Study singularities of constant curvature surfaces and harmonic maps.
problem Understanding singularities and bifurcations of constant curvature surfaces.
method Loop group methods to construct bifurcations and analyze harmonic maps.
result Determine which map germs can be represented by harmonic maps.
Gradient of harmonic functions tied to level hypersurface geometry.
problem Understanding how the gradient of harmonic functions changes.
method Analyzes how the gradient of harmonic functions changes along gradient flows.
result The gradient's change is determined by the mean curvature of level hypersurfaces.
Study on determining metrics via Dirichlet-to-Neumann map for harmonic maps.
problem Determining Riemannian metrics from boundary measurements.
method Higher linearization method, integral identities, energy rigidity.
result Metrics on the target manifold are equal if the target is analytic.
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.
Study finds all 4D Lie groups with harmonic curvature.
problem Finding Lie groups with harmonic curvature in 4D.
method Analyzing pp-wave Lie groups and determining harmonic curvature conditions.
result Description of 4D pp-wave Lie groups obtained.
Study Dirac-harmonic maps on Riemann surfaces and their relation to J-holomorphic curves.
problem Understanding critical points of fermionic action functionals on Riemann surfaces.
method Analyzing Dirac-harmonic maps and their relation to J-holomorphic curves on Kaehler manifolds.
result The tangent bundle to the moduli space of J-holomorphic curves consists of Dirac-harmonic maps.
Study shows volume density in central harmonic spaces can vary arbitrarily.
problem Volume density in central harmonic spaces can vary arbitrarily.
method Analyzes asymptotics of volume density function in central harmonic manifolds.
result Volume density in central harmonic spaces can be specified arbitrarily and does not determine geometry.
We determine the harmonic volumes for all the hyperelliptic curves. This gives a geometric interpretation of a theorem established by A. Tanaka.
In this article I show that every special Lagrangian cone in C^3 determines, and is determined by, a primitive harmonic surface in the 6-symmetric space SU_3/SO_2. For cones over tori, this allows us to use the classification theory of harmonic tori to describe the construction of all the corresponding special Lagrangi…
The study examines eigenfunctions on lens spaces using harmonic-counting measures.
problem Understanding the asymptotic behavior of eigenfunctions on lens spaces.
method Introduced harmonic-counting measures and applied them to lens spaces.
result Determined the asymptotic behavior of eigenfunctions associated with lattice points.
We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm ∣∇R∣ of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic spheres in a harmonic space determi…
Harmonic maps from the plane to SO(3) are studied via integral iterations.
problem Minimal harmonic maps from the plane to SO(3) with polynomial cubic differentials.
method Fixed-point problem for integral operator, spectral networks, BPS state counts.
result Determining the asymptotic structure of g by a convex polygon Y(P) in RP2. Study geometrical properties of oscillator group with a Lorentzian metric.
problem Geometrical analysis of oscillator group.
method Bi-invariant Lorentzian metric, homogeneous Ricci solitons, harmonicity properties, energy functional.
result Determination of critical points for energy functional and explicit calculation of their energy.
The paper identifies manifolds with harmonic complex structures.
problem Understanding harmonic almost complex structures on Riemannian manifolds.
method Analyzing the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures on twistor spaces.
result Characterizes manifolds with harmonic complex structures.
New findings on isospectral tori and harmonic maps between flat tori.
problem Determining if isospectral tori are isometric using harmonic maps.
method Examined harmonic maps between flat tori, focusing on Milnor's isospectral tori.
result Milnor's isospectral tori cannot be distinguished by harmonic maps from lower-dimensional tori, but can be distinguished by higher-dimensional ones.
The paper studies how hyperbolic surfaces degenerate along harmonic map rays.
problem The degeneration of hyperbolic surfaces along harmonic map rays.
method Using Teichmüller space and holomorphic quadratic differentials, the authors show convergence of rescaled distance functions to the intersection number with a vertical measured foliation.
result Hyperbolic surfaces along the ray converge to the dual R-tree of the vertical measured foliation in the sense of Gromov-Hausdorff.
Characterizes level-set families of harmonic functions without critical points.
problem Understanding level-set families of harmonic functions without critical points.
method Characterization via local differential-geometric condition and construction from geometric data.
result Evolution of gradient of harmonic functions determined by mean curvature of level sets.
Statistical hyperbolicity proven for Teichmüller space.
problem Harmonic measures from random walks on mapping class groups.
method Proving statistical hyperbolicity using Teichmüller metric.
result Teichmüller space is statistically hyperbolic for certain harmonic measures.
Method constructs harmonic immersions in R^3 using Enneper-type representation.
problem Constructing harmonic immersions in R^3.
method Enneper-type representation approach.
result Any harmonic immersion in R^3 can be obtained using this approach.
The absence of interesting harmonic sections for the Sasaki and Cheeger-Gromoll metrics has led to the consideration of alternatives, for example in the form of a two-parameter family of natural metrics shown to relax existence conditions for harmonicity. This article investigates harmonic Killing vector fields, proves…
Minimal surfaces in harmonic conformally flat space are studied.
problem Minimal surfaces in harmonic conformally flat space.
method Variational geometry approach focusing on mean curvature and Willmore functionals.
result Critical points of mean curvature functional are homeomorphic to the sphere.
We give a classification of quadratic harmonic morphisms between Euclidean spaces (Theorem 2.4) after proving a Rank Lemma. We also find a correspondence between umbilical (Definition 2.7) quadratic harmonic morphisms and Clifford systems. In the case R4⟶R3, we determine all quadr…
The study finds nondegenerate harmonic 1-forms using symmetry conditions.
problem Existence of nondegenerate harmonic 1-forms over Riemannian manifolds.
method Utilizing Z3 symmetry to establish topological conditions. result Found nondegenerate Z2 harmonic 1-forms over branched coverings of links. Study connects landslide flow to integrable systems for harmonic maps.
problem Understanding the holonomy of complex landslide flow.
method Integrable systems approach to harmonic maps into symmetric spaces.
result Holonomy of complex landslide flow derived from harmonic map holonomy.
Researchers compute determinants and torsions of Rumin complex in specific Lie group representations.
problem Computing determinants and torsions of Rumin complex in specific Lie group representations.
method Analyzing Schrodinger and generic representations of the (2,3,5) nilpotent Lie group.
result Computed the spectrum and zeta regularized determinant of Rumin differentials in Schrodinger representations and evaluated their alternating product in generic representations.
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…
We study asymptotically harmonic manifolds of negative curvature, without any cocompactness or homogeneity assumption. We show that asymptotic harmonicity provides a lot of information on the asymptotic geometry of these spaces: in particular, we determine the volume entropy, the spectrum and the relative densities of …
We show that under certain symmetry, the images of complete harmonic embeddings from the complex plane into the hyperbolic plane is completely determined by the geometric information of the vertical measured foliation and is independent of the horizontal measured foliation of the corresponding Hopf differentials.
The paper explores α−harmonic maps and their stability, proving key properties and conditions.
problem Existence and stability of α−harmonic maps between Riemannian manifolds. method Analysis of α−energy functional, construction of α−harmonic maps, and stability conditions. result Conditions for the stability of α−harmonic maps and their instability from compact manifolds. The study finds infinitely many p-harmonic maps between spheres for specific p and m.
problem Investigating p-harmonic maps between spheres for different dimensions and p-values.
method Analyzing rotationally symmetric p-harmonic maps and their stability.
result Existence of infinitely many p-harmonic self-maps of spheres for given p and m.
We consider four dimensional lie groups equipped with left invariant Lorentzian Einstein metrics, and determine the harmonicity properties of vector fields on these spaces. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. We also classify vector fields de…
Study on conformal harmonic coordinates on manifolds, proving existence and properties.
problem Existence and properties of conformal harmonic coordinates on Riemannian manifolds.
method Solutions to the conformal Laplace equation, proving up to boundary regularity results, elliptic regularity, and unique continuation results.
result Proves conformal harmonic coordinates are a close conformal analogue of harmonic coordinates.
The study classifies harmonic cubic polynomials in up to 4 dimensions.
problem Describing harmonic cubic polynomials with specific Hessian properties.
method Construction and classification in all dimensions; techniques for inequivalence determination.
result Classification of solutions in dimensions up to 4.
Study characterizes martingales on fiber bundles for harmonic map analysis.
problem Characterizing martingales on fiber bundles for harmonic maps.
method Investigated ablaP-martingales on principal fiber bundles P(M,G) with a projectable connection. result New characterizations of harmonic maps established.
The paper extends energy identities and neck existence for ε-harmonic maps.
problem Understanding the energy identity and neck formation for ε-harmonic maps.
method Finding analogues of energy identities and neck existence results for ε-harmonic maps.
result Specific quantities determine energy identity and neck formation for ε-harmonic maps.
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.
In this article, we use the harmonic sequence associated to a weakly conformal harmonic map f:S→S6 in order to determine explicit examples of linearly full almost complex 2-spheres of S6 with at most two singularities. We prove that the singularity type of these almost complex 2-spheres has an extra symmetry a…
Develops relative harmonic metrics and deformation theory for Higgs and flat bundles on compact Kähler manifolds.
problem Analyzing harmonic metrics and deformations on Higgs and flat bundles on compact Kähler manifolds.
method Relative analytic theory, Sobolev completions, elliptic regularity, normalized gluing, plotwise smoothness, heat flow, harmonic filtrations, obstruction theory.
result Global smooth harmonic metrics exist for smooth stable Higgs families under certain conditions, and this theory extends to reduced singular parameter spaces.
Study analyzes convergence of harmonic maps into compact locally CAT(1) spaces.
problem Analyzing convergence of harmonic maps with energy bounds.
method Bubble tree convergence, exploiting local convexity properties of CAT(1) spaces.
result Energy quantization and no-neck property demonstrated for harmonic maps.
A metric is formal if all products of harmonic forms are again harmonic. The existence of a formal metric implies Sullivan formality of the manifold, and hence formal metrics can exist only in presence of a very restricted topology. We show that a warped product metric is formal if and only if the warping function is c…
Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalise harmonic maps. We consider the Hopf map $ψ:\s^3\to \s^2$ and modify it into a nonharmonic biharmonic map $φ:\s^3\to \s^3$. We show φ to be unstable and estimate its biharmonic index and nullity. Resolving the s…
New proof of harmonic map existence from punctured surfaces to crowned hyperbolic targets.
problem Existence of harmonic maps from punctured surfaces to specific hyperbolic targets.
method Using Teichmüller theory and Minsky's work on limiting harmonic maps, constructing the conformal limit and proving existence.
result Existence of harmonic maps from any punctured Riemann surface to a given crowned hyperbolic target.
Study on determinant properties of elliptic operators with counterexamples and positive results.
problem Does the determinant of a matrix solution to a second order elliptic equation satisfy the unique continuation property?
method Analyzes counterexamples and positive results for various operators, including reductions to special cases.
result Partial answers and counterexamples provided, with positive results for specific cases.
This note is concerned in so called harmonic complex structures introduced by the author previously. I will recall some previous results and emphasize the motivation: Provide an attempt to a fundamental problem in geometry--determining the complex structures on an almost complex manifold. I also discuss the almost-Herm…
We consider harmonic immersions in RN 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…
Our aim in this paper is to investigate some geometrical properties of Berger Spheres i.e. homogeneous Ricci solitons and harmonicity properties of invariant vector fields. We determine all vector fields which are critical points for the energy functional restricted to vector fields. We also see that do not exist any v…
For a particular class of pseudo manifolds, we show that the intersection cohomology groups for any perversity may be naturally represented by extended weighted L2 harmonic forms for a complete metric on the regular stratum with respect to some weight determined by the perversity. Extended weighted L2 harmonic fo…
We consider the energy functional on the space of sections of a sphere bundle over a Riemannian manifold (M, <,>) equipped with the Sasaki metric and we discuss the characterising condition for critical points. Likewise, we provide a useful method for computing the tension field in some particular situations. Such a me…