arXiv research

We say that a distribution is harmonic if it is harmonic when considered as a section of a Grassmann bundle. We find new examples of harmonic distributions and show nonexistense of harmonic distrubutions on some Riemannian manifolds by two different approaches. Firstly, we lift distributions to the second tangent bundl…

Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.

Kaimanovich and Masur showed that a random walk on the mapping class group for an initial distribution with finite first moment and whose support generates a non-elementary subgroup, converges almost surely to a point in the space PMF of projective measured foliations on the surface. This defines a harmonic measure on …

A new method estimates marginal likelihood using normalizing flows.

Pseudo-harmonic morphisms give rise on the domain space to a distribution which admits an almost complex structure compatible with the given Riemannian metric. We shall show that this property, together with the harmonicity, are preserved by a biconformal change of the domain metric. The special case of the pseudo-hori…

Statistical hyperbolicity proven for Teichmüller space.

Let $π:(E,\nabla^{E}) \to (M,g)$ be an affine submersion with horizontal distribution, where $\nabla^{E}$ is a symmetric connection and $M$ is a Riemannian manifold. Let $σ$ be a section of $π$, namely, $π\circ σ= Id_{M}$. It is possible to study the harmonic property of section $σ$ in two ways. First, we see $σ$ as a …

Given a $C^1$ planes distribution $P_T$ on all ${\mathbb R}^m$ we consider {\em horizontal $α$-harmonic maps}, $α\ge 1/2$, with respect to such a distribution. These are maps $u\in H^α({\mathbb R}^k,{\mathbb R}^m)$ satisfying $P_T\nabla u=\nabla u$ and $P_T(u)(-Δ)^αu=0$ in ${\mathcal D}'({\mathbb R}^k).$ If the distrib…

The study shows that the visible range from a point on harmonic manifolds follows an exponential distribution.

In a range of fields including the geosciences, molecular biology, robotics and computer vision, one encounters problems that involve random variables on manifolds. Currently, there is a lack of flexible probabilistic models on manifolds that are fast and easy to train. We define an extremely flexible class of exponent…

Harmonic maps study on surfaces with non-positive curvature.

In this note, we consider a fixed vector field $V$ on $S^2$ and study the distribution of points which lie on the nodal set (of a random spherical harmonic) where $V$ is also tangent. We show that the expected value of the corresponding counting function is asymptotic to the eigenvalue with a leading coefficient that i…

This work aims to create a large-scale model for critical care time series data.

GH-PID uses guided harmonic paths for efficient SOT with interpretable diagnostics.

The paper defines and explores Clairaut conformal submersions in Riemannian geometry.

Study on Riemannian submersions from nearly Kaehler manifolds.

This paper shows that when the Riemannian metric on a contact manifold is blown up along the direction orthogonal to the contact distribution, the corresponding harmonic forms rescaled and normalized in the $L^2$-norms will converge to Rumin's harmonic forms. This proves a conjecture in Gromov `` Carnot-Caratheodory sp…

Let $M$ be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by $σ$. Let $f: M \rightarrow \mathbb{R}$ be a non-constant eigenfunction of the Laplacian. The random wave conjecture suggests that in certain situations, the value distribution of $f$ under $σ$ is approximately Gaussian. Wr…

Given a measure on the Thurston boundary of Teichmueller space, one can pick a geodesic ray joining some basepoint to a randomly chosen point on the boundary. Different choices of measures may yield typical geodesics with different geometric properties. In particular, we consider two families of measures: the ones whic…

Study cohomology classes related to harmonic maps on submersions.

Study on harmonic maps on weighted Riemannian foliations.

In this article we study the role of the Green function for the Laplacian in a compact Riemannian manifold as a tool for obtaining well-distributed points. In particular, we prove that a sequence of minimizers for the Green energy is asymptotically uniformly distributed. We pay special attention to the case of locally …

The paper examines triviality of Ricci-Bourguignon harmonic solitons.

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…

Study cohomology classes related to $n$-harmonic morphisms and $F$-harmonic maps.

Study on CPSRM from/to Kähler manifolds, deriving integrability and geodesic results.

The paper proves nonexistence of harmonic and bi-harmonic maps under specific conditions.

We study the energy distribution of harmonic 1-forms on a compact hyperbolic Riemann surface $S$ where a short closed geodesic is pinched. If the geodesic separates the surface into two parts, then the Jacobian torus of $S$ develops into a torus that splits. If the geodesic is nonseparating then the Jacobian torus of $…

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…

Finite energy solutions of 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 $p\to \infty$. Infinity harmoncity appears in many familiar contexts. For example,…

Study examines maximal domains of radial harmonic functions across different curvature types.

New p-harmonic and harmonic morphisms found on Lie groups.

Constructs explicit harmonic functions and morphisms on complex and quaternionic Grassmannians.

Bayesian framework for sphere regression using Gaussian fields.

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"…

Extends p-harmonic map theory for new properties.

New method uses spherical harmonics to simplify learning single-index models.

$\infty$-Harmonic maps are a generalization of $\infty$-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 $\infty$-harmonic maps from and into a sphere, quadratic $\infty$-harmonic maps between E…

Dirac-harmonic maps are uncoupled under certain conditions.

The paper studies Hodge Laplacians from point clouds, proving spectral convergence and harmonic form consistency.

The paper investigates Liouville type theorems for various harmonic forms on Riemannian manifolds.

Study existence of harmonic and Dirac-harmonic maps from degenerating surfaces.

Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and applications to several areas in mathematics (see the book by Baird and Wood for details)…

Analyzes harmonic and biharmonic maps from gradient Ricci solitons.

The paper studies harmonic map heat flow to flat tori, proving ergodic behavior and convergence to hyperbolic measure.

Study heat flow for half-harmonic maps and harmonic maps with free boundary.

The paper explores harmonic and asymptotically harmonic Finsler manifolds and their properties.