Recalls intrinsically harmonic forms and open problems.
problem Open problems related to intrinsically harmonic forms.
method Recalling definitions and known results.
result Clarification of intrinsically harmonic forms and open questions.
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48 results for “intrinsically harmonic forms”
Criterion for flat circle bundles using intrinsically harmonic forms.
problem Characterizing flat circle bundles.
method Criterion based on intrinsic harmonicity of a specific form.
result Flatness of a principal circle bundle is equivalent to intrinsic harmonicity of a certain form.
The n-dimensional torus is uniquely characterized by specific harmonic forms.
problem Characterizing the n-dimensional torus via harmonic forms.
method Analyzing closed 1-forms on the torus to determine unique properties.
result The n-dimensional torus is the unique manifold supporting a linearly independent set of (n-1) closed 1-forms whose product determines a non-zero cohomological class.
We discuss the construction of Sp(2)Sp(1)-structures whose fundamental form is closed. In particular, we find 10 new examples of 8-dimensional nilmanifolds that admit an invariant closed 4-form with stabiliser Sp(2)Sp(1). Our constructions entail the notion of SO(4)-structures on 7-manifolds. We present a thorough inve…
Sharp Veronese rigidity theorem for submanifolds of unit ball.
problem Veronese rigidity of submanifolds under harmonic structure.
method Intrinsic harmonic structure assumptions, Bochner-Gauss mechanism, shape operators.
result Sharp lower bound on maximal normal curvature for specific submanifolds.
Every harmonic map is an intrinsic bi-harmonic map as an absolute minimizer of the intrinsic bi-energy functional, therefore intrinsic bi-harmonic map and its heat flow are more geometrically natural to study, but they are also considerably more difficult analytically than the extrinsic counterparts due to the lack of …
The paper studies harmonic graphs in the Heisenberg group and their properties.
problem No analogous theorem exists for -minimal surfaces in the Heisenberg group.
method Introduced intrinsic Dirichlet energy and studied its critical points (contact harmonic graphs).
result Calibration condition and construction of energy-minimizing graphs with various singularities.
Bayesian framework for sphere regression using Gaussian fields.
problem Nonparametric regression on the sphere with Gaussian priors.
method Isotropic Gaussian field priors, harmonic structure, exact posterior distributions, optimal spectral truncation, posterior contraction rates.
result Sharp posterior contraction rates for Gaussian priors with polynomially decaying angular power spectra.
Let be a closed oriented manifold of dimension and a closed 1-form on it. We discuss the question whether there exists a Riemannian metric for which is co-closed. For closed 1-forms with nondegenerate zeros the question was answered completely by Calabi in 1969. The goal of this paper is to give an answ…
Proves harmonic coordinates for weak immersions in even dimensions.
problem Existence of harmonic coordinates for weak immersions in Sobolev spaces.
method Analyzes weak immersions in critical Sobolev spaces and uses smallness conditions on the second fundamental form.
result Global harmonic coordinates exist for weak immersions in even dimensions under certain conditions.
We go further on the study of harmonicity for almost contact metric structures already initiated by Vergara-Diaz and Wood. By using the intrinsic torsion, we characterise harmonic almost contact metric structures in several equivalent ways and show conditions relating harmonicity and classes of almost contact metric st…
Study topological properties of foliations induced by closed 1-forms on orbifolds.
problem Characterize the topology of foliation leaves induced by closed 1-forms on orbifolds.
method Establish criteria for the compactness of foliation leaves and extend a topological result to orbifolds.
result Criteria for the compactness and coexistence of foliation leaves are established.
We show how the tangent bundle decomposition generated by a system of ordinary differential equations may be generalized to the case of a system of second order PDEs `of connection type'. Whereas for ODEs the decomposition is intrinsic, for PDEs it is necessary to specify a closed 1-form on the manifold of independent …
Let (M,I, ω, Ω) be a nearly Kaehler 6-manifold, that is, an SU(3)-manifold with the (3,0)-form Ωand the Hermitian form ωwhich satisfies , for a non-zero real constant λ. We develop an analogue of Kaehler relations on M, proving several useful identities for various intrinsic Laplacians on M. Wh…
Study sesqui-harmonic map flow from Riemannian surfaces
problem Investigate sesqui-harmonic map flow from Riemannian surfaces
method L2-gradient flow of an energy functional
result Generalizes Struwe's regularity result for harmonic maps
Develops a framework for generating harmonic maps from a unit ball to a sphere.
problem Creating families of harmonic maps from a unit ball to a sphere.
method Based on Toth's machinery for generating eigenmaps, combined with a generalized radial projection.
result Provides theoretical background for constructing solutions to variational problems.
