arXiv research

The paper examines properties of generalized τ-quasi Ricci-harmonic metrics and proves rigidity results.

The study of harmonic diffeomorphisms reduces to solving a specific Beltrami equation.

Study on linear stability of Schwarzschild spacetime using harmonic gauge.

Twistor methods provide a powerful tool in the study of harmonic maps and harmonic morphisms. Indeed, their use has enabled us to produce a variety of examples of harmonic morphisms defined on 4-dimensional manifolds, and a complete classification in some cases. In the first part of this work, we generalize those const…

Proves part of the singular set of energy minimizing harmonic maps is a topological manifold.

Obtains a condition for complex deformations of Kähler-Ricci solitons.

Abstract operator calculus solves fermionic quantum harmonic oscillator problems.

In this paper, the author discusses the eigenvalues and entropies under the harmonic-Ricci flow, which is the Ricci flow coupled with the harmonic map flow. We give an alternative proof of results for compact steady and expanding harmonic-Ricci breathers. In the second part, we derive some monotonicity formulas for eig…

In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work of Lopez-Rio imply rigidity of gradient shrinking Ricci solitons with harmonic Weyl tensor. In the second part of the paper we address the issue of existence of harmonic …

Paper bounds local curvature for Ricci-harmonic flow under Ricci bounded condition.

Harmonic maps from complex plane to hyperbolic space constructed using heat flow.

We continue our discussion from part I.

Harmonic functions on Calabi-Yau manifolds with maximal volume growth are studied.

BacHMMachine harmonizes Baroque chorales using theory-driven principles and Hidden Markov Models.

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

We study a functional, whose critical points couple Dirac-harmonic maps from surfaces with a two form. The critical points can be interpreted as coupling the prescribed mean curvature equation to spinor fields. On the other hand, this functional also arises as part of the supersymmetric sigma model in theoretical physi…

Random spherical harmonics on $S^3$ have a single nodal component with expected genus proportional to $mN$.

The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank~1. This conjecture has been proved by Z. Szabó \cite{Sz} for harmonic manifolds with compact universal cover. E. Damek and F. Ricci \cite{DR} provided examples showing that in the noncompact case the conj…

Study on harmonic flow of Spin(7)-structures in 8D manifolds.

Establishes regularity of conformal harmonic maps.

Harmonic maps link Teichmüller spaces to framed representations.

In this paper we give some results on the topology of manifolds with $\infty$-Bakry-Émery Ricci tensor bounded below, and in particular of steady and expanding gradient Ricci solitons. To this aim we clarify and further develop the theory of f-harmonic maps from non-compact manifolds into non-positively curved manifold…

We characterize general pseudo-harmonic morphisms from a Riemannian manifold to a Hermitian manifold as pseudo horizontally weakly conformal maps with an additional property. We study to what extent we can (locally) describe these submersive pseudo-harmonic morphisms via the foliation given by the kernel of the associa…

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

We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemann geometry of the target. These maps are called affine harmonic. We show an existence result for affine harmonic…

We consider harmonic immersions in $\R^{\N}$ of compact Riemann surfaces with finitely many punctures where the harmonic coordinate functions are given as real parts of meromorphic functions. We prove that such surfaces have finite total Gauss curvature. The contribution of each end is a multiple of $2π$, determined by…

The paper explores eigenfunctions of spherical conical metrics using harmonic maps to spheres.

Optimal Liouville theorem for minimal disks in any codimension.

Extremal metrics found for Laplace eigenvalues in nearby conformal classes.

Study harmonic measures and rigidity in Seifert 3-manifolds using $S^1$-connections.

We consider noncompact complete manifolds with Spin(9) holonomy and proved an one end result and a splitting type theorem under different conditions on the bottom of the spectrum. We proved that any harmonic functions with finite Dirichlet integral must be Cayley-harmonic, which allowed us to conclude an one end result…

Embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$ are reasonably well understood: From far away, they look like intersecting catenoids and planes, suitably desingularized. We consider the larger class of harmonic embeddings in $\mathbb{R}^{3}$ of compact Riemann surfaces with finitely many punctures…

Study pinches Weyl curvature on 4-manifolds, proving anti-self-duality.

The paper studies steady solitons with curvature decay and proves their smoothness.

The paper classifies Willmore 2-spheres in $S^n$.

Let $(X,g)$ be a compact Riemannian stratified space with simple edge singularity. Thus a neighbourhood of the singular stratum is a bundle of truncated cones over a lower dimensional compact smooth manifold. We calculate the various polynomially weighted de Rham cohomology spaces of $X$, as well as the associated spac…

Random walks on mapping class group lead to recurrent geodesics in quadratic differentials.

Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.

Extends Teichmüller space parametrization using poles of higher order.

Paper answers part of a conjecture about submersions from 3D spaces.

For any n>1 we give an explicit example of an n-axially symmetric Cartesian current in B^3 x S^2 with non-trivial vertical part and non-constant graph part minimizing the relaxed Dirichlet energy among the n-axially symmetric Cartesian currents with the same boundary. This stands in sharp contrast with a results of Har…

The Ooguri-Vafa space is linked to framed wild harmonic bundles via hyperkähler geometry.

Study on stability of half-harmonic maps from R to S, proving non-degeneracy and quantitative stability.

The paper studies the Teichmüller harmonic map flow and its limits.

Study the energy distribution of harmonic 1-forms on Riemann surfaces with a short geodesic.

Harmonic analysis on directed graphs for signal modeling and semi-supervised learning.

For a Kähler manifold endowed with a weighted measure $e^{-f}\,dv,$ the associated weighted Hodge Laplacian $Δ_{f}$ maps the space of $(p,q)$-forms to itself if and only if the $(1,0)$-part of the gradient vector field $\nabla f$ is holomorphic. We use this fact to prove that for such $f$, a finite energy $f$ harmonic …

Study classifies triharmonic surfaces in 3D homogeneous spaces.