New proof of Yamabe invariant for RP^3 using harmonic functions.
problem Yamabe invariant of
method Using harmonic functions
result New proof of Yamabe invariant for
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48 results for “harmonic invariance”
New forms of symmetric shift-invariant subspaces found for harmonic maps.
problem Understanding harmonic maps into symmetric and -symmetric spaces.
method Imposing a symmetry condition on shift-invariant subspaces of a Hilbert space.
result Obtained new general forms for symmetric shift-invariant subspaces and extended solutions.
Study on harmonic spinors on specific Lie groups.
problem Existence of left-invariant harmonic spinors on 3D Lie groups.
method Revised spin Dirac operator formula for left-invariant spinors, identified constraints on Lie algebras, and classified metrics with harmonic spinors.
result Identified conditions and metrics for left-invariant harmonic spinors on 3D Lie groups.
Researchers compute the ν-invariant for specific G2-structures on nilmanifolds.
problem Detecting connected components of G2-structure moduli spaces.
method Defined and computed the ν-invariant using Mathai-Quillen currents, harmonic spinors, and η-invariants.
result Determined the parity of harmonic spinor dimensions and deduced ν vanishing on invariant spinors.
Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
problem Characterize left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
method Analyze left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups, classify Lie algebras and groups.
result New results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator.
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.
Study links harmonic maps to shift-invariant subspaces in complex function spaces.
problem Understanding the relationship between harmonic maps and shift-invariant subspaces.
method Operator-theoretic methods to derive a criterion for the finiteness of the uniton number.
result Derives a criterion for the finiteness of the uniton number in harmonic maps.
Harmonization schemes limit accuracy due to domain information.
problem Harmonization schemes lead to inaccurate predictions due to domain information.
method Analysis of mutual information and real label value informativeness.
result Accuracy is limited by the domain with least information.
Study invariant operators and vanishing theorems in CR geometry.
problem Analyzing invariant operators and vanishing theorems in CR geometry.
method Investigates Kohn-Dirac operators and derives CR invariant twistor operators.
result Proves vanishing theorems for harmonic spinors and Kohn-Rossi groups.
The study explores harmonic vector fields on a specific type of Riemannian Lie group.
problem Characterizing harmonic vector fields on a warped product of a line and a 3D Riemannian Lie group.
method Using a characteristic variational condition, the study applies to the case of a 3D Riemannian Lie group equipped with a left-invariant metric.
result Examples of harmonic vector fields on the warped product that are not left-invariant are provided.
The paper introduces a new complex analytic invariant called the pointed harmonic volume and its relation to the Johnson homomorphism.
problem Exploring new complex analytic invariants related to the complex structure of Riemann surfaces.
method Defining and computing the pointed harmonic volume as a natural extension of Chen's iterated integrals.
result Established a relationship between the harmonic volume and the first extended Johnson homomorphism.
Study on Lie groups' conformal foliations and harmonic morphisms.
problem Characterizing minimal conformal foliations on Lie groups.
method Analyzing left-invariant metrics and foliations on Lie groups.
result Minimal conformal foliations on Lie groups are fibres of harmonic morphisms.
Consider an asymptotically flat Riemannian manifold of dimension with nonempty compact boundary. We recall the harmonic conformal class of the metric, which consists of all conformal rescalings given by a harmonic function raised to an appropriate power. The geometric significance is that eve…
Study Dolbeault harmonic forms on Lie group quotients with specific structures.
problem Characterize the space of Dolbeault harmonic (1,1)-forms on compact Lie group quotients.
method Analyze left invariant almost Hermitian structures on 4D Lie groups and their quotients.
result Dimension of Dolbeault harmonic (1,1)-forms depends on existence of a specific anti-self-dual form.
Study on vector fields on Lie groups reveals surprising algebraic coincidences.
problem Characterizing vector fields on Lie groups with Riemannian metrics.
method Algebraic and geometric analysis of left-invariant vector fields on nilpotent Lie groups.
result Spaces of Killing, one-harmonic, and conformal vector fields coincide with the center of the Lie algebra on nilpotent Lie groups.
Minimal conformal foliations on Lie groups are shown to be fibres of harmonic morphisms.
problem Characterizing minimal conformal foliations on Lie groups.
method Analyzing left-invariant semi-Riemannian metrics and harmonic morphisms.
result Minimal conformal foliations of codimension two are fibres of complex-valued harmonic morphisms.
Harmonic 3-forms defined on compact homogeneous spaces are studied and conditions for their harmonicity are provided.
problem Analyzing harmonic 3-forms on compact homogeneous spaces.
method Investigating bi-invariant symmetric bilinear forms and their associated closed 3-forms, determining conditions for these forms to be harmonic under various metrics.
result Conditions for the harmonicity of 3-forms are given, and specific behaviors are observed depending on the structure of the space.
We consider four dimensional Lie groups with left-invariant Riemannian metrics. For such groups we classify left-invariant conformal foliations with minimal leaves of codimension two. These foliations produce local complex-valued harmonic morphisms.
Seiberg-Witten invariants match Gromov invariants for self-dual forms.
problem Equivalence of Seiberg-Witten and Gromov invariants for specific forms.
method Extension of Taubes' theorem to non-symplectic 4-manifolds, focusing on self-dual harmonic 2-forms.
result Seiberg-Witten invariants are equivalent to Gromov invariants for self-dual forms.
Classifies submaximally symmetric vector ODEs of C-class.
problem Classifying vector ODEs of C-class with specific invariants.
method Local classification using point transformations and harmonic theory.
result Generalizations of classical results for scalar ODEs.
New mathematical tools for studying knots and links.
problem Understanding knot and link diagrams using topological invariants.
method Introducing Khovanov Laplacian and Khovanov Dirac to study diagrams.
result The harmonic spectrum retains Khovanov homology invariants, while non-harmonic spectra reveal additional information.
We consider 5-dimensional Lie groups with left-invariant Riemannian metrics. For such groups we give a partial classification of left-invariant conformal foliations with minimal leaves of codimension 2. These foliations produce local complex-valued harmonic morphisms.
The aim of this paper is to calculate the eta invariants and the dimensions of the spaces of harmonic spinors of an infinite family of closed flat manifolds. It consits of some flat manifolds M with cyclic holonomy groups.
Paper proves existence of Dirac-harmonic maps with trivial index.
problem Finding Dirac-harmonic maps with trivial index.
method Defining a new quantity and proving its homotopy invariance.
result Existence of Dirac-harmonic maps from closed Riemann surfaces to Kähler manifolds.
New metrics with special curvature properties are shown to be parallel in certain Lie groups.
problem Characterizing metrics with harmonic curvature in Lie groups.
method Analyzing left invariant metrics on solvable and low-dimensional Lie groups.
result Left invariant metrics with harmonic curvature are Ricci-parallel in solvable Lie groups and Lie groups of dimension ≤6.
New spectral sequence for -manifolds, computing cohomology and harmonic forms.
problem Computing cohomology and harmonic forms of -manifolds.
method Introducing a new spectral sequence and using it to generalize theorems from -contact geometry.
result Computed cohomology ring and harmonic forms of -manifolds.
Left-invariant almost complex structures on the higher dimensional Kodaira-Thurston manifoldsmath.DG
Computational techniques calculate dimensions of complex structures.
problem Calculating dimensions of complex structures on manifolds.
method Developed computational techniques to calculate Kodaira dimension and Dolbeault harmonic forms.
result Computed dimensions of left-invariant almost complex structures.
We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation of a compact Riemannian manifold with coefficients in a leafwise U(p,q)-flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted higher Betti classes. Consequences …
Harmonic maps studied in sub-Riemannian geometry for Lie groups.
problem Defining and characterizing harmonic maps in sub-Riemannian settings.
method Generalization of Riemannian harmonic maps to sub-Riemannian manifolds and Lie groups.
result Conditions for sub-Riemannian harmonic maps and their classification.
Conformal harmonic maps from a 4-dimensional conformal manifold to a Riemannian manifold are maps satisfying a certain conformally invariant fourth order equation. We prove a general existence result for conformal harmonic maps, analogous to the Eells-Sampson theorem for harmonic maps. The proof uses a geometric flow a…
Biharmonic and harmonic homomorphisms between Riemannian three dimensional unimodular Lie groupsmath.DG
Study classifies biharmonic and harmonic homomorphisms between specific Lie groups.
problem Classifying biharmonic and harmonic homomorphisms between Riemannian three-dimensional unimodular Lie groups.
method Classification based on left invariant Riemannian metrics.
result Classification of biharmonic and harmonic homomorphisms between specific Lie groups.
