Harmonic extension of Weil-Petersson circle homeomorphisms
problem Harmonic maps from the Weil-Petersson disk to the hyperbolic disk
method Anti-holomorphic L2-energy result Harmonic extension of Weil-Petersson circle homeomorphisms minimizes the L2-energy Quasiregular maps get harmonic extensions in hyperbolic space.
problem Existence of harmonic extensions for quasiregular maps.
method Proving harmonic extensions for non-constant quasiregular maps on spheres.
result Non-constant quasiregular maps on spheres have harmonic extensions in hyperbolic space.
Develops relative harmonic metrics and deformation theory for Higgs and flat bundles on compact Kähler manifolds.
problem Analyzing harmonic metrics and deformations on Higgs and flat bundles on compact Kähler manifolds.
method Relative analytic theory, Sobolev completions, elliptic regularity, normalized gluing, plotwise smoothness, heat flow, harmonic filtrations, obstruction theory.
result Global smooth harmonic metrics exist for smooth stable Higgs families under certain conditions, and this theory extends to reduced singular parameter spaces.
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 study k-harmonic immersion into a sphere, and get the rerationship between radious and "k"…
DELIMIT PyTorch enhances deep learning for diffusion imaging.
problem Applying deep learning to spherical diffusion imaging data.
method Added spherical harmonic interpolation and local convolution layers to PyTorch.
result Deep learning can now be applied conveniently to diffusion imaging data.
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…
Conservation law for weakly harmonic mappings in high dimensions.
problem Conservation law for harmonic mappings in supercritical dimensions.
method Partial extension of Rivière's conservation law with Lorentz integrability condition.
result Conservation law for weakly harmonic mappings in supercritical dimensions.
Certain solvable extensions of H-type groups provide noncompact counterexamples to the so-called Lichnerowicz conjecture, which asserted that ``harmonic'' Riemannian spaces must be rank 1 symmetric spaces.
Study surfaces in Half-Pipe space and vector fields on hyperbolic plane.
problem Mapping surfaces in Half-Pipe space to vector fields on hyperbolic plane.
method Use harmonic Lagrangian vector fields and infinitesimal Douady-Earle extension.
result Prove existence and uniqueness of harmonic Lagrangian extensions with Zygmund conditions.
Extends functions on symmetric spaces to analytic functions.
problem Extending functions on symmetric spaces to analytic functions.
method Harmonic analysis on symmetric spaces and representation theory of groups.
result Proves Whitney type extension theorems for symmetric spaces.
Maximizes capacity of extensions with fixed boundary data.
problem Maximizing the capacity of extensions with nonnegative scalar curvature.
method Using the method of Lagrange multipliers on the constraint space of scalar-flat extensions.
result Derives variational condition for maximal capacity extensions and proves they have constant scalar curvature.
We prove that a quasiconformal map of the 2-sphere admits a harmonic quasi-isometric extension to the 3-dimensional hyperbolic space, thus confirming the well known Schoen Conjecture in dimension 3.
Paper proves existence of smooth nontrivial Dirac-harmonic maps.
problem Existence of nontrivial Dirac-harmonic maps from closed surfaces.
method Proves existence using ε-regularity and perturbations.
result Existence of smooth nontrivial Dirac-harmonic maps.
New bounds on singular set size for harmonic maps into 2-sphere in higher dimensions.
problem Bounding the size of singular set for harmonic maps into 2-sphere.
method Extending previous results to higher dimensions, proving new inequalities.
result Stable bounds on singular set size under small perturbations.
Paper proposes a harmonized approach to multimodal learning using GPLVMs.
problem Modality heterogeneity in multimodal data.
method Develops a novel learning scheme called Harmonization to jointly learn latent model parameters from different modalities.
result Experimental results show superior performance in cross-modal retrieval tasks.
We continue our discussion from part I.
Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature h. In this article we present results for harmonic functions on rank one asymptotically harmonic manifolds X with mild curvature boundedness c…
The paper studies p-harmonic functions and their conjugates, showing they converge to calibrations of laminations.
problem Behavior of q-harmonic functions and their conjugates in the limit as qo1. method Analysis of p-harmonic conjugates and their convergence to calibrations of laminations. result The laminations calibrated by the limiting p-harmonic conjugates are exactly those arising from the 1-Laplacian. 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.
Researchers extend regularity of p-harmonic maps into spheres for a new range of p.
problem Establishing regularity of p-harmonic maps for a broader range of p. method Combining Morrey's methods with Hardt and Lin's Extension Theorem, and proving a sharp Kato inequality.
result Regularity for p∈[2.961,3] and p∈[2,p0] with p0≈2.366. 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.
Extends Eells-Sampson theorem for manifolds with positive sectional curvature bounds.
problem Rigidity of harmonic maps between manifolds with curvature constraints.
method Proves an extension of Eells-Sampson theorem under positive sectional curvature upper bounds.
result Recover Hamilton's rigidity result for positive Ricci curvature.
New non-existence results for harmonic maps into perturbed cones.
problem Proper harmonic maps into perturbed cones in \(\mathbb{R}^n\), horospheres in \(\mathbb{H}^n\).
method Extension of foliated maximum principle to non-compact settings.
result New non-existence results for proper harmonic maps.
The paper explores unique continuation properties for polyharmonic maps between Riemannian manifolds.
problem Investigating unique continuation principles for polyharmonic maps.
method Analyzing critical points of higher order functionals to prove extensions of known results in harmonic and biharmonic cases.
result Proving extensions of unique continuation principles for k-harmonic maps.
