Study calculates spectra of Hodge-Laplace operators on flat tori and spheres.
problem Calculating the spectrum of Hodge-Laplace operators on specific manifolds.
method Explicit calculation of spectra for flat tori and round spheres.
result Operators on flat tori and spheres are isospectral under certain conditions.
Researchers describe Hodge-Laplace spectrum on lens spaces.
problem Characterizing lens spaces based on their form spectra.
method Explicit description and rational function encoding of spectra.
result Geometric characterization of lens spaces with identical spectra.
Computational study finds isospectral lens spaces and orbifolds for smooth p-forms.
problem Identifying isospectral lens spaces and orbifolds for smooth p-forms.
method Computational study of Hodge--Laplace operators on lens spaces and orbifolds.
result Evidence of isospectral lens spaces and orbifolds for smooth p-forms.
The study bounds Riesz transforms on manifolds with controlled curvature.
problem Bounding Riesz transforms on manifolds with controlled curvature.
method Established Lp-boundedness of local covariant Riesz transforms for differential forms. result Calderón-Zygmund estimates for manifolds with bounded Riemannian curvature.
This paper presents the construction of parametrices for the Gauss-Bonnet and Hodge Laplace operators on noncompact manifolds modelled on Q-rank 1 locally symmetric spaces. These operators are, up to a scalar factor, φ-differential operators, that is, they live in the generalised φ-calculus studied by the authors i…
The goal of the paper is to calculate the limit sectrum of the Hodge-Laplace operator under the perturbation of collapse of one part of a connected sum. This gives some new results concerning the 'conformal spectrum' on differential forms.
Standard Laplace operator extends Hodge and Casimir operators to broader geometric contexts.
problem Extending Laplace operator to vector bundles and Riemannian manifolds.
method Functorial approach, showing commutation with homomorphisms and differential operators.
result Standard Laplace operator commutes with a wide range of differential operators.
We discuss a peculiar interplay between the representation theory of the holonomy group of a Riemannian manifold, the Weitzenboeck formula for the Hodge-Laplace operator on forms and the Lichnerowicz formula for twisted Dirac operators. For quaternionic Kaehler manifolds this leads to simple proofs of eigenvalue estima…
In this paper, we develop a method of solving the Poincaré-Lelong equation, mainly via the study of the large time asymptotics of a global solution to the Hodge-Laplace heat equation on (1,1)-forms. The method is effective in proving an optimal result when M has nonnegative bisectional curvature. It also provides …
A manifold with fibered cusp metrics X can be considered as a geometrical generalization of locally symmetric spaces of Q-rank one at infinity. We prove a Hodge-type theorem for this class of Riemannian manifolds, i.e. we find harmonic representatives of the de Rham cohomology Hp(X). Similar to the situ…
We study the p-spectrum of a locally symmetric space of constant curvature Γ\X, in connection with the right regular representation of the full isometry group G of X on L2(Γ\G)τp, where τp is the complexified p-exterior representation of O(n) on $\bigwedge^p(\mathbb{R}…
We obtain a simple formula for the multiplicity of eigenvalues of the Hodge-Laplace operator, Δf, acting on sections of the full exterior bundle over an arbitrary compact flat Riemannian n-manifold M with holonomy group Z_2^k, with 0<k<n. This formula implies that any two compact flat manifolds with holonomy group Z…
Rust library solves complex equations on abstract simplicial complexes.
problem Solving partial differential equations on abstract simplicial complexes.
method Finite Element Exterior Calculus, intrinsic Riemannian metric, first-order Whitney basis functions.
result Verification through convergence studies on elliptic Hodge-Laplace eigenvalue and source problems.
Harmonic metallic structures on compact manifolds are equivalent to vanishing of their metallic structure.
problem Characterizing harmonic metallic structures on compact manifolds.
method Proving equivalence of harmonicity to dJ=0 and conditions for preservation by harmonic maps. result Conditions for harmonic metallic structures on compact manifolds.
