Harmonic maps from hyperbolic planes to hyperbolic space exist with given boundary data.
problem Existence of harmonic maps from product of hyperbolic planes to hyperbolic space.
method Existence result for asymptotic Dirichlet problem.
result Existence of harmonic maps with given boundary data.
Study of harmonic Riemannian submersions from 3D geometries.
problem Characterizing harmonic Riemannian submersions from specific 3D geometries.
method Using generalized integrability data and classifications of Thurston's 3D geometries, 3D BCV spaces, and Berger sphere.
result Complete classifications and explicit constructions of harmonic Riemannian submersions.
Paper introduces MSP Network for harmonizing images from different scanners.
problem Improving predictive performance and learning efficiency in data harmonization.
method Multi-Stage Prediction (MSP) Network integrating neural networks of different architectures.
result MSP Network shows 20% improvement in patch-based mean-squared error over state-of-the-art methods.
Harmonic functions stable under small changes.
problem Stability of multivalued harmonic functions under deformations.
method Application of Nash-Moser implicit function theorem.
result Stability of harmonic sections under small deformations.
BacHMMachine harmonizes Baroque chorales using theory-driven principles and Hidden Markov Models.
problem Algorithmic harmonization of Baroque chorales.
method Theory-driven approach guided by music composition principles, combined with data-driven learning of key and chord transitions.
result BacHMMachine generates musically coherent harmonizations with reduced computational burden and greater interpretability.
This is an expository article which describes one approach to the construction and classification of harmonic tori "of finite type", namely, via their ring of polynomial Killing fields. To keep the discussion focussed, the first section is devoted entirely to non-conformal harmonic tori in the 2-sphere. The second sect…
Method computes harmonic and conformal maps from point clouds.
problem Computing maps from irregular point cloud data.
method Meshless method using cubic lattice approximations.
result Harmonic and conformal maps computed accurately.
We find algebraic parametrizations of extended solutions of harmonic maps of finite uniton number from a surface to the orthogonal group O(n) in terms of free holomorphic data which lead to formulae for all such harmonic maps. Our work reveals an interesting correspondence between certain harmonic maps and the free Wei…
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,…
Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of …
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.
Study on harmonic functions in spaces with collapsing behaviors.
problem Harmonic functions on spaces with inhomogeneous collapsing behaviors at infinity.
method Analysis of complete and incomplete spaces with nonnegative Ricci curvature.
result Any nonconstant harmonic function yields a definite exponential growth rate.
A formula connects discrete harmonic surfaces to holomorphic functions.
problem Creating smooth discrete harmonic surfaces from holomorphic data.
method Weierstrass representation formula for discrete harmonic surfaces.
result Smooth converging sequence of discrete harmonic surfaces converges to a minimal surface.
Paper harmonizes medical data using flow-based causal inference.
problem Heterogeneity in medical data from different sites and protocols.
method Flow-based normalizing method for counterfactual inference on structural causal models.
result Better cross-domain generalization compared to state-of-the-art algorithms.
We establish explicit formulae for canonical factorizations of extended solutions corresponding to harmonic maps of finite uniton number into the exceptional Lie group G2 in terms of the Grassmannian model for the group of based algebraic loops in G2. A description of the ``Frenet frame data" for such harmonic ma…
The method of geometric harmonics is adapted to the situation of incomplete data by means of the iterated geometric harmonics (IGH) scheme. The method is tested on natural and synthetic data sets with 50--500 data points and dimensionality of 400--10,000. Experiments suggest that the algorithm converges to a near optim…
Study Lorentz harmonic maps and spacelike surfaces in anti-de Sitter space.
problem Analyzing the relationship between Lorentz harmonic maps and spacelike surfaces.
method Using loop group techniques, develop DPW-type representations and solve Cauchy problems.
result Establish a correspondence between Lorentz harmonic maps and spacelike immersions, leading to families of surfaces of constant Gauss curvature.
In this note, we show that for any harmonic map into a non-compact symmetric space one can find naturally a "dual" harmonic map into a compact symmetric space which can be constructed from the same basic data (called "potentials" in the loop group formalism). Locally also the inverse/converse duality theorem holds.
We classify all harmonic maps of finite uniton number from a Riemann surface into SU(n) in terms of certain pieces of the Bruhat decomposition of the subgroup of algebraic loops in SU(n). We give a description of the "Frenet frame data" for such harmonic maps in a given class.
Vortex solutions on flat surfaces map to harmonic spinors on Nappi-Witten space.
problem Constructing Abelian magnetic zero-modes on flat spacetime.
method Establishing a correspondence between vortex equations and harmonic spinors on the Nappi-Witten space.
result Explicit solutions of a twisted Dirac equation induce harmonic spinors on Minkowski space.
PEHRT harmonizes EHR data for translational research.
problem Barriers in using EHR data for translational research.
method Common pipeline including open-source code, visualization tools, and detailed documentation.
result PEHRT harmonizes EHR data to standardized ontologies and generates robust embeddings.
We show that given a harmonic map φ from a Riemann surface to a classical compact simply connected inner symmetric space, there is a J2-holomorphic twistor lift of φ (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae i…
Estimates bandwidth for CMC initial data sets.
problem Estimating bandwidth for constant mean curvature (CMC) initial data sets.
method Three independent proofs: stability of null expansion, spacetime harmonic function perturbation, Dirac operator.
result Generalized Gromov's band width estimate to CMC initial data sets.
Paper bounds mass of 3D Einstein data using spacetime harmonic functions.
problem Calculating the mass of 3D asymptotically flat initial data for the Einstein equations.
method Uses spacetime harmonic functions to give a lower bound for the ADM mass.
result New proof of spacetime positive mass theorem and rigidity statement.
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.
This paper improves boundary regularity of harmonic maps in metric measure spaces.
problem Improving boundary regularity of harmonic maps in non-smooth spaces.
method Developed a Gauss-Green formula for RCD(K,N) spaces and applied it to harmonic maps. result Optimal boundary regularity of harmonic maps from RCD(K,N)-spaces to CAT(0)-spaces. The paper studies constant harmonic mean curvature surfaces in Schwarzschild spaces, proving they foliate the space.
problem Investigating constant harmonic mean curvature surfaces in Schwarzschild spaces.
method Volume-preserving harmonic mean curvature flow in asymptotically Schwarzschild spaces.
result These surfaces form a foliation of the space outside a large ball.
New complex-valued maps found on complex geometries.
problem Finding new maps on complex geometries.
method Solving non-linear PDEs based on manifold geometry.
result Constructed new proper biharmonic and (2,1)-harmonic maps.
Study cohomology classes related to harmonic maps on submersions.
problem Understanding harmonic maps on submersions and their cohomology.
method Extending previous results on Riemannian submersions and p-harmonic morphisms to F-harmonic and f-harmonic maps.
result Extend results on Riemannian submersions to F-harmonic and f-harmonic maps.
Study on harmonic maps on weighted Riemannian foliations.
problem Characterize harmonic maps on weighted foliations.
method Analyze transversally f-harmonic and (F,F′)f-harmonic maps. result Equivalence of transversally f-harmonic and (F,F′)f-harmonic maps in minimal foliations. We investigate the well-posedness of (i) the heat flow of harmonic maps from Rn to a compact Riemannian manifold without boundary for initial data in BMO; and (ii) the hydrodynamic flow (u,d) of nematic liquid crystals on Rn for initial data in BMO−1×BMO.
Harmonic maps study on surfaces with non-positive curvature.
problem Characterize harmonic maps on surfaces with non-positive curvature.
method Transversality theory for Banach manifolds.
result Set of somewhere injective harmonic maps is open, dense, and connected.
The paper examines triviality of Ricci-Bourguignon harmonic solitons.
problem Investigating triviality of Ricci-Bourguignon harmonic solitons.
method Utilizing results from V-harmonic map to study Ricci harmonic soliton properties.
result Triviality of Ricci-Bourguignon harmonic solitons examined.
The paper constructs non-convergent solutions to Vafa-Witten equations with specific harmonic 2-form limits.
problem Constructing solutions to Vafa-Witten equations with non-zero mass term.
method Constructs divergent sequences of solutions, renormalizes them, and defines harmonic 2-form data sets.
result Defines an 'interesting' harmonic 2-form data set with specific properties.
