Harmonic Map Heat Flow Research Articles

Flows of geometric structures

Abstract We develop an abstract theory of flows of geometric 𝐻-structures, i.e., flows of tensor fields defining 𝐻-reductions of the frame bundle, for a closed and connected subgroup H ⊂ SO ⁢ ( n ) H\subset\mathrm{SO}(n) , on any connected and oriented 𝑛-manifold with sufficient topology to admit such structures. The first part of the article sets up a unifying theoretical framework for deformations of 𝐻-structures, by way of the natural infinitesimal action of GL ⁢ ( n , R ) \mathrm{GL}(n,\mathbb{R}) on tensors combined with various bundle decompositions induced by 𝐻-structures. We compute evolution equations for the intrinsic torsion under general flows of 𝐻-structures and, as applications, we obtain general Bianchi-type identities for 𝐻-structures, and, for closed manifolds, a general first variation formula for the L 2 L^{2} -Dirichlet energy functional ℰ on the space of 𝐻-structures. We then specialise the theory to the negative gradient flow of ℰ over isometric 𝐻-structures, i.e., their harmonic flow. The core result is an almost-monotonocity formula along the flow for a scale-invariant localised energy, similar to the classical formulas by Chen–Struwe [M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Differential Geom. 28 (1988), 3, 485–502; Y. M. Chen and M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201 (1989), 1, 83–103] for the harmonic map heat flow. This yields an 𝜀-regularity theorem and an energy gap result for harmonic structures, as well as long-time existence for the flow under small initial energy, relative to the L ∞ L^{\infty} -norm of initial torsion, in the spirit of Chen–Ding [Y. M. Chen and W. Y. Ding, Blow-up and global existence for heat flows of harmonic maps, Invent. Math. 99 (1990), 3, 567–578]. Moreover, below a certain energy level, the absence of a torsion-free isometric 𝐻-structure in the initial homotopy class imposes the formation of finite-time singularities. These seemingly contrasting statements are illustrated by examples on flat 𝑛-tori, so long as the set [ S n , SO ⁢ ( n ) / H ] [\mathbb{S}^{n},\mathrm{SO}(n)/H] of homotopy classes of maps S n → SO ⁢ ( n ) / H \mathbb{S}^{n}\to\mathrm{SO}(n)/H contains more than one element and the universal cover of SO ⁢ ( n ) / H \mathrm{SO}(n)/H is a sphere, e.g. when n = 7 n=7 and H = G 2 H=\mathrm{G}_{2} , or n = 8 n=8 and H = Spin ⁢ ( 7 ) H=\mathrm{Spin}(7) .

Finite Time Blow-up for Heat Flows of Self-induced Harmonic Maps

Finite Time Blow-up for Heat Flows of Self-induced Harmonic Maps

Existence and stability of shrinkers for the harmonic map heat flow in higher dimensions

We study singularity formation for the heat flow of harmonic maps from Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^d$$\\end{document}. For each d≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 4$$\\end{document}, we construct a compact, d-dimensional, rotationally symmetric target manifold that allows for the existence of a corotational self-similar shrinking solution (shortly shrinker) that represents a stable blowup mechanism for the corresponding Cauchy problem.

Diffusive stability and self-similar decay for the harmonic map heat flow

In this paper we study the harmonic map heat flow on the euclidean space Rd and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space B˙p,∞dp(Rd) where d<p<∞. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form ‖∇u(t)‖L∞(Rd)≤Ct−12.Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity.

Bubble decomposition for the harmonic map heat flow in the equivariant case

We consider the harmonic map heat flow for maps mathbb {R}^{2} rightarrow mathbb {S}^2, under equivariant symmetry. It is known that solutions to the initial value problem can exhibit bubbling along a sequence of times—the solution decouples into a superposition of harmonic maps concentrating at different scales and a body map that accounts for the rest of the energy. We prove that this bubble decomposition is unique and occurs continuously in time. The main new ingredient in the proof is the notion of a collision interval from Jendrej and Lawrie (J Amer Math Soc).

A New Conformal Heat Flow of Harmonic Maps

A New Conformal Heat Flow of Harmonic Maps

Localized Sequential Bubbling for the Radial Energy Critical Semilinear Heat Equation

In this expository note, we prove a localized bubbling result for solutions of the energy critical nonlinear heat equation with bounded H˙1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dot{H} ^1$$\\end{document} norm. The proof uses a combination of Gérard’s profile decomposition (ESAIM Control Optim. Calc. Var. 3: 213–233, 1998), concentration compactness techniques in the spirit of Duyckaerts, Kenig, and Merle’s seminal work (Geom. Funct. Anal. 22: 639–698, 2012), and a virial argument in the spirit of Jia and Kenig’s work (Amer. J. Math. 139: 1521–1603, 2017) to deduce the vanishing of the error in the neck regions between the bubbles. The argument is based closely on an analogous lemma proved in the author’s recent work with Jendrej (arXiv:2210.14963, 2022) on the equivariant harmonic map heat flow in dimension two.

Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints

Abstract An error estimate for a canonical discretization of the harmonic map heat flow into spheres is derived. The numerical scheme uses standard finite elements with a nodal treatment of linearized unit-length constraints. The analysis is based on elementary approximation results and only uses the discrete weak formulation.

Nematic Liquid Crystal Flow with Partially Free Boundary

We study a simplified Ericksen-Leslie system modeling the flow of nematic liquid crystals with partially free boundary conditions. It is a coupled system between the Navier-Stokes equation for the fluid velocity with a transported heat flow of harmonic maps, and both of these parabolic equations are critical for analysis in two dimensions. The boundary conditions are physically natural and they correspond to the Navier slip boundary condition with zero friction for the velocity field and a Plateau-Neumann type boundary condition for the map. In this paper we construct smooth solutions of this coupled system that blow up in finite time at any finitely many given points on the boundary or in the interior of the domain.

Matrix-valued Allen–Cahn equation and the Keller–Rubinstein–Sternberg problem

In this paper, we consider the sharp interface limit of a matrix-valued Allen–Cahn equation, which takes the form: $$\begin{aligned} \partial _t{\textbf{A}}=\Delta {\textbf{A}}-\varepsilon ^{-2}({\textbf{A}}{\textbf{A}}^{\textrm{T}}{\textbf{A}}-{\textbf{A}})\quad \text {with}\quad {\textbf{A}}:\Omega \subset {\mathbb {R}}^m\rightarrow {\mathbb {R}}^{n\times n}. \end{aligned}$$ We show that the sharp interface system is a two-phases flow system: the interface evolves according to the motion by mean curvature; in the two bulk phase regions, the solution obeys the heat flow of harmonic maps with values in $$O^+(n)$$ and $$O^-(n)$$ (represent the sets of $$n\times n$$ orthogonal matrices with determinant $$+1$$ and $$-1$$ respectively); on the interface, the phase matrices on two sides satisfy a novel mixed boundary condition. The above result provides a solution to the Keller–Rubinstein–Sternberg’s problem in the O(n) setting. Our proof relies on two key ingredients. First, in order to construct the approximate solutions by matched asymptotic expansions, as the standard approach does not seem to work, we introduce the notion of quasi-minimal connecting orbits. They satisfy the usual leading order equations up to some small higher order terms. In addition, the linearized systems around these quasi-minimal orbits needs to be solvable up to some good remainders. These flexibilities are needed for the possible “degenerations” and higher dimensional kernels for the linearized operators on matrix-valued functions due to intriguing boundary conditions at the sharp interface. The second key point is to establish a spectral uniform lower bound estimate for the linearized operator around approximate solutions. To this end, we introduce additional decompositions to reduce the problem into the coercive estimates of several linearized operators for scalar functions and some singular product estimates which are accomplished by exploring special cancellation structures between eigenfunctions of these linearized operators.

Short-Time Existence for Harmonic Map Heat Flow with Time-Dependent Metrics

In this work, we obtain a short time existence result for harmonic map heat flow coupled with a smooth family of complete metrics in the domain manifold. Our results generalize short time existence results for harmonic map heat flow by Li-Tam [The heat equation and harmonic maps of complete manifolds, Invent. Math., 1991] and Chen-Zhu [Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geometry, 2006]. In particular, we prove the short time existence of harmonic map heat flow along a complete Ricci flow $g(t)$ on $M$ into a complete manifold with curvature bounded from above with a smooth initial map of uniformly bounded energy density, under the assumptions that $|\text{Rm}(g(t))|\leq a/t$ and $g(t)$ is uniformly equivalent to $g(0)$.

Equivariant heat and Schrödinger flows from Euclidean space to complex projective space

We study the equivariant harmonic map heat flow, Schrödinger maps equation, and generalized Landau–Lifshitz equation from \mathbb{C}^n to \mathbb{CP}^n . By means of a careful geometric analysis, we determine a new, highly useful representation of the problem in terms of a PDE for radial functions from \mathbb{C}^n to \mathbb{S}^2 . Using this new representation, we are able to write explicit formulas for the harmonic maps in this context, and prove that they all have infinite energy.We show that the PDEs admit a family of self-similar solutions with smooth profiles; these solutions again have infinite energy, and give an example of regularity breakdown. Then, using a variant of the Hasimoto transformation applied to our new equation for the dynamics, we prove a small-data global well-posedness result when n=2 . This is, to the best of our knowledge, the first global well-posedness result for Schrödinger maps when the complex dimension of the target is greater than 1. In the final section we study a special case of the harmonic map heat flow corresponding to initial data valued in one great circle. We show that the n=2 case of this problem is a borderline case for the standard classification theory for PDEs of its type.

