The study proves Liouville-type theorems and Bochner formulas for harmonic maps into CAT(κ) spaces.
problem Analyzing harmonic maps from Riemannian polyhedra into CAT(κ) spaces.
method Computing a target variation formula to derive Liouville-type theorems and Bochner formulas.
result Proves Liouville-type theorems and Bochner formulas for harmonic maps into CAT(1) spaces.
Proves regularity of harmonic maps into Teichmüller space.
problem Harmonic maps into Teichmüller space and their singularities.
method Analyzes harmonic maps from Riemannian domains to Teichmüller space with specific conditions.
result If a harmonic map intersects a stratum, it is entirely contained in that stratum.
Study analyzes convergence of harmonic maps into compact locally CAT(1) spaces.
problem Analyzing convergence of harmonic maps with energy bounds.
method Bubble tree convergence, exploiting local convexity properties of CAT(1) spaces.
result Energy quantization and no-neck property demonstrated for harmonic maps.
Harmonic maps from surfaces to spheres constructed and studied.
problem Existence of harmonic maps into spheres of arbitrary genus.
method Relates to convex geometry of spheres, uses heat flow approach.
result New examples of harmonic maps and regions without closed geodesics.
Stability for ΦS,F,H harmonic map and ΦT,F,H harmonic mapmath.DG The paper examines stability of harmonic maps on specific manifolds.
problem Stability of harmonic maps on compact convex hypersurfaces.
method Analyzes stability conditions for ΦS,F,H and ΦT,F,H harmonic maps. result Provides theorems to determine stability of ΦS,F,H and ΦT,F,H harmonic maps. Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.
problem Proving Lipschitz regularity for harmonic map heat flows into CAT(0) spaces.
method Elliptic approximation method
result Every weak solution of the harmonic map heat flow into CAT(0) spaces is Lipschitz continuous in both space and time.
This paper proves energy convexity for bi-harmonic maps into spheres, with applications to heat flow and uniqueness.
problem Analyzing the geometric properties of bi-harmonic maps and their heat flow.
method Energy convexity and ε-regularity of bi-harmonic maps.
result Uniqueness of weakly intrinsic bi-harmonic maps and long-time existence of the heat flow.
Effective methods compute equivariant harmonic maps from surfaces to nonpositively curved spaces.
problem Computing equivariant harmonic maps from surfaces to nonpositively curved spaces.
method Discretization of the theory, strong convexity of energy functional, convergence of discrete heat flow, center of mass methods.
result Explicit convergence rate and numerical computation with Harmony software.
The study describes the geometry of surfaces and their representations in SL(3,R).
problem Understanding the geometry of surface group representations into SL(3,R).
method Proving asymptotic formulas and harmonic map convergence for equivariant maps.
result The geometry of the image is weakly convex and a (one-third) translation surface.
Existence of harmonic maps near projections in hyperbolic spaces for large convex sets.
problem Existence of harmonic maps near projections in hyperbolic spaces.
method Weak quasi-isometry condition, non-collapsing property, harmonic measures.
result Existence of harmonic maps at finite distance from projections of large convex sets in hyperbolic spaces.
Constructs harmonic maps near retractions in hyperbolic spaces.
problem Finding harmonic maps near retractions in hyperbolic spaces.
method Constructs harmonic maps to the hyperbolic plane from quasidisks, and to convex hulls from sets in the boundary at infinity of pinched Hadamard manifolds.
result Harmonic maps are bounded from nearest-point retractions in hyperbolic spaces.
New principle for harmonic maps helps study higher-dimensional submanifolds.
problem Understanding unboundedness of totally geodesic projections in higher codimension.
method Introducing a flexible notion of convexity and applying it to harmonic and conformal maps.
result New maximum principle for harmonic maps applicable to various geometric settings.
The paper extends gradient estimates for harmonic maps into singular spaces.
problem Estimating gradients for harmonic maps into singular spaces.
method Using Alexandrov curvature bounds and extending Cheng and Choi's work.
result Yau's gradient estimates for harmonic maps into singular spaces.