The paper extends the Cheeger-Müller theorem to spaces with conical singularities.
problem Analyzing the L2-analytic torsion and intersection torsion on spaces with conical singularities. method Developed a combinatorial cellular theory and spectral theory for Hodge-Laplace operator on spaces with conical singularities.
result The L2-analytic torsion coincides with the Ray-Singer intersection torsion under certain conditions. By solving the Cauchy problem for the Hodge-Laplace heat equation for d-closed, positive (1,1)-forms, we prove an optimal gap theorem for Kähler manifolds with nonnegative bisectional curvature which asserts that the manifold is flat if the average of the scalar curvature over balls of radius r centered at any f…
Optimal gap theorem proven for CR manifolds with curvature conditions.
problem Proving optimal gap theorem for CR manifolds with specific curvature conditions.
method Applying Li-Yau-Hamilton inequality and heat equation deformation monotonicity.
result Proves flatness of CR manifold under certain curvature decay conditions.
Study Hodge theory on non-compact Riemannian manifolds with Lr estimates.
problem Solving Hodge Laplace equation on p forms in non-compact Riemannian manifolds. method Generalization of Raising Steps Method for non-compact manifolds, spectral gap hypothesis.
result Non-classical Lr Hodge decomposition theorems without bounded Riesz transforms. To every n-dimensional lens space L, we associate a congruence lattice L in Zm, with n=2m−1 and we prove a formula relating the multiplicities of Hodge-Laplace eigenvalues on L with the number of lattice elements of a given ∥⋅∥1-length in L. As a consequence, we show th…
The paper constructs arbitrary order conformally invariant operators in higher spin spaces.
problem Classifying and constructing conformally invariant differential operators in higher spin spaces.
method Explicit expressions and convolution type operators, intertwining operators, and representation theory.
result Explicit expressions and properties of conformally invariant differential operators in higher spin spaces.
New operators for Paneitz energy on manifolds with boundary.
problem Energy functional for Paneitz operator on compact manifolds with boundary.
method Conformally covariant boundary operators associated to Paneitz operator.
result Agreement with fractional GJMS operators in Poincaré-Einstein manifolds.
New boundary operators for sixth-order GJMS operator on manifolds.
problem Developing boundary operators for sixth-order GJMS operator.
method Conformally covariant boundary operators and fractional GJMS operators.
result New realization of fractional GJMS operators and Sobolev trace inequalities.
The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.
problem Eigenvalue problems and heat trace asymptotics for different operators.
method Establishes connections and inequalities for eigenvalues and heat traces.
result Eigenvalue inequalities and three-term asymptotic formulas for heat traces of various operators.
Introduces a new elliptic operator with positive eigenvalue.
problem None explicitly stated in the abstract.
method Introduces a new elliptic operator called the two-radical Laplace operator.
result The eigenvalue of the new operator is the positive square root of the Laplace operator's eigenvalue.
Proves Kato inequalities for various conformal operators.
problem Proving inequalities for differential operators.
method Analyzes a class of first order differential operators, including Dirac and Penrose twistor operators.
result Derives Kato inequalities that interpolate between classical and refined versions.
Study higher order fermionic and bosonic operators on cylinders and Hopf manifolds.
problem Understanding higher order higher spin operators on specific manifolds.
method Analysis of higher order operators on cylinders and Hopf manifolds.
result Construction of kernels for these operators on the studied manifolds.
Study on biharmonic hypersurfaces with specific recurrent operators in Euclidean space.
problem Characterizing biharmonic hypersurfaces with recurrent operators.
method Analysis of various recurrent operators and their impact on biharmonic hypersurfaces.
result Some well-known recurrent operators play a significant role in making biharmonic hypersurfaces minimal.
The k-Dirac operator is defined with initial conditions.
problem Defining the k-Dirac operator with initial conditions.
method Adapting Cartan-Kahler theorem to weighted differential operators.
result Initial conditions for the k-Dirac operator established.
Study the heat operator of a transversally elliptic operator on Lie groups.
problem Spectral properties and convergence of a heat operator on Lie groups.
method Review spectral properties, define character, estimate heat operator convergence.
result Estimate of fα(t) determines convergence of the character. Local index theorem for chiral geometric operators proved using heat kernel.
problem Proving a local index theorem for geometric first-order differential operators.
method Using Gilkey's invariance theory and heat kernel techniques.
result Supertrace of heat kernel converges to Chern-Weil form.
Study estimates eigenvalues for concave Hessian operators on convex domains.
problem Estimating eigenvalues for concave elliptic Hessian operators.
method Investigates Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators.
result Existence and properties of the first nonzero eigenvalue and eigenfunction.
The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.
problem Spaces of unbounded Fredholm operators and their properties.
method Analyzing the spaces and proving homotopy equivalences.
result Natural maps between four spaces of unbounded Fredholm operators are homotopy equivalences.