Heat flow for harmonic maps from graphs into Riemannian manifolds

We introduce the notion of harmonic map from a graph into a Riemannian manifold via a discrete version of the energy density. Existence and basic properties are established. Global existence and convergence of the associated heat flow are proved without any assumption on the curvature of the target manifold. We discuss a variant of the Steiner problem which replaces length by elastic energy.

Convergence of Renormalized Finite Element Methods for Heat Flow of Harmonic Maps

Convergence of Renormalized Finite Element Methods for Heat Flow of Harmonic Maps

Ancient gradient flows of elliptic functionals and Morse index

We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions and codimensions, we classify ancient mean curvature flows in ${\bf S}^n$ with low area: they are steady or shrinking equatorial spheres. In the mean curvature flow case in ${\bf S}^3$, we classify ancient flows with more relaxed area bounds: they are steady or shrinking equators or Clifford tori. In the embedded curve shortening case in ${\bf S}^2$, we completely classify ancient flows of bounded length: they are steady or shrinking circles.

Convex hull property for ancient harmonic map heat flows

Convex hull property for ancient harmonic map heat flows

Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometry

We study harmonic map heat flow along ancient super Ricci flow, and derive several Liouville theorems with controlled growth from Perelman’s reduced geometric viewpoint. For non-positively curved target spaces, our growth condition is sharp. For positively curved target spaces, our Liouville theorem is new even in the static case (i.e., for harmonic maps); moreover, we point out that the growth condition can be improved, and almost sharp in the static case. This fills the gap between the Liouville theorem of Choi and the example constructed by Schoen–Uhlenbeck.

Nematic\u2013Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective Geometric Motions

In this work, we study the nematic–isotropic phase transition based on the dynamics of the Landau–De Gennes theory of liquid crystals. At the critical temperature, the Landau–De Gennes bulk potential favors the isotropic phase and nematic phase equally. When the elastic coefficient is much smaller than that of the bulk potential, a scaling limit can be derived by formal asymptotic expansions: the solution gradient concentrates on a closed surface evolving by mean curvature flow. Moreover, on one side of the surface the solution tends to the nematic phase which is governed by the harmonic map heat flow into the sphere while on the other side, it tends to the isotropic phase. To rigorously justify such a scaling limit, we prove a convergence result by combining weak convergence methods and the modulated energy method. Our proof applies as long as the limiting mean curvature flow remains smooth.

The harmonic heat flow of almost complex structures

We define and study the harmonic heat flow for almost complex structures which are compatible with a Riemannian structure ( M , g ) (M, g) . This is a tensor-valued version of harmonic map heat flow. We prove that if the initial almost complex structure J J has small energy (depending on the norm | ∇ J | |\nabla J| ), then the flow exists for all time and converges to a Kähler structure. We also prove that there is a finite time singularity if the initial energy is sufficiently small but there is no Kähler structure in the homotopy class. A main technical tool is a version of monotonicity formula, similar as in the theory of the harmonic map heat flow. We also construct an almost complex structure on a flat four tori with small energy such that the harmonic heat flow blows up at finite time with such an initial data.

Limiting behaviors of high dimensional stochastic spin ensembles

Lattice spin models in statistical physics are used to understand magnetism. Their Hamiltonians are a discrete form of a version of a Dirichlet energy, signifying a relationship to the Harmonic map heat flow equation. The Gibbs distribution, defined with this Hamiltonian, is used in the Metropolis-Hastings (M-H) algorithm to generate dynamics tending towards an equilibrium state. In the limiting situation when the inverse temperature is large, we establish the relationship between the discrete M-H dynamics and the continuous Harmonic map heat flow associated with the Hamiltonian. We show the convergence of the M-H dynamics to the Harmonic map heat flow equation in two steps: First, with fixed lattice size and proper choice of proposal size in one M-H step, the M-H dynamics acts as gradient descent and will be shown to converge to a system of Langevin stochastic differential equations (SDE). Second, with proper scaling of the inverse temperature in the Gibbs distribution and taking the lattice size to infinity, it will be shown that this SDE system converges to the deterministic Harmonic map heat flow equation. Our results are not unexpected, but show remarkable connections between the M-H steps and the SDE Stratonovich formulation, as well as reveal trajectory-wise out of equilibrium dynamics to be related to a canonical PDE system with geometric constraints.