Estimates for complex Hessian equations on Hermitian manifolds.
problem Establishing estimates for solutions to complex Hessian equations.
method Using concavity inequality for complex sum-of-Hessian operators.
result Second-order estimates for admissible solutions on Hermitian manifolds.
New operators generalize Michelsohn's on almost Hermitian manifolds.
problem Generalizing differential operators to almost Hermitian manifolds.
method Introducing two differential operators on sections of the complex Clifford bundle over compact almost Hermitian manifolds.
result Surprising Kähler-like symmetries in the kernel of the Laplacians of these operators.
Study differential operators on specific manifolds and their harmonic forms.
problem Understanding harmonic forms on almost-Hermitian manifolds.
method Analysis of differential operators, Hodge Theory, and cohomologies.
result Comparison of harmonic forms and cohomologies with classical ones.
The study identifies Hermitian metrics preserving the total Monge-Ampere volume.
problem Understanding Hermitian metrics preserving volume invariance.
method Characterizations and comparison principles for complex Monge-Ampere operator.
result Several characterizations of Hermitian metrics satisfying the comparison principle.
Researchers found a canonical form for pairs of Hermitian and antilinear operators.
problem Simultaneous normalization of pairs of Hermitian and antilinear operators in differential geometry.
method Finding a canonical form for pairs of Hermitian and antilinear operators.
result Generalized previous results on simultaneous normalization of such pairs.
We describe the shape of the symplectic Dirac operators on Hermitian symmetric spaces. For this, we consider these operators as families of operators that can be handled more easily than the original ones.
This work reconsiders the holomorphic and anti-holomorphic Dirac operators of Hermitian Clifford analysis to determine whether or not they are the natural generalization of the orthogonal Dirac operator to spaces with complex structure. We argue the generalized gradient construction of Stein and Weiss based on represen…
Researchers solve the Calderón problem for fractional Dirac operators.
problem Determining the metric and structure from boundary measurements.
method Analyzing the fractional Dirac operator on vector bundles.
result The Calderón problem is solved uniquely for the fractional Dirac operator.
Quillen proved that repeated multiplication of the standard sesquilinear form to a positive Hermitian bihomogeneous polynomial eventually results in a sum of Hermitian squares, which was the first Hermitian analogue of Hilbert's seventeenth problem in the nondegenerate case. Later Catlin-D'Angelo generalized this posit…
New operators help focus on specific areas in complex math problems.
problem Concentration in complex mathematical structures.
method Construct conjugate-linear perturbations of twisted spinc Dirac operators using the conjugate-linear Hodge star operator.
result These perturbations satisfy the concentration principle.
Quillen proved that, if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D'Angelo and Varolin deduced this positivstellensatz of Quillen from the eventu…
Let J be a unitary almost complex structure on a Riemannian manifold (M,g). If x is a unit tangent vector, let P be the associated complex line spanned by x and by Jx. We show that if (M,g) is Hermitian or if (M,g) is nearly Kaehler, then either the complex Jacobi operator (JC(P)y=R(y,x)x+R(y,Jx)Jx) or the complex curv…
Introduces Chern-Dirac bundles on non-Kähler Hermitian manifolds.
problem No specific problem stated; focuses on new mathematical structures.
method Introduces Chern-Dirac bundles and operators, showing isomorphisms with cohomology.
result Spaces of harmonic spinors on V-spinor bundle isomorphic to Dolbeault cohomology.
Estimates eigenvalue for Hermitian manifolds using curvature.
problem Estimating the first eigenvalue of Hermitian manifolds.
method Using holomorphic Ricci and sectional curvatures.
result Established estimates for the first eigenvalue.
Let (E,h) be a holomorphic Hermitian vector bundle over a polarized manifold. We provide a canonical quantization of the Laplacian operator acting on sections of the bundle of Hermitian endomorphisms of E. If E is simple we obtain an approximation of the eigenvalues and eigenspaces of the Laplacian.