Characterizes operations on contact manifold differential forms.
problem Understanding natural operations on contact manifold differential forms.
method Introduces algebraic operators and the exterior derivative to characterize operations.
result All natural operations are built from introduced algebraic operators and the exterior derivative.
GJMS operators connect geometry, analysis, and physics.
problem None explicitly stated; focus on operators and their impact.
method Construction of conformally invariant differential operators.
result GJMS operators have significant impact in geometry, analysis, and physics.
Extends Calabi operator to Riemannian locally symmetric spaces.
problem Local integrability conditions on Riemannian locally symmetric spaces.
method Generalizes Calabi operator to Riemannian locally symmetric spaces.
result Generalised operator works in irreducible case and fails in products.
Method uses neural networks to fit nonlinear operators from data.
problem Finding nonlinear integro-differential operators from data.
method Parametrizes spatial operator with neural networks and Fourier transforms.
result Can recover spatial operators in fractional heat and Kuramoto-Sivashinsky equations.
Mathai, Melrose, and Singer compute the index of projective elliptic operators.
problem Computing the index of projective elliptic operators on manifolds with Azumaya bundles.
method Equivariant index of transversally elliptic operators as pullbacks of projective elliptic operators.
result Comprehensive fractional index formula for projective elliptic operators.
The paper classifies higher order fermionic and bosonic operators in spin spaces.
problem Understanding higher order conformally invariant differential operators.
method Using higher spin theory in Clifford analysis, constructing operators as generalizations of the Euclidean Dirac operator.
result These operators act on functions in homogeneous harmonic or monogenic polynomial spaces, and are classified as fermionic or bosonic based on spin.
New spectral torsion defined for rescaled Dirac operators.
problem Defining spectral torsion for rescaled Dirac operators.
method Using three vector fields and noncommutative residue.
result Computed spectral torsion for one form rescaled Dirac operators.
Researchers create new operators from Riemannian invariants.
problem Developing new mathematical tools for Riemannian geometry.
method Introducing formally self-adjoint conformally covariant polydifferential operators.
result Found a fourth-order, conformally covariant tridifferential operator.
Constructs conformal boundary operators and fractional Laplacians.
problem Developing conformally invariant boundary operators and fractional Laplacians.
method Constructs continuously parametrised families of conformally invariant boundary operators on densities.
result Constructs odd-order conformally invariant fractional Laplacian pseudo-differential operators.
Proves formal self-adjointness of certain differential operators.
problem Verifying conjectures about differential operators.
method Proving formal self-adjointness through mathematical proof.
result Proves two conjectures about differential operators.
Study describes how operator properties depend on smoothness on surfaces.
problem Understanding operator properties on surfaces with Morse-Smale diffeomorphisms.
method Analyzes pseudodifferential operators and shift operators on closed smooth surfaces.
result Fredholm property of operators depends on Sobolev smoothness exponent.
The paper proves new theorems about specific types of operator perturbations.
problem Analyzing conformal perturbations of Dirac and signature operators.
method Developed Kastler-Kalau-Walze type theorems for specific operator types.
result Established new theorems for six-dimensional manifolds with boundary.
Extends index theorem to uniformly elliptic operators on manifolds.
problem Generalizing index theorem to uniformly elliptic operators.
method Local index theorem on manifolds of bounded geometry.
result Validates multigraded elliptic uniform pseudodifferential operators.
Study of Dirac-like operators on spin manifolds with large mass parameters.
problem Understanding spectra of Dirac-like operators with piecewise constant mass terms.
method Analysis of asymptotic regimes to derive effective operators.
result Extension of MIT Bag operator concept to spin geometry.
The paper decomposes and analyzes the higher spin Laplace operator.
problem Understanding the properties and solutions of the higher spin Laplace operator.
method Decomposition into Rrita-Schwinger operators, proving conformal invariance, establishing integral formulas.
result Established a Borel-Pompeiu type formula and a Green type integral formula for the higher spin Laplace operator.
The paper extends higher-order operators in higher spin spaces.
problem Generalizing higher-order operators in higher spin spaces.
method Constructing 3rd order fermionic and 4th order bosonic operators in higher spin spaces.
result Fundamental solutions and intertwining operators of 3rd order fermionic and 4th order bosonic operators are presented.