Study Hamiltonian stationary Lagrangian surfaces in complex space forms.
problem Characterize Lagrangian surfaces with harmonic mean curvature in complex space forms.
method Analyze surfaces with constant and harmonic mean curvature, using second fundamental form parallelism and Gaussian curvature constancy.
result Complete classification of Lagrangian surfaces with harmonic mean curvature and constant Gaussian curvature.
Study estimates self-shrinker index with conical ends, proving index bound.
problem Estimating the index of self-shrinkers with asymptotically conical ends.
method Constructing Gaussian Harmonic forms and extending index estimates.
result Proves Morse index of self-shrinkers is at least (2g+r-1)/3.
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.
We consider 2-dimensional orientable self-shrinkers Σ for the Mean Curvature Flow of polynomial volume growth immersed in Rn. We look at closed one forms minimizing the norm $\int_Σ\eterm |ω|^2$ in their cohomology class. Any closed form satisfying the Euler-Lagrange equation for this minimization will be …
It was proved that the fundamental group of the space of harmonic polynomials of degree n(n≥2), with the same Gaussian curvature is not trivial. Furthermore, we give an example of topologically nonequivalent conjugate harmonic functions having the same Gaussian curvature.
Combines Gaussian process and Geometric Harmonics for better uncertainty estimation.
problem Uncertainty estimation in kernel-based methods.
method Combines Gaussian process and Geometric Harmonics.
result Alternative interpretations of uncertainty and accelerated Bayesian Optimization.
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.
Wave maps connect to constant curvature surfaces, studying singularities and bifurcations.
problem Understanding singularities and bifurcations in wave maps and pseudospherical surfaces.
method Constructing germs of wave maps from their jets and using loop groups to construct pseudospherical surfaces.
result Obtained bifurcations in generic 1-parameter families of pseudospherical surfaces.
Revisits Gaussian process model with spherical harmonics for scalable deep learning.
problem Scaling Gaussian process models to large input dimensions with high frequency learning.
method Introduces new kernels related to deep models, variational learning of spherical harmonic phases, and sparseness in eigenbasis.
result Enables scaling to larger input dimensions and learning of high frequency variations.
The paper proves cohomology vanishing for a specific type of minimal submanifolds in a weighted Euclidean ball.
problem Proving cohomology vanishing for free boundary f-minimal submanifolds in Gaussian-weighted Euclidean balls. method The proof uses a weighted Hardy inequality, cancellation in the weighted Weitzenböck curvature operator, and a boundary reduction.
result The space of tangential f-harmonic p-forms vanishes, leading to Hp(M;R)=0. A new method using spherical harmonics approximates the Sliced-Wasserstein distance.
problem Approximating the Sliced-Wasserstein distance between probability measures.
method Spherical Harmonics Control Variates (SHCV) method for Monte Carlo approximation of the SW distance.
result SHCV method provides an improved rate of convergence compared to Monte Carlo for general measures.
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 p-harmonic function, p-harmonic 1 form, and harmonic q form (with q≥2). Harmonic spinors linked to twistor spinors via potential forms and conformal Killing-Yano forms.
problem Connecting harmonic spinors and twistor spinors using mathematical operators.
method Constructing symmetry operators from conformal Killing-Yano forms and finding transformation operators between twistor and harmonic spinors in terms of potential forms.
result Operators transforming gauged twistor spinors to gauged harmonic spinors and algebraic conditions for Seiberg-Witten equations solutions.
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 Z2-harmonic 1-forms on compact manifolds. method Develops a gluing theorem for non-degenerate Z2-harmonic 1-forms on compact manifolds. result Proves the existence of non-degenerate Z2-harmonic 1-forms on compact manifolds with positive first Betti number. Classifies surfaces in hyperbolic space with constant Gaussian curvature.
problem Classifying surfaces in hyperbolic space with specific curvature.
method Loop group method, spectral parameter deformation, holomorphic quadratic differentials.
result Weakly complete constant Gaussian curvature surfaces are in one-to-one correspondence with holomorphic quadratic differentials.
