The paper studies 1-equivariant harmonic map flow behavior from R² to S².
problem Examining the long-term dynamics of 1-equivariant harmonic map flow.
method Constructing and analyzing global solutions for the flow equation.
result Global solutions exhibit trichotomy in long-time asymptotic behavior.
We present effective methods to compute equivariant harmonic maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive convergence of the discrete heat flow to the energy minimizer, with explic…
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…
Harmonic maps link Teichmüller spaces to framed representations.
problem Connecting Teichmüller spaces with framed representations.
method Uses harmonic map heat flow to find unique maps.
result Unique harmonic maps exist under specified conditions.
Constructs harmonic maps between special geometric shapes.
problem Creating harmonic maps between specific types of geometric shapes.
method Equivariant harmonic maps constructed between cohomogeneity one manifolds.
result Developed a method to construct harmonic maps.
This survey studies equivariant harmonic maps arising from Higgs bundles. We explain the non-abelian Hodge correspondence and focus on the role of equivariant harmonic maps in the correspondence. With the preparation, we review current progress towards some open problems in the study of equivariant harmonic maps.
The study constructs equivariant harmonic maps into symmetric spaces with applications to Willmore surfaces.
problem Constructing harmonic maps into symmetric spaces.
method Equivariant primitive harmonic maps construction.
result Examples of S1-equivariant Willmore Moebius strips in S3. Proves unique maps from certain spaces to others.
problem Uniqueness of equivariant harmonic maps into specific spaces.
method Analyzes maps into irreducible symmetric spaces and Euclidean buildings.
result Proves uniqueness of maps for certain actions.
Researchers found uncountable harmonic self-maps in complex projective spaces.
problem Harmonic maps between complex projective spaces.
method Constructing two families of harmonic self-maps using equivariant maps.
result Explicit harmonic self-maps of complex projective spaces constructed and analyzed.
Investigates harmonic self-maps' stability on cohomogeneity one manifolds.
problem Stability of harmonic self-maps on cohomogeneity one manifolds.
method Systematic study of Jacobi equation for harmonic self-maps.
result Explicit solutions for specific cases show identity map's stability.
Harmonic maps depend analytically on representations.
problem Analyzing harmonic maps into symmetric spaces.
method Construction of deformation maps to transform equivariant harmonic maps into a fixed target space.
result Equivariant harmonic maps depend real analytically on the representation.
We prove that a primitive harmonic map is equivariant if and only if it admits a holomorphic potential of degree one. We investigate when the equivariant harmonic map is periodic, and as an application discuss constant mean curvature cylinders with screw motion symmetries.
Solves initial value problem for harmonic maps on specific manifolds.
problem Initial value problem for harmonic maps on cohomogeneity one manifolds.
method Setup and solve the initial value problem using equivariant harmonic maps and regular-singular systems.
result Local existence of harmonic maps in a neighborhood of singular orbits.
We show that the Korevaar-Schoen limit of the sequence of equivariant harmonic maps corresponding to a sequence of irreducible SL2(C) representations of the fundamental group of a compact Riemannian manifold is an equivariant harmonic map to an R-tree which is minimal and whose length function …
Study heat flow for half-harmonic maps and harmonic maps with free boundary.
problem Integrability and regularity of half-harmonic maps and harmonic maps with free boundary.
method Introduced a heat flow associated to half-harmonic maps and constructed weak solutions via Ginzburg-Landau approximation.
result Proved partial regularity of weak solutions in space and time.
Study sesqui-harmonic map flow from Riemannian surfaces
problem Investigate sesqui-harmonic map flow from Riemannian surfaces
method L2-gradient flow of an energy functional
result Generalizes Struwe's regularity result for harmonic maps
New rigidity estimate derived via harmonic map flow.
problem Rigidity of maps from S2 to S2. method Harmonic map flow approach.
result Rigidity estimate derived.
The Liouville theorem is proven for V T-harmonic map heat flow.
problem Proving Liouville theorems for V T-harmonic maps.
method Analyzing heat flow on manifolds with specific properties.
result Liouville theorems established for V T-harmonic maps.
Study proves existence of non-trivial harmonic map flows to hemispheres.
problem Existence of non-trivial harmonic map flows to hemispheres.
method Construction of infinitely many weak solutions to harmonic map flow starting from non-minimizing but stationary maps.
result Proves existence of non-trivial self-expanding harmonic map flows to hemispheres.
Paper studies heat flow for maps on manifolds, avoiding singularities.
problem Avoiding singularities in heat flow for maps on manifolds.
method Introduces regularized conformal heat flow for n-harmonic maps. result Regularized n-conformal heat flow does not develop finite time singularities. Paper proves energy identity and no-neck property for special harmonic maps.
problem Analyzing special harmonic maps with homogeneous targets.
method Introduced equivariant embedding for ε-harmonic case. result Energy identity and no-neck property established for ε- and α-harmonic maps. The paper describes a parametrisation of harmonic maps from a 2-torus to the 3-sphere.
problem Understanding the moduli space of equivariant harmonic maps from a 2-torus to the 3-sphere.
method Explicit parametrisation using spectral data and line bundles.
result The space of spectral data is a fibre bundle over the space of spectral curves, with nontrivial structure for certain invariance groups.
Exponential rate of convergence for harmonic heat flow maps.
problem Analyzing the convergence rate of harmonic heat flow maps.
method Proving exponential convergence rate for harmonic heat flow maps.
result Exponential convergence rate of the harmonic heat flow.
Harmonic map flow preserves almost-holomorphic maps without singularities.
problem Preserving almost-holomorphic maps without singularities.
method Harmonic map flow applied to almost-holomorphic maps.
result No singularities or necks appear in the limit at singular times.