The paper investigates Liouville type theorems for various harmonic forms on Riemannian manifolds.
problem Investigating Liouville type properties of harmonic forms on Riemannian manifolds.
method Normalized integral Ricci curvature and BiRic curvature.
result Established Liouville theorems for -harmonic function, -harmonic 1 form, and harmonic form (with ).
The study of harmonicity for almost contact metric structures was initiated by Vergara-Díaz and Wood and continued by González-Dávila and the present author. By using the intrinsic torsion and some restriction on the type of almost contact metric structure, González-Dávila and the present author have characterised harm…
New metrics produce discrete zero sets for nondegenerate harmonic forms.
problem Creating metrics to produce discrete zero sets for nondegenerate harmonic forms.
method Metric perturbation to produce new nondegenerate harmonic forms with discrete zero sets.
result Existence of metrics producing discrete zero sets for nondegenerate harmonic forms.
The study resolves a conjecture about harmonic forms on compact manifolds.
problem Finding non-degenerate -harmonic 1-forms on compact manifolds.
method Develops a gluing theorem for non-degenerate -harmonic 1-forms on compact manifolds.
result Proves the existence of non-degenerate -harmonic 1-forms on compact manifolds with positive first Betti number.
The paper examines metrics on foliated manifolds that have special geometric properties.
problem Characterizing metrics on foliated manifolds with specific harmonic properties.
method Examining the properties of bundle-like metrics on foliated manifolds.
result The interior product of basic harmonic forms is basic harmonic under certain conditions.
The study examines harmonic forms on almost Hermitian manifolds and complex surfaces.
problem Analyzing harmonic forms on almost Hermitian manifolds and complex surfaces.
method Using techniques from Bott-Chern and Aeppli numbers, the study generalizes harmonic forms from complex and symplectic manifolds to almost Hermitian manifolds.
result Bott-Chern and Aeppli numbers of compact complex surfaces depend only on the topology of the underlying manifold.
Study primitive decompositions for harmonic forms on almost Kähler manifolds.
problem Decomposing harmonic forms on almost Kähler manifolds.
method Proved primitive decompositions for Bott-Chern and Aeppli harmonic forms in specific bidegrees.
result Optimal bidegrees for primitive decompositions of harmonic forms.
The study calculates harmonic functions and 1-forms on specific 4D spaces.
problem Computing harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
method Computed the expansion of harmonic functions and 1-forms.
result Computed the expansion of harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
The paper proves vanishing and finiteness theorems for p-harmonic 1-forms.
problem The vanishing and finiteness of p-harmonic 1-forms on submanifolds.
method Using BiRic curvature conditions to prove theorems.
result Theorems on vanishing and finiteness of p-harmonic 1-forms.
The study finds nondegenerate harmonic 1-forms using symmetry conditions.
problem Existence of nondegenerate harmonic 1-forms over Riemannian manifolds.
method Utilizing symmetry to establish topological conditions.
result Found nondegenerate harmonic 1-forms over branched coverings of links.
Conditions for a soliton's dual form to be harmonic or Ricci harmonic are derived.
problem Characterizing solitons and their dual forms.
method Necessary and sufficient conditions for the dual form to be harmonic or Ricci harmonic are derived.
result Conditions for the dual form of a soliton to be harmonic or Ricci harmonic are provided.
The study shows that the second fundamental form is intrinsic under certain conditions in space forms.
problem Understanding the intrinsic nature of the second fundamental form in space forms.
method Proving the intrinsic nature of the normalized second fundamental form under specific conditions.
result The normalized second fundamental form is intrinsic if for some .
We consider interpolating sesqui-harmonic Legendre curves in Sasakian space forms. We find the necessary and sufficient conditions for Legendre curves in Sasakian space forms to be interpolating sesqui-harmonic. Finally, we obtain an example for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.
We characterize the harmonic forms on a flag manifold defined by Kostant in 1963 in terms of a Poisson structure. Namely, they are ``Poisson harmonic" with respect to the so-called Bruhat Poisson structure on . This enables us to give Poisson geometrical proofs of many of the special properties of these harm…
Symmetry operators of twistor spinors and harmonic spinors can be constructed from conformal Killing-Yano forms. Transformation operators relating twistors to harmonic spinors are found in terms of potential forms. These constructions are generalized to gauged twistor spinors and gauged harmonic spinors. The operators …
In general, the product of harmonic forms is not harmonic. We study the top exterior power of harmonic two-forms on compact Kaehler manifolds. Often, it is not harmonic. This phenomenon is related to the geometry of the manifold and to the existence of rational curves in particular. K3 surfaces and hyperkaehler manifol…
The paper constructs harmonic 1-forms on 3-manifolds with cylindrical necks.
problem Stabilizing Z/2-harmonic 1-forms on closed 3-manifolds.
method Explicit construction of harmonic 1-forms by modifying metrics near links.
result Construction of harmonic 1-forms degenerating to manifolds with cylindrical ends.
