arXiv research

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

Uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures established.

Harmonic extension of Weil-Petersson circle homeomorphisms

In a family of compact, canonically polarized, complex manifolds equipped with Kähler-Einstein metrics the first variation of the lengths of closed geodesics was previously shown in by the authors in [arXiv:0808.3741v2] to be the geodesic integral of the harmonic Kodaira-Spencer form. We compute the second variation. F…

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

We define a discrete Laplace-Beltrami operator for simplicial surfaces. It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called ``cotan formula'') except that it is based on the intrinsic Delau…

Uniform bounds are developed for derivatives of solutions of the $2$-dimensional constant negative curvature equation and the Weil-Petersson metric for the Teichmüller and moduli spaces. The dependence of the bounds on the geometry of the underlying Riemann surface is studied. The comparisons between the $C^0$, $C^{2,α…

We study the soliton flow on the domain of a twistorial harmonic morphism between Riemannian manifolds of dimensions four and three. Assuming real-analyticity, we prove that, for the Gibbons-Hawking construction, any soliton flow is uniquely determined by its restriction to any local section of the corresponding harmon…

For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace-Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differ…

Bayesian framework for sphere regression using Gaussian fields.

Study improves Beltrami fields with nonconstant proportionality factors.

We use Beltrami's theorem as an excuse to present some arguments from parabolic differential geometry without any of the parabolic machinery.

Harmonic functions on compact symmetric spaces exhibit strong convexity properties.

We give a proof of the Donnelly-Fefferman growth bound of Laplace-Beltrami eigenfunctions which is probably the easiest and the most elementary one. Our proof also gives new quantitative geometric estimates in terms of curvature bounds which improve and simplify previous work by Garofalo and Lin. The proof is based on …

Using the symmetry properties of two-dmensional sigma models, we introduce a notion of the Beltrami-Courant differential, so that there is a natural homotopy Gerstenhaber algebra related to it. We conjecture that the generalized Maurer-Cartan equation for the corresponding $L_{\infty}$ subalgebra gives solutions to the…

Geometrically connects Laplace eigenfunctions to Borel-Weil theory on symmetric spaces.

The paper finds rotationally symmetric critical metrics for Laplace eigenvalues on tori.

We introduce a general notion of twistorial map and classify twistorial harmonic morphisms with one-dimensional fibres from self-dual four-manifolds. Such maps can be characterised as those which pull back Abelian monopoles to self-dual connections. In fact, the constructions involve solving a generalised monopole equa…

Geometric theory connects machine learning classifiers to differential geometry.

Generalizes quantum integrability to all signatures for projectively equivalent metrics.

Holomorphic solutions vary in Sobolev spaces for Beltrami equations.

This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We give a constructive proof of a critical metric which is smooth except at some coni…

We prove that the nodal set (zero set) of a solution of a generalized Dirac equation on a Riemannian manifold has codimension 2 at least. If the underlying manifold is a surface, then the nodal set is discrete. We obtain a quick proof of the fact that the nodal set of an eigenfunction for the Laplace-Beltrami operator …

Method finds domain of Laplace-Beltrami operator on 2D almost-Riemannian manifolds.

Let $Ł$ be the Laplace operator on $\R ^d$, $d\geq 3$ or the Laplace Beltrami operator on the harmonic $NA$ group (in particular on a rank one noncompact symmetric space). For the equation $Łu - \varphi(\cdot,u)=0$ we give necessary and sufficient conditions for the existence of entire bounded or large solutions under…

We establish dimension-independent estimates related to heat operators e^{tL} on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates on the norm behavior of harmonic and non-negative subharmonic functions. We apply…

In this article, associated with each lattice $T\subseteq \mathbb{Z}^n$ the concept of a harmonic-counting measure $ν_T$ on a sphere $S^{n-1}$ is introduced and it is applied to determine the asymptotic behavior of the eigenfunctions of the Laplace-Beltrami operator on a lens space. In fact, the asymptotic behavior of …

On a compact Kähler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eig…

The Navier-Stokes equations on certain manifolds can perform universal computation.

Optimizes maps with controlled distortion for geometric tasks.

In this note we determine the first two derivatives of the classical Boltzmann-Shannon entropy of the conjugate heat equation on general evolving manifolds. Based on the second derivative of the Boltzmann-Shannon entropy, we construct Perelman's F and W entropy in abstract geometric flows. Monotonicity of the entropies…

In this paper, we investigate the first eigenvalues of two closed eigenvalue problems of the bi-Beltrami-Laplacian on minimal embedded isoparametric hypersurface in the unit sphere $\mathbb{S}^{n+1}(1)$. Although many mathematicians want to derive the corresponding results for the first eigenvalues of bi-Beltrami-Lapla…

Study Chern number inequalities for negative curvature Kähler manifolds.

This paper presents a method to compute the {\it quasi-conformal parameterization} (QCMC) for a multiply-connected 2D domain or surface. QCMC computes a quasi-conformal map from a multiply-connected domain $S$ onto a punctured disk $D_S$ associated with a given Beltrami differential. The Beltrami differential, which me…

This is an expository article on the question of whether zero lies in the spectrum of the Laplace-Beltrami operator acting on differential forms on a manifold.

The paper explores self-adjointness of Laplace-Beltrami operator on special geometric manifolds.

GNPs learn operators on non-Euclidean geometries using neural networks.

Study differential operators on specific manifolds and their harmonic forms.

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

Study Z2 harmonic functions with singularities on flat space.

Introduces a new Hodge theory using vector fields on manifolds.

Study differential operators on non-compact harmonic manifolds, finding conditions for radial fundamental solutions and dense heat-semigroups.

Non-polynomial growth harmonic maps from the complex plane to the hyperbolic space are studied. Some non-surjectivity results are obtained. Moreover, images of such harmonic maps are investigated with reference to their Hopf differentials.

The study examines surfaces in isotropic space with specific Gauss map properties.

Unified geometric framework for Brownian motion on various manifolds.

In this paper, we introduce a new concept so called harmonic complex structure by using harmonic theory for vector bundle-valued differential forms. It is a new structure intermediates between complex structure and Kähler structure. From differential geometric viewpoint, it is a natural generalization of Kähler structu…

Finite energy solutions of 4-harmonic and ES-4-harmonic maps are trivial.

New formulas for Bertrand curves lead to harmonicity conditions.