Harmonic maps between symmetric spaces have duals.
problem Understanding harmonic maps between symmetric spaces.
method Constructing dual harmonic maps using potentials.
result Duality theorem for harmonic maps into inner symmetric spaces.
In this paper we prove the existence of a solution to the Dirichlet problem for harmonic maps into a geodesic ball on which the squared distance function from the origin is strictly convex. This improves a celebrated theorem obtained by S. Hildebrandt, H. Kaul and K. Widman in 1977. In particular no curvature assumptio…
The study proves local and global Holder continuity of quasi-n-harmonic mappings into spaces with non-positive curvature.
problem Holder continuity of quasi-n-harmonic mappings into metric spaces with non-positive curvature.
method Local and global Holder continuity proved using Euclidean domains and bounded Lipschitz domains.
result Local and global Holder continuity of quasi-n-harmonic mappings.
We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasi-linear e…
In this paper, we study harmonic functions on weighted manifolds and harmonic maps from weighted manifolds into Hadamard spaces introduced by Korevaar and Schoen. We prove Liouville theorems for these harmonic maps with finite energy.
Harmonic mappings into Teichmuller spaces appear in the study of manifolds which are fibrations whose fibers are Riemann surfaces. In this article we will study the existence and uniquenesses questions of harmonic mappings into Teichmuller spaces, as well as some local and global behavior of the harmonic images induced…
Study on harmonic maps in special geometric spaces.
problem Harmonic maps from rectifiable spaces into $\CAT(1)$ balls.
method Proving the existence and uniqueness of minimizers for energy function.
result Existence and uniqueness of minimizers for Korevaar-Schoen energy.
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.
Non-convex extremal length found in surface metrics.
problem Extremal length functions on surfaces are not always convex.
method Used harmonic maps to R-trees and minimal surfaces in Rn. result Found measured foliations with non-convex extremal length functions.
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. In \cite{kamz} the author proved that every quasiconformal harmonic mapping between two Jordan domains with C1,α, 0<α≤1, boundary is bi-Lipschitz, providing that the domain is convex. In this paper we avoid the restriction of convexity. More precisely we prove: any quasiconformal harmonic mapping between two …
The study proves existence of harmonic maps into CAT(1) spaces.
problem Existence of harmonic maps into compact locally CAT(1) spaces.
method Harmonic replacement technique and compactness for energy minimizers.
result Either a harmonic map homotopic to φ or a conformal harmonic map on the sphere.
∞-Harmonic maps are a generalization of ∞-harmonic functions. They can be viewed as the limiting cases of p-harmonic maps as p goes to infinity. In this paper, we give complete classifications of linear and quadratic ∞-harmonic maps from and into a sphere, quadratic ∞-harmonic maps between E…
The energy function associated to harmonic maps between surfaces is convex at critical points.
problem Proving convexity of the energy function for harmonic maps between surfaces.
method Analyzing the energy function on Teichmüller space and proving convexity at critical points.
result The energy function is convex at critical points and strictly convex under certain conditions.
On some specified convex supporting sets of spheres, we find a generalized longitude function whose level sets are totally geodesic. Given an arbitrary (weakly) harmonic map into spheres, the composition of the generalized longitude function and harmonic map satisfies an elliptic equation of divergence type. With the a…
We show that a harmonic map from a Riemann surface into the exceptional symmetric space G2/SO(4) has a J2-holomorphic twistor lift into one of the three flag manifolds of G2 if and only if it is `nilconformal', i.e., has nilpotent derivative. Then we find relationships with almost complex maps from a…
Harmonic maps pull convex functions on metric spaces to subharmonic ones.
problem Understanding how convex functions behave under harmonic maps on metric spaces.
method Proving subharmonicity of pullbacks of convex functions by harmonic maps in metric spaces.
result The pullback of a convex function by a harmonic map is subharmonic in metric spaces.
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.
J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed the first variational formula. When, k=2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, we consider the relationship between biharmonic maps and k-harmonic maps, and show non-existe…
Rectifies singular set of harmonic maps into complex.
problem Regularity of harmonic maps into complex manifolds.
method Proves (m−2)-rectifiability of singular set. result Singular set is (m−2)-rectifiable. Study shows how maps from certain geometric spaces behave near their edges.
problem Boundary regularity of harmonic maps in specific geometric spaces.
method Analysis of RCD(K,N) and CAT(0) spaces. result Established boundary regularity of harmonic maps.
We study harmonic maps from Riemannian manifolds into arbitrary non-positively curved and CAT(-1) metric spaces. First we discuss the domain variation formula with special emphasis on the error terms. Expanding higher order terms of this and other formulas in terms of curvature, we prove an analogue of the Eels-Sampson…
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.
Local Lipschitz continuity of sub-elliptic harmonic maps into CAT(0) spaces proved.
problem Proving Lipschitz continuity of sub-elliptic harmonic maps between singular spaces.
method Analyzing sub-elliptic harmonic maps from the Heisenberg group into CAT(0) spaces.
result Local Lipschitz continuity established for sub-elliptic harmonic maps.
Harmonic maps from curved spaces to hyperbolic ones exist and are Lipschitz.
problem Existence and properties of harmonic maps between curved and hyperbolic spaces.
method Energy minimization and finite distance analysis.
result Harmonic maps from Hadamard manifolds to Gromov hyperbolic mCAT(0)-spaces are Lipschitz. Quantifies uniqueness of conformal-harmonic maps on 4-manifolds.
problem Quantifying uniqueness of conformal-harmonic maps on 4-manifolds.
method Proves a quantitative uniqueness result using convexity and second order Hardy inequality.
result Proves a version of second order Hardy inequality on manifolds.
We generalize the results of Song-Zelditch on geodesics in spaces of Kahler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kahler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated…
This article shows that every non-isotropic harmonic 2-torus in complex projective space factors through a generalised Jacobi variety related to the spectral curve. Each map is composed of a homomorphism into the variety and a rational map off it. The same ideas allow one to construct (pluri)-harmonic maps of finite ty…
One-harmonic maps from a curved surface to hyperbolic plane have specific interior properties.
problem Characterizing one-harmonic maps from curved surfaces to hyperbolic spaces.
method Using Minkowski geometry and interpreting maps as Gauss maps of convex surfaces.
result One-harmonic maps have images confined to the interior of convex hulls.
Inspired by work of Colding-Minicozzi on mean curvature flow, Zhang introduced a notion of entropy stability for harmonic map flow. We build further upon this work in several directions. First we prove the equivalence of entropy stability with a more computationally tractable F-stability. Then, focusing on t…
Study of moduli spaces for special harmonic maps from projective line to quadrics.
problem Characterizing moduli spaces of Einstein-Hermitian harmonic maps.
method Using vector bundles and representation theory, the authors describe the moduli spaces of these maps.
result The dimension of the moduli spaces is independent of the Einstein-Hermitian constant.
Our purpose is to use a Darboux homogenous derivative to understand the harmonic maps with values in homogeneous space. We present a characterization of these harmonic maps from the geometry of homogeneous space. Furthermore, our work covers all type of invariant geometry in homogeneous space.
We define two transforms between non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between non-conformal harmonic maps into the 3-sphere, H-surfaces in Euclidean 3-space…
The paper explores harmonic maps and their properties in symmetric spaces.
problem Understanding harmonic maps in symmetric spaces.
method Discussion of associated family of harmonic maps from Riemann surfaces into symmetric spaces.
result Comparison and conjecture on harmonic maps and totally symmetric harmonic maps.
The paper proves local Lipschitz regularity for harmonic map heat flows in RCD spaces into CAT(0) spaces.
problem Understanding the regularity of harmonic map heat flows in nonsmooth spaces.
method Combines metric Hamilton-Jacobi argument with elliptic contact estimates.
result Local Lipschitz regularity for the flow from RCD spaces into CAT(0) spaces.