New type of maps studied with non-vanishing torsion.
problem Harmonic maps with non-trivial torsion.
method Introduced and analyzed new type of maps between manifolds with torsion connections.
result First mathematical analysis of harmonic maps with torsion.
In this paper we construct a new family of harmonic morphisms $\varphi:V^5\to\s^2$, where V5 is a 5-dimensional open manifold contained in an ellipsoidal hypersurface of $\c^4=\r^8$. These harmonic morphisms admit a continuous extension to the completion V∗5, which turns out to be an explicit real algebra…
The paper studies rigidity results for harmonic forms on Kähler manifolds.
problem Understanding harmonic forms on Kähler manifolds.
method Analyzes rigidity results for harmonic (p,q)-forms in complete Kähler manifolds. result Shows several rigidity results and applications to non-compact Kähler manifolds.
Every connected, weighted graph with non-negative curvature has exactly two ends.
problem Characterizing the structure of connected, weighted graphs with non-negative curvature.
method Extremal Lipschitz extensions, variational principle, study of harmonic functions.
result Every salami has exactly two ends and no vertices with positive curvature.
The paper introduces a new complex analytic invariant called the pointed harmonic volume and its relation to the Johnson homomorphism.
problem Exploring new complex analytic invariants related to the complex structure of Riemann surfaces.
method Defining and computing the pointed harmonic volume as a natural extension of Chen's iterated integrals.
result Established a relationship between the harmonic volume and the first extended Johnson homomorphism.
The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Groetz…
The paper extends harmonic map theory to higher dimensions and fractional spaces.
problem Understanding the singular set of minimizing harmonic maps in various dimensions and spaces.
method Analytical proofs and extensions of previous theorems.
result Stability of the singular set under small perturbations in specific conditions.
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.
The paper studies harmonic forms on submanifolds with finite total curvature.
problem Understanding harmonic forms on submanifolds with finite total curvature.
method Analyzes L2-harmonic p-forms on complete submanifolds with flat normal bundle in spheres. result Shows triviality of Hp(L2(M)) if total curvature is less than a positive constant. We give exposition of a Liouville theorem established in \cite{Li3} which is a novel extension of the classical Liouville theorem for harmonic functions. To illustrate some ideas of the proof of the Liouville theorem, we present a new proof of the classical Liouville theorem for harmonic functions. Applications of the …
In this paper, we introduce the stress-energy tensors of the partial energies E'(f) and E"(f) of maps between Kaehler manifolds. Assuming the domain manifolds poss some special exhaustion functions, we use these stress-energy tensors to establish some monotonicity formulae of the partial energies of pluriharmonic maps …
The Black-Scholes model anticipates rather well the observed prices for options in the case of a strike price that is not too far from the current price of the underlying asset. Some useful extensions can be obtained by an adequate modification of the coefficients in the Black-Scholes equation. We investigate from a ma…
We extend harmonic map techniques to the setting of more general differential equations in conformal geometry. We obtain an extension of Siu's rigidity to Kahler-Weyl geometry and apply the latter to Vaisman's conjecture. Other applications include topological obstructions to the existence of Kahler-Weyl structures. Fo…
Riemann surfaces are two-dimensional manifolds with a conformal class of metrics. It is well known that the harmonic action functional and harmonic maps are tools to study the moduli space of Riemann surfaces. Super Riemann surfaces are an analogue of Riemann surfaces in the world of super geometry. After a short intro…
The paper explores new structures on cotangent bundles induced by natural Riemann extensions.
problem Investigating new geometric structures on cotangent bundles.
method Constructing and analyzing almost para-Hermitian and paracontact metric structures.
result Conditions for paracontact metric, K-paracontact metric, and para-Sasakian structures.
Study dynamics of Lp-multipliers on harmonic manifolds with exponential volume growth.
problem Characterize the behavior of Lp-multipliers on harmonic manifolds of purely exponential volume growth. method Analyzing the dynamics of Lp-multipliers on non-compact harmonic manifolds, using Fourier transformation and properties of radial functions. result Show that Lp-multipliers acting nicely on smooth functions with compact support for p≤2 cannot be chaotic. 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.
Novel approach to instantons on quaternionic Kähler manifolds.
problem Understanding Yang-Mills instantons on quaternionic Kähler manifolds.
method Extension of harmonic space method for hyperkähler manifolds.
result Established a bijection between instantons and holomorphic maps.
We derive conservation laws for Dirac-harmonic maps and their extensions to manifolds that have isometries, where we mostly focus on the spherical case. In addition, we discuss several geometric and analytic applications of the latter.
We give a new construction of Ricci-flat self-dual metrics which is a natural extension of the Gibbons--Hawking ansatz. We also give characterisations of both these constructions, and explain how they come from harmonic morphisms.
Improved lower bound for first eigenvalue of minimal hypersurfaces in spheres.
problem Finding a tighter bound for the first eigenvalue of minimal hypersurfaces in spheres.
method Rayleigh quotient estimate for a harmonic extension of an eigenfunction.
result Proved a new lower bound for the first eigenvalue of minimal hypersurfaces in spheres.
In this paper we develop new methods for studying the convergence problem for the heat flow on negatively curved spaces and prove that any quasiconformal map of the sphere Sn−1, n≥3, can be extended to the n-dimensional hyperbolic space such that the heat flow starting with this extension converge…
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 show that the family of probability measures on the n-dimensional unit sphere, having density proportional to: \[ S^n \ni y \mapsto \frac{1}{|y - x|^{n+α}}, \] satisfies the Curvature-Dimension condition CD(n−1−4n+α,−α), for all ∣x∣<1, α≥−n and n≥2. The case α=1 corresponds to the hit…