arXiv research

Harmonic extension of Weil-Petersson circle homeomorphisms

We prove that every non-constant quasiregular selfmap of the $n$-sphere $\mathbb{S}^{n}$ admits a harmonic extension to the hyperbolic space $\mathbb{H}^{n+1}$ for $n\ge 2$.

Develops relative harmonic metrics and deformation theory for Higgs and flat bundles on compact Kähler manifolds.

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"…

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.

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.

Extends functions on symmetric spaces to analytic functions.

Study surfaces in Half-Pipe space and vector fields on hyperbolic plane.

Maximizes capacity of extensions with fixed boundary data.

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.

DELIMIT is a framework extension for deep learning in diffusion imaging, which extends the basic framework PyTorch towards spherical signals. Based on several novel layers, deep learning can be applied to spherical diffusion imaging data in a very convenient way. First, two spherical harmonic interpolation layers are a…

$α$-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $α$-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For $α>1$, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in …

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…

Vortex solutions on flat surfaces map to harmonic spinors on Nappi-Witten space.

The paper studies $p$-harmonic functions and their conjugates, showing they converge to calibrations of laminations.

Researchers extend regularity of $p$-harmonic maps into spheres for a new range of $p$.

Global existence and convergence of heat flow for p-harmonic maps.

Extends Eells-Sampson theorem for manifolds with positive sectional curvature bounds.

New non-existence results for harmonic maps into perturbed cones.

The paper explores unique continuation properties for polyharmonic maps between Riemannian manifolds.

In this paper we construct a new family of harmonic morphisms $\varphi:V^5\to\s^2$, where $V^5$ 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 ${V^{\ast}}^5$, which turns out to be an explicit real algebra…

Multimodal learning aims to discover the relationship between multiple modalities. It has become an important research topic due to extensive multimodal applications such as cross-modal retrieval. This paper attempts to address the modality heterogeneity problem based on Gaussian process latent variable models (GPLVMs)…

The paper studies rigidity results for harmonic forms on Kähler manifolds.

In this article we introduce a natural extension of the well-studied equation for harmonic maps between Riemannian manifolds by assuming that the target manifold is equipped with a connection that is metric but has non-vanishing torsion. Such connections have already been classified in the work of Cartan. The maps unde…

Every connected, weighted graph with non-negative curvature has exactly two ends.

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 period for a compact Riemann surface, defined by the integral of differential 1-forms, is a classical complex analytic invariant, strongly related to the complex structure of the surface. In this paper, we treat another complex analytic invariant called the pointed harmonic volume. As a natural extension of the per…

The paper examines curvature and stability in quasi-geostrophic motions using spherical harmonics.

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 …

We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions $n \geq 4$. For minimizing harmonic maps $u\in W^{1,2}(Ω,\mathbb{S}^2)$ from $n$-dimensional domains into the two dimensional sphere we prove: (1) An extension of Almgren and Lieb's linear law, namely \[\mathcal{H}^{n-3}(\textrm{sin…

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…

Study dynamics of $L^p$-multipliers on harmonic manifolds with exponential volume growth.

The paper constructs non-convergent solutions to Vafa-Witten equations with specific harmonic 2-form limits.

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.

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 $\mathbb{S}^{n-1}$, $n\geq 3$, 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.

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 $CD(n-1-\frac{n+α}{4},-α)$, for all $|x| < 1$, $α\geq -n$ and $n\geq 2$. The case $α= 1$ corresponds to the hit…

The paper solves the Dirichlet problem at infinity and defines Poisson boundaries for certain manifolds.

Minimal harmonic maps proved for specific manifolds.

The period is a classical complex analytic invariant for a compact Riemann surface defined by integration of differential 1-forms. It has a strong relationship with the complex structure of the surface. In this chapter, we review another complex analytic invariant called the harmonic volume. It is a natural extension o…

Researchers create integral representations for two-layer ReLU networks with quantitative bounds.