Computed formulas for curvature operators and Poincaré polynomials of symmetric spaces.
problem Calculating curvature operators and Poincaré polynomials for symmetric spaces.
method Explicit formulas derived using quantum numbers and eigenvalue analysis.
result Maximum eigenvalue of curvature operators bounded by Einstein constant, with equality for Hermitian spaces.
We show that the non Hermitian Black-Scholes Hamiltonian and its various generalizations are eta-pseudo Hermitian. The metric operator eta is explicitly constructed for this class of Hamitonians. It is also shown that the effective Black-Scholes Hamiltonian and its partner form a pseudo supersymmetric system.
Fractional Laplacian inverse problem solved for connection Laplacians.
problem Determining structures from metric, bundle, and map knowledge.
method Local knowledge of metric, bundle, and map determines global structures.
result Global structures determined from local knowledge of metric, bundle, and map.
Study the Bochner-Schrödinger operator's trace in semiclassical limit.
problem Trace formula for Bochner-Schrödinger operator on tensor powers of line and vector bundles.
method Semiclassical analysis of the Bochner-Schrödinger operator Hp on tensor powers of a Hermitian line bundle and vector bundle. result Complete asymptotic expansion of the trace of φ(Hp) in the semiclassical limit po∞. Introduces new Hermitian metrics linking to Gauduchon and balanced metrics.
problem Finding conditions for compact complex manifolds to be Kähler.
method Introducing pluriclosed star split metrics and studying their properties.
result Affirmative answer to Fino-Vezzoni conjecture under extra assumptions.
Let X=G/P be a homogeneous space of a complex semisimple Lie group G equipped with a hermitian metric. We study the action of the Hodge star operator on the space of harmonic differential forms on X. We obtain explicit combinatorial formulas for this action when X is an irreducible hermitian symmetric space of compact …
We prove the rigidity and vanishing of several indices of "geometrically natural" twisted Dirac operators on almost even-Clifford Hermitian manifolds admitting circle actions by automorphisms.
Develops noncommutative Cowen-Douglas theory for noncommuting operators.
problem Exploring noncommutative analogues of classical Cowen-Douglas theory.
method Defining noncommutative Cowen-Douglas class using matricial joint eigenvalues and showing equivalence classes are determined by associated noncommutative vector bundles.
result Unitary equivalence class of a tuple in the noncommutative Cowen-Douglas class is determined by the equivalence class of its associated noncommutative vector bundle.
The paper studies spectral convergence of Hodge-Kodaira Laplacians on degenerating Hermitian metrics.
problem Spectral convergence of Hodge-Kodaira Laplacians on degenerating Hermitian metrics.
method Proves spectral convergence theorems for Hodge-Kodaira Laplacians Δ∂,m,0,s under general assumptions. result Eigenvalues, heat operators, and heat kernels converge to those of a self-adjoint operator Δ∂,m,0,abs. Characterizes integrability of generalized structures on Courant algebroids.
problem Integrability of generalized structures on Courant algebroids.
method Characterization via torsion-free generalized connections and Dirac generating operators.
result Criterion for integrability of generalized almost Hermitian structures and hyper-Hermitian structures.
Exponential localization of eigensections for Bochner-Schrödinger operator.
problem Understanding spectral properties of Bochner-Schrödinger operator on high tensor powers of Hermitian line bundles.
method Approximation of operator by model Schrödinger operator with constant magnetic field, analysis of spectrum.
result Spectrum of Bochner-Schrödinger operator in gaps is discrete and eigensections decay exponentially.
Study curvatures on compact pseudo-Hermitian manifolds using special methods.
problem Prescribing Webster scalar curvatures on compact pseudo-Hermitian manifolds.
method Upper and lower solutions, perturbation theory of self-adjoint operators, CR conformal deformations.
result Described sets of Webster scalar curvature functions that can be realized.
Study local commutation relation on almost complex manifolds.
problem Local commutