Constructs examples of centrally harmonic spaces and shows they are not generically harmonic.
problem Understanding conditions for centrally harmonic spaces.
method Generalizing work of Copson and Ruse to construct examples.
result Examples of centrally harmonic spaces are not generically harmonic.
Study shows volume density in central harmonic spaces can vary arbitrarily.
problem Volume density in central harmonic spaces can vary arbitrarily.
method Analyzes asymptotics of volume density function in central harmonic manifolds.
result Volume density in central harmonic spaces can be specified arbitrarily and does not determine geometry.
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.
The conformal Laplacian's algebraic structure is explored in 2D, revealing a central charge.
problem Exploring the algebraic structure of the conformal Laplacian in 2D.
method Using prefactorization algebras and Green functions.
result In 2D, the conformal Laplacian's algebraic structure is revealed through a central charge.
New map connects stability conditions to Teichmüller space.
problem Stability conditions and Teichmüller space relationship.
method Using harmonic maps and 3-Calabi-Yau categories.
result Natural map between stability conditions and Teichmüller space.
Study on vector fields on Lie groups reveals surprising algebraic coincidences.
problem Characterizing vector fields on Lie groups with Riemannian metrics.
method Algebraic and geometric analysis of left-invariant vector fields on nilpotent Lie groups.
result Spaces of Killing, one-harmonic, and conformal vector fields coincide with the center of the Lie algebra on nilpotent Lie groups.
A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, t…
Survey on random walks on mapping class groups and their properties.
problem Understanding random walks on mapping class groups.
method Analyzing actions on Teichmüller spaces and curve complexes.
result Laws of large numbers and central limit theorems for random walks.
Solutions to the n-dimensional Laplace equation which are constant on a central quadric are found. The associated twistor description of the case n=3 is used to characterise Gibbons-Hawking metrics with tri-holomorphic $SL(2, \C)$ symmetry.
In this paper, we consider the motion of a particle on a surface of revolution under the influence of a central force field. We prove that there are at most two analytic central potentials for which all the bounded, nonsingular orbits are closed and that there are exactly two on some surfaces with constant Gaussian cur…
Extended current algebra on S^3 with new bilinear form and 2-cocycle.
problem Investigate the current algebra on S^3 of complex Lie algebras.
method Defined a new quaternion-valued 2-cocycle and symmetric invariant bilinear form.
result Extended affine Kac-Moody algebra to Lie algebra of smooth mappings.
The paper examines curvature and stability in quasi-geostrophic motions using spherical harmonics.
problem Analyzing the curvature and stability of quasi-geostrophic motions.
method Utilizing spherical harmonics and structure constants, the curvature of the L2 metric on the central extension is computed. result A lower bound for weather prediction error in a simplified model is suggested.
In this article, we initiate a geometric measure theoretic approach to symplectic Hodge theory. In particular, we apply one of the central results in geometric measure theory, the Federer-Fleming deformation theorem, together with the cohomology theory of normal cur- rents on a differential manifold, to establish a fun…
New random walk results on rank one symmetric spaces.
problem Analyzing random walks on noncompact rank one symmetric spaces.
method Unified algebraic framework using Möbius addition and harmonic analysis of spherical functions.
result Renormalized walk converges to heat kernel on Laplace-Beltrami operator.
A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass--Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian…
This paper gives an exposition of the authors' harmonic deformation theory for 3-dimensional hyperbolic cone-manifolds. We discuss topological applications to hyperbolic Dehn surgery as well as recent applications to Kleinian group theory. A central idea is that local rigidity results (for deformations fixing cone angl…
Estimates eigenvalues of Jacobi operator for harmonic spaces.
problem Estimating eigenvalues of Jacobi operator.
method Using density function of a harmonic space.
result Sharp estimates imply symmetric Osserman space.
Dataset for rainfall modeling in central Europe from 1981-2011.
problem Improving rainfall streamflow modeling beyond simple catchments.
method Spatially resolved meteorological and ancillary data compilation.
result Dataset for neural network-driven hydrological modeling.
∞-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…
We classify noncompact homogeneous spaces which are Einstein and asymptotically harmonic. This completes the classification of Riemannian harmonic spaces in the homogeneous case: Any simply connected homogeneous harmonic space is flat, or rank-one symmetric, or a nonsymmetric Damek-Ricci space. Independently, Y. Nikola…
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.
Compact RCD spaces are proven to be smooth manifolds under specific harmonicity conditions.
problem Characterizing compact RCD spaces as harmonic manifolds.
method Analyzing heat kernel and geodesic ball volumes for harmonicity.
result Compact RCD spaces are isometric to smooth manifolds under given conditions.
Extends harmonic maps compactification to punctured Riemann surfaces.
problem Compactifying Teichmüller spaces for punctured Riemann surfaces.
method Using harmonic maps rays to extend compactification.
result The compactification still coincides with Thurston's compactification.
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…
New submersion proves complex-valued harmonic map existence.
problem Existence of non-constant harmonic morphisms.
method Constructing harmonic Riemannian submersions from symmetric spaces.
result Existence of non-constant, globally defined complex-valued harmonic morphism.
The study characterizes harmonic spaces and their radial eigen-functions and vector fields.
problem Characterizing harmonic spaces and their geometric properties.
method Examining radial eigen-spaces of Laplacians and using duality.
result Results extend to spaces harmonic with respect to a single point.
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.
We consider interpolating sesqui-harmonic Legendre curves in Sasakian space forms. We find the necessary and sufficient conditions for Legendre curves in Sasakian space forms to be interpolating sesqui-harmonic. Finally, we obtain an example for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.
Discretizes diffusions and harmonic functions on covering spaces.
problem Harmonic functions on covering spaces with bounded growth.
method Lyons-Sullivan discretizations of diffusion operators.
result Equivalence of discretized and continuous harmonic functions.
Geometric theory connects machine learning classifiers to differential geometry.
problem Classifying data points in machine learning.
method Mapping binary classification to vector bundles and differential geometry.
result Harmonic interpolation solves RKHS interpolation problems.
Constructs explicit harmonic functions and morphisms on complex and quaternionic Grassmannians.
problem Creating explicit solutions for p-harmonic functions and harmonic morphisms. method Using joint eigenfunctions of the Laplace-Beltrami and conformality operators.
result Induces solutions on dual non-compact Riemannian symmetric spaces.
In this paper, we give complete classifications of linear ∞-harmonic maps between Euclidean and Heisenberg spaces, between Nil and Sol spaces. We also classify all ∞-harmonic linear endomorphisms of Sol space and show that there is a subgroup of ∞-harmonic linear automorphisms in the group of linea…
P. Baird and the second author studied harmonic morphisms from a three-dimensional simply-connected space form to a surface and obtained a complete local and global classification of them. In this paper, we obtain a description of all harmonic morphisms from any three-dimensional Euclidean and spherical space form to a…
Smooth Busemann functions found in harmonic Finsler spaces.
problem Analyzing Busemann functions in Finsler manifolds.
method Investigation of Busemann functions in general and asymptotically harmonic Finsler manifolds.
result Smoothness of Busemann functions on asymptotically harmonic Finsler manifolds.
Paper proves weak unique continuation for harmonic functions on RCD spaces but finds counterexample for strong uniqueness.
problem Unique continuation of harmonic functions on RCD spaces, especially strong uniqueness.
method Establishes weak unique continuation theorem and provides counterexample for strong uniqueness.
result Found counterexample for strong unique continuation in RCD(K,N) spaces for N≥4 and K∈R.
Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.
problem Understanding harmonic maps between singular spaces.
method Analyzing homogeneous harmonic maps between simplicial cones and their degrees.
result Degrees of homogeneous harmonic maps are related to eigenvalues of discrete graph Laplacians.
The paper classifies vacuum static spaces with harmonic curvature.
problem Classifying vacuum static spaces with harmonic curvature.
method Extending the 4-dimensional work by Kim-Shin, the paper classifies n-dimensional spaces (n≥5). result New counterexamples to the Fischer-Marsden conjecture on compact vacuum static spaces.
The spaces of harmonic maps of the projective plane to the four-dimensional sphere are investigated in this paper by means of twistor lifts. It is shown that such spaces are empty in case of even harmonic degree. In case of harmonic degree less than 6 it was shown that such spaces are path-connected and an explicit par…
Harmonic functions on compact symmetric spaces exhibit strong convexity properties.
problem Understanding the convexity of harmonic functions on compact symmetric spaces.
method Analyzing the nonnegativity of the Laplacian powers of harmonic functions.
result Harmonic functions on compact symmetric spaces have nonnegative Laplacian powers, demonstrating strong convexity.
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.
Survey on harmonic maps in non-smooth spaces, focusing on rigidity.
problem Rigidity phenomena in non-smooth spaces.
method Regularity theory of harmonic maps to non-smooth targets.
result Generalizations of Margulis superrigidity and holomorphic rigidity of Teichmüller space.
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…
The paper proves a Liouville theorem for specific harmonic maps with free boundary.
problem Analyzing harmonic maps with free boundary conditions.
method Developed Liouville theorem for φ-F-symphonic, φ-F-harmonic, and φ-ΦS,p,ε harmonic maps. result Established Liouville theorem for the specified harmonic maps with free boundary.
Harmonic manifolds of hypergeometric type have entropy bounds related to real hyperbolic spaces.
problem Bounding the volume entropy of harmonic manifolds of hypergeometric type.
method Normalized Ricci curvature and entropy analysis.
result Upper and lower bounds for volume entropy of harmonic manifolds of hypergeometric type.
Extends Taubes's theorem to non-compact manifolds with harmonic forms.
problem Generalizing Taubes's \(SW=Gr\) theorem to non-compact settings.
method Proves variants of Taubes's convergence theorem for non-compact manifolds with harmonic 2-forms.
result Extends Taubes's theorem to non-compact manifolds with cylindrical ends and self-dual harmonic 2-forms.
Constructs explicit p-harmonic functions on specific Lie groups.
problem Finding explicit p-harmonic functions on a specific class of Lie groups.
method Constructs explicit p-harmonic functions on rank-one Lie groups of Iwasawa type.
result Proves existence of proper p-harmonic functions on these groups.
The paper classifies harmonic and biharmonic submersions from Sol space.
problem Classifying harmonic and biharmonic submersions from Sol space.
method Complete classification by proving non-existence and existence conditions.
result No harmonic or biharmonic submersion exists from Sol space to any surface.
Study proves rigidity of harmonic maps from 2-torus to complex projective space.
problem Rigidity of isotropic harmonic maps from a 2-torus to a complex projective space.
method Proves rigidity through holomorphic embeddings and complete linear systems.
result Ensures rigidity of harmonic bands in condensed matter physics.