Study cohomology classes related to n-harmonic morphisms and F-harmonic maps.
problem Understanding cohomology classes associated with n-harmonic morphisms and F-harmonic maps. method Utilizing the n-conservation law (2.6) to obtain sharp results. result Sharp results on cohomology classes related to n-harmonic morphisms and F-harmonic maps. The paper proves nonexistence of harmonic and bi-harmonic maps under specific conditions.
problem Existence of harmonic and bi-harmonic maps into certain Riemannian manifolds.
method Analysis of manifolds with conformal vector fields or Ricci solitons.
result Nonexistence of harmonic and bi-harmonic maps in specified conditions.
The Ooguri-Vafa space is linked to framed wild harmonic bundles via hyperkähler geometry.
problem Relating the Ooguri-Vafa space to Hitchin moduli spaces.
method Interpreting the Ooguri-Vafa space as framed wild harmonic bundles over CP1.
result The Ooguri-Vafa space is a moduli space of framed wild harmonic bundles.
A new mathematical approach detects frequency-based alterations in brain networks.
problem Understanding disease-relevant brain alterations through network analysis.
method Proposes a novel connectome harmonic analysis framework using common harmonic waves learned from Stiefel manifolds.
result Identifies more significant and reproducible network dysfunction patterns in Alzheimer's disease.
The paper studies a heat flow for almost complex structures and proves convergence under certain conditions.
problem The study of harmonic heat flow for almost complex structures compatible with a Riemannian metric.
method Definition and analysis of the harmonic heat flow, proving existence and convergence under small energy conditions.
result The flow converges to a Kähler structure if the initial energy is small, but there are finite time singularities for small enough initial energy.
In this paper, we consider a Riemannian foliation whose normal bundle carries a parallel or harmonic basic form. We estimate the norm of the O'Neill tensor in terms of the curvature data of the whole manifold. Some examples are then given.
f-Harmonic maps were first introduced and studied by Lichnerowicz in \cite{Li} (see also Section 10.20 in Eells-Lemaire's report \cite{EL}). In this paper, we study a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-harmonic functions. We prove that a map betwe…
Proves density and mass theorems for specific initial data sets.
problem Initial data sets with boundary in spacetime.
method Harmonic asymptotics and dominant energy condition.
result Spacetime positive mass theorem for initial data sets with apparent horizon boundary.
New methods using spacetime harmonic functions solve geometric inequalities.
problem Geometric inequalities involving mass in spacetime.
method Utilizing spacetime harmonic functions and other elliptic equations.
result Novel concept of total mass and proof of positive mass theorem.
Finite energy solutions of 4-harmonic and ES-4-harmonic maps are trivial.
problem Characterizing the qualitative behavior of 4-harmonic and ES-4-harmonic maps.
method Proving triviality of finite energy solutions for both maps.
result Finite energy solutions of both 4-harmonic and ES-4-harmonic maps are trivial.
We propose a new notion called \emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of p% -harmonic maps as p→∞. Infinity harmoncity appears in many familiar contexts. For example,…
The paper examines properties of generalized τ-quasi Ricci-harmonic metrics and proves rigidity results.
problem Characterizing and proving rigidity of generalized τ-quasi Ricci-harmonic metrics.
method Exploring conditions for harmonic-Einstein metrics, obtaining rigidity results, and proving gap theorems.
result Rigidity results for compact generalized τ-quasi Ricci-harmonic metrics.
Study examines maximal domains of radial harmonic functions across different curvature types.
problem Understanding maximal domains of radial harmonic functions in various curvature settings.
method Analysis of harmonic spaces with positive, zero, and negative curvature.
result Characterization of maximal domains for radial harmonic functions in different curvature contexts.
New p-harmonic and harmonic morphisms found on Lie groups.
problem Constructing explicit p-harmonic and harmonic morphisms on Lie groups.
method Using the method of eigenfamilies to construct explicit complex-valued p-harmonic functions and harmonic morphisms.
result Explicit complex-valued p-harmonic functions and harmonic morphisms constructed on non-compact classical Lie groups.