Global existence and convergence of heat flow for p-harmonic maps.
problem Global existence and convergence of heat flow for p-harmonic maps between manifolds.
method Analysis of heat flow equations for p-harmonic maps.
result Global existence and convergence of heat flow for p-harmonic maps under certain conditions.
Study proves short-term existence for harmonic maps under evolving metrics.
problem Analyzing harmonic maps under time-dependent metrics.
method Proves short-term existence for harmonic map heat flow coupled with a smooth family of complete metrics.
result Generalizes short-term existence results for harmonic map heat flow.
Strict type-II blowup in harmonic map flow is proven to have Hölder continuous body map.
problem Finite-time singularity of harmonic map flow.
method Analysis of outer energy scale and Hölder continuity proof.
result Strictly type-II blowup body map is Hölder continuous.
In the previous paper [GLM2018], we showed that the theory of harmonic maps between Riemannian manifolds may be discretized by introducing triangulations with vertex and edge weights on the domain manifold. In the present paper, we study convergence of the discrete theory to the smooth theory when taking finer and fine…
The paper studies harmonic map flows and proves rectifiability of singular sets.
problem Understanding the structure of singular sets in harmonic map flows.
method Investigates the stratification theory for suitable solutions using tangent measures.
result Each time slice of the singular set is rectifiable.
Paper studies heat flow for VT harmonic maps on compact manifolds.
problem Existence of VT harmonic maps and geodesics on compact manifolds.
method Heat flow method to solve Dirichlet problem and existence of geodesics.
result Existence of VT harmonic maps and geodesics under certain conditions.
New approach to heat flow for half-harmonic maps, related to minimal surfaces.
problem Heat flow for half-harmonic maps from S1 to closed target manifolds. method Classical approach using Dirichlet-to-Neumann operator for the Laplace equation.
result Analogous results to 1985 harmonic map heat flow, valid for finite-energy data.
Paper shows no finite time singularities for smooth conformal heat flow of harmonic maps.
problem Smoothness of conformal heat flow of harmonic maps.
method Combines harmonic map flow with metric evolution in conformal direction.
result No finite time singularity occurs for the flow, and under certain conditions, maps converge to a point.
Study connects landslide flow to integrable systems for harmonic maps.
problem Understanding the holonomy of complex landslide flow.
method Integrable systems approach to harmonic maps into symmetric spaces.
result Holonomy of complex landslide flow derived from harmonic map holonomy.
Harmonic maps from complex plane to hyperbolic space constructed using heat flow.
problem Constructing harmonic maps from complex plane to hyperbolic space.
method Heat flow method to construct harmonic maps.
result Harmonic maps are unique once the principal part of their Hopf differential is prescribed.
Harmonic map flow's singularity properties proven with Lojasiewicz inequalities.
problem Finite-time singularities of harmonic map flow in critical dimensions.
method Proving a weighted Lojasiewicz inequality.
result Continuity of body map and no-neck property for bubble-tree decompositions.
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…
Study Liouville theorems for harmonic maps along ancient super Ricci flows.
problem Proving Liouville theorems for harmonic maps under specific geometric conditions.
method Using Perelman's reduced geometric viewpoint, derive Liouville theorems with controlled growth.
result Sharp growth conditions and new Liouville theorems for both non-positively and positively curved target spaces.
The purpose of the article is to study a foliation associated to a lattice-equivariant harmonic map of small rank from a complex ball to another. The result is related to rigidity of some complex ball quotients.
In this work we study properties of stability and non-stability of harmonic maps under the homogeneous Ricci flow. We provide examples where the stability (non-stability) is preserved under the Ricci flow and an example where the Ricci flow does not preserve the stability of an harmonic map.
In this paper we study the Teichmüller harmonic map flow as introduced by Rupflin and Topping [15]. It evolves pairs of maps and metrics (u,g) into branched minimal immersions, or equivalently into weakly conformal harmonic maps, where u maps from a fixed closed surface M with metric g to a general target manif…
Translating or rotating an input image should not affect the results of many computer vision tasks. Convolutional neural networks (CNNs) are already translation equivariant: input image translations produce proportionate feature map translations. This is not the case for rotations. Global rotation equivariance is typic…
Compactifies character varieties for group actions.
problem Compactify character varieties for better topological analysis.
method Real spectrum compactification of character varieties in semisimple Lie groups.
result Provides a compactification with good topological properties.
Paper studies Laplace operator estimates in harmonic map heat flows.
problem Estimating Laplace operator in harmonic map heat flows outside singularities.
method Investigates estimates using spherical coordinates for T2 and T3 boundary conditions. result Provides higher-order estimates for the Ericksen--Leslie system.
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,…
Improved Liouville theorems for ancient solutions to V-harmonic map heat flows.
problem Establishing Liouville theorems for ancient solutions to V-harmonic map heat flows.
method Refined gradient estimates and exponential growth conditions.
result Better Liouville theorems for ancient solutions to V-harmonic map heat flows.
The paper proves short-time existence of α-Dirac-harmonic map flow and applications.
problem Existence of α-Dirac-harmonic maps from closed surfaces. method Heat flow for α-Dirac-harmonic maps, blow-up analysis, density of maps with minimal kernel. result Existence of nontrivial α-Dirac-harmonic maps (α≥1) from closed surfaces. New heat flow for harmonic maps avoids singularities but not bubbles.
problem Finite time singularities in harmonic maps.
method Introduces a conformal heat flow for harmonic maps defined by an evolution equation.
result Global weak solution exists, smooth except at most finitely many points.
We analyse finite-time singularities of the Teichmüller harmonic map flow -- a natural gradient flow of the harmonic map energy -- and find a canonical way of flowing beyond them in order to construct global solutions in full generality. Moreover, we prove a no-loss-of-topology result at finite time, which completes th…