Study metric perturbations to make degenerate harmonic forms non-degenerate.
problem Dealing with degenerate harmonic 1-forms in Riemannian geometry.
method Combining analysis of local expansions with Nash-Moser implicit function theorem.
result Proves deformation to nearby non-degenerate Z/2-harmonic 1-forms.
Study -harmonic forms on almost Kähler manifolds, extending vanishing theorems.
problem Analyzing -harmonic forms on complete almost Kähler manifolds.
method Decomposing -harmonic forms into Lefschetz powers of primitive forms, extending vanishing theorems.
result Spaces of harmonic -forms on vanish unless .
Analyzes -harmonic forms on curved manifolds, proving integrability conditions.
problem Analyzing integrability of -harmonic forms on curved manifolds.
method Established -estimate via Moser iteration, proved vanishing of integrable forms.
result Proves that -harmonic forms on non-positively curved manifolds are integrable if and only if they vanish.
Novel singularity models for 4D harmonic forms and spinors from polytopes.
problem Understanding harmonic forms and spinors in 4D.
method Homogeneous singularity models based on regular 4-polytopes.
result Models describe cones on the 1-skeletal of polytopes.
P. Baird and the second author studied harmonic morphisms from a three-dimensional simply-connected space form to a surface and obtained a complete local and global classification of them. In this paper, we obtain a description of all harmonic morphisms from any three-dimensional Euclidean and spherical space form to a…
Harmonic forms and Rumin complex linked on Sasakian manifolds.
problem Relationship between harmonic forms and Rumin complex on Sasakian manifolds.
method Analytic torsion function and Rumin complex analysis.
result Kernel of Rumin Laplacian matches Hodge-de Rham Laplacian on compact Sasakian manifolds.
The structure equations for a surface are introduced and two required results based on the Codazzi equations are obtained from them. Important theorems pertaining to isometric surfaces are stated and a theorem of Bonnet is obtained. A tranformation formula for the connection forms is developed. It is proved that the an…
New examples of Z/2 harmonic 1-forms and their branching sets are explored.
problem Exploring the properties and examples of Z/2 harmonic 1-forms and their branching sets.
method Elementary constructions and families of harmonic 1-forms.
result The branching set of a harmonic 1-form can exhibit various features including non-trivial links, multiple covers, and immersed structures.
Quantifies uniqueness of conformal-harmonic maps on 4-manifolds.
problem Quantifying uniqueness of conformal-harmonic maps on 4-manifolds.
method Proves a quantitative uniqueness result using convexity and second order Hardy inequality.
result Proves a version of second order Hardy inequality on manifolds.
The paper introduces new cohomologies and studies harmonic forms on almost complex manifolds.
problem Understanding cohomologies and harmonic forms on almost complex manifolds.
method Introducing new cohomologies (Bott-Chern and Aeppli) and studying associated harmonic forms.
result Bott-Chern cohomology of 1-forms is finite-dimensional on compact manifolds and provides an invariant.
We prove that manifolds admitting a Riemannian metric for which products of harmonic forms are harmonic satisfy strong topological restrictions, some of which are akin to properties of flat manifolds. Others are more subtle, and are related to symplectic geometry and Seiberg-Witten theory. We also prove that a manifold…
Study on harmonic forms on K3 surfaces converging to a flat 4D orbifold.
problem Behavior of harmonic 2-forms on K3 surfaces with Ricci-flat metrics.
method Analysis of convergence of harmonic forms to flat 4D orbifold.
result Decomposition of harmonic 2-forms into converging subspaces.
New type of harmonic Finsler manifolds created.
problem Creating new types of harmonic Finsler manifolds.
method Defined by a Riemannian metric and a special 1-form.
result Constructed harmonic and asymptotically harmonic Finsler manifolds of -type.
The paper connects harmonic forms to tree maps and character varieties.
problem Analytic compactification of SL2(C) moduli spaces.
method Explicit correspondence between Z/2 harmonic 1-forms and tree maps.
result Existence of Z/2 harmonic 1-forms on all Haken manifolds.
Study existence of harmonic 1-forms on Calabi-Yau manifolds.
problem Tackles existence of harmonic 1-forms on Calabi-Yau manifolds.
method Uses neural networks to approximate metrics and harmonic 1-forms.
result Suggests existence of harmonic 1-forms on some Calabi-Yau manifolds.