Study finds conditions for Legendre curves to be interpolating sesqui-harmonic in Sasakian space forms.
problem Characterizing Legendre curves in Sasakian space forms.
method Analyzes necessary and sufficient conditions for Legendre curves to be interpolating sesqui-harmonic.
result Obtains an example of an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.
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.
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.
This paper constrains Gaussian processes to arbitrary domains using harmonic features.
problem Constraining Gaussian processes to arbitrary domains with boundary conditions.
method Solves a Fourier-like generalised harmonic feature representation of the GP prior, scaling as O(nm^2) in prediction and O(m^3) in hyperparameter learning.
result The method allows for efficient inference and handling of non-Gaussian likelihoods.
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.
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 Z3 symmetry to establish topological conditions. result Found nondegenerate Z2 harmonic 1-forms over branched coverings of links. Paper estimates Gaussian curvature of minimal graphs in a specific manifold.
problem Estimating Gaussian curvature of minimal graphs in MimesR. method Using Weierstrass representation via ℘−harmonic mappings and Schwarz lemma type results. result Proves Schwarz lemma type and Heinz type results for harmonic mappings.
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.
Calabi surgery modifies Z/2 harmonic 1-forms using 2-valued 1-forms.
problem Modifying Z/2 harmonic 1-forms under weak regularity assumptions.
method Calabi surgery method, involving cutting and pasting closed 2-valued 1-forms.
result Flexible construction and modification of Z/2 harmonic 1-forms.
We characterize the harmonic forms on a flag manifold K/T 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 K/T. This enables us to give Poisson geometrical proofs of many of the special properties of these harm…
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.
The paper studies harmonic symmetric bilinear forms on Riemannian manifolds and proves properties of the Bourguignon Laplacian.
problem Analyzing harmonic symmetric bilinear forms on Riemannian manifolds.
method Developed the theory of harmonic symmetric bilinear forms and proved properties of the Bourguignon Laplacian.
result The kernel of the Bourguignon Laplacian is a finite-dimensional vector space of harmonic symmetric bilinear forms on a compact Riemannian manifold.
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.
Harmonic forms on complex manifolds satisfy special symmetries.
problem Understanding symmetries in harmonic forms on complex manifolds.
method Representation of sl(2,C) to generalize Kähler manifold properties. result Harmonic forms on Hermitian manifolds exhibit hard Lefschetz duality.
Study L2-harmonic forms on almost Kähler manifolds, extending vanishing theorems.
problem Analyzing L2-harmonic forms on complete almost Kähler manifolds. method Decomposing L2-harmonic forms into Lefschetz powers of primitive forms, extending vanishing theorems. result Spaces of harmonic (p,q)-forms on X vanish unless p+q=n. Analyzes L2-harmonic forms on curved manifolds, proving integrability conditions.
problem Analyzing integrability of L2-harmonic forms on curved manifolds. method Established L∞-estimate via Moser iteration, proved vanishing of integrable forms. result Proves that L2-harmonic forms on non-positively curved manifolds are integrable if and only if they vanish. Compactifies curves in symplectizations with harmonic forms.
problem Handle compactification of H−holomorphic curves. method Modification of pseudoholomorphic curves equation, harmonic 1-forms.
result A moduli space with bounds on harmonic 1-forms.
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.
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 Z2 harmonic 1-forms. result The branching set Σ of a Z2 harmonic 1-form can exhibit various features including non-trivial links, multiple covers, and immersed structures. 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…
The paper studies harmonic forms and spinors on Taub-bolt space.
problem Analyzing harmonic forms and spinors on Taub-bolt, a Ricci-flat ALF space.
method Proving dimensions of harmonic 2-forms, constructing zero modes of Dirac operator, comparing with known results.
result Explicitly found a 2-parameter family of L2 zero modes of the Dirac operator. 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.
A new GP model uses spherical harmonics for faster inference.
problem Efficiently fitting large datasets with Gaussian processes.
method Sparse Gaussian processes with spherical harmonic features.
result Significant speed-up in inference for large datasets.
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.