N is either a unit sphere $S^{k-1}$ or a compact Riemannian homogeneous manifold without boundary. For such a class of solutions, we also establish the convexity propert…

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 stable, and satisfy the local energy inequality. We also discuss how uniqueness can…

In this paper we develop new methods for studying the convergence problem for the heat flow on negatively curved spaces and prove that any quasiconformal map of the sphere $\mathbb{S}^{n-1}$, $n\geq 3$, can be extended to the $n$-dimensional hyperbolic space such that the heat flow starting with this extension converge…

The paper establishes a series of gradient estimates for positive solutions to the heat equation on a manifold $M$ evolving under the Ricci flow, coupled with the harmonic map flow between $M$ and a second manifold $N$. We prove Li-Yau type Harnack inequalities and we consider the cases when $M$ is a complete manifold …

The paper explores heat flow and constants on graphs, proving properties and proposing new concepts.

Study mean curvature flow into evolving manifold with coupled flows.

We estimate the heat kernel on a closed Riemannian manifold $M$, with $dim(M)\geq 3$, evolving under the Ricci-harmonic map flow and the result depends on some constants arising from a Sobolev imbedding theorem. In a special case, when the scalar curvature satisfies a certain natural inequality, we obtain, as a corolla…

Study on contracting maps and their rigidity under curvature constraints.

Proves rigidity of maps between manifolds with scalar curvature constraints.

In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang extended thi…

We study the Ricci flow for initial metrics which are C^0 small perturbations of the Euclidean metric on R^n. In the case that this metric is asymptotically Euclidean, we show that a Ricci harmonic map heat flow exists for all times, and converges uniformly to the Euclidean metric as time approaches infinity. In provin…

Proves uniqueness of Ricci flow with scaling invariant estimates.

Harmonic maps link Teichmüller spaces to framed representations.

Every harmonic map is an intrinsic bi-harmonic map as an absolute minimizer of the intrinsic bi-energy functional, therefore intrinsic bi-harmonic map and its heat flow are more geometrically natural to study, but they are also considerably more difficult analytically than the extrinsic counterparts due to the lack of …

The paper investigates subelliptic harmonic maps with potential using heat flow.

We study harmonic maps from surfaces coupled to a scalar and a two-form potential, which arise as critical points of the action of the full bosonic string. We investigate several analytic and geometric properties of these maps and prove an existence result by the heat flow method.