Minimal vector fields on oscillator groups studied, with specific conditions for minimality.
problem Characterizing minimal left-invariant unit vector fields on oscillator groups.
method Analyzing structure constants and harmonic maps into the unit tangent bundle.
result Minimal vector fields defined by specific conditions on oscillator groups.
Investigates Schouten-Weyl tensor on 3D Lie groups with specific metrics.
problem Analyzing the Schouten-Weyl tensor on 3D Lie groups with special metrics.
method Examines left-invariant Lorentzian metrics and investigates harmonicity of the tensor.
result Identifies specific Lie groups with zero Schouten-Weyl tensor.
The paper develops a regularity theory for O'hara knot energies, focusing on Möbius energy.
problem Developing a regularity theory for extremal knots of scale invariant knot energies defined by J. O'hara.
method Reinterpreting O'hara knot energies as a nonlinear, nonlocal -energy acting on the unit tangent of the knot parametrization, drawing a connection to the theory of (fractional) harmonic maps into spheres.
result Proves regularity for minimizers and critical knots of the scale-invariant O'hara knot energies.
The article defines and analyzes Clairaut anti-invariant maps between Riemannian and trans-Sasakian manifolds.
problem Characterizing Clairaut anti-invariant Riemannian maps between specific types of manifolds.
method Deriving conditions for Clairaut maps, discussing integrability, and establishing harmonicity.
result Necessary and sufficient conditions for Clairaut anti-invariant maps are derived.
We are studying the harmonic and twistor equation on Lorentzian surfaces, that is a two dimensional orientable manifold with a metric of signature . We will investigate the properties of the solutions of these equations and try to relate the conformal invariant dimension of the space of harmonic and twistor spin…
Our purpose is to use a Darboux homogenous derivative to understand the harmonic maps with values in homogeneous space. We present a characterization of these harmonic maps from the geometry of homogeneous space. Furthermore, our work covers all type of invariant geometry in homogeneous space.
We present a new method for manufacturing complex-valued harmonic morphisms from a wide class of Riemannian Lie groups. This yields new solutions from an important family of homogeneous Hadamard manifolds. We also give a new method for constructing left-invariant foliations on a large class of Lie groups producing harm…
We generalize the Uhlenbeck-Segal theory for harmonic maps into compact semi-simple Lie groups to general Lie groups equipped with torsion free bi-invariant connection.
CR-harmonic mapsmath.DG
CR-harmonic maps defined for pseudoconvex manifolds.
problem Defining CR-harmonic maps in CR geometry.
method Developing renormalized energy and CR covariant subelliptic PDE.
result CR-harmonic maps satisfy a CR covariant subelliptic PDE.
Unified approach to conformal and modular invariants on surfaces.
problem Constructing a general family of conformal invariants on surfaces.
method Using an identification of Teichmüller space and rigged moduli space, and analytic work on harmonic functions.
result Unified conformal and modular invariants can be viewed as generalized modular invariants and functions on the rigged moduli space.
Scattering theory for harmonic one-forms on Riemann surfaces.
problem Understanding harmonic one-forms on Riemann surfaces.
method Constructing scattering theory through boundary value problems and integral operators.
result Explicit expression for the scattering matrix and proof of unitarity.
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…
Classifies sections of Riemannian bundles on Lie groups.
problem Classifying sections of Riemannian bundles on Lie groups.
method Developed variational theory of higher-power energy for mappings and sections.
result Complete classification of left-invariant vector fields on 3D Lie groups.
Study finds bound on energy of minimal spheres on complex manifolds.
problem Finding bounds on energy of minimal spheres on complex manifolds.
method Proving existence of harmonic spheres with Morse index bound one.
result Sum of energies of minimal spheres realizes a geometric invariant width.
The paper classifies Ricci solitons and studies harmonic vector fields on a specific Thurston geometry.
problem Classifying Ricci solitons and studying harmonic vector fields in a specific Thurston geometry.
method Left-invariant Riemannian metric classification and analysis of harmonic maps and vector fields.
result All Ricci solitons on are expanding and non-gradient.
New method corrects MRI biases across scanners and sites.
problem Site and scanner biases in diffusion MRI data.
method Learning invariant representations using variational auto-encoders (VAE).
result Improvements on test data relative to a baseline method.
We consider the oscillator group equipped with a bi-invariant Lorentzian metric, and then some geometrical properties of this group i.e. homogeneous Ricci solitons and harmonicity properties of invariant vector fields are obtained. We also determine all vector fields which are critical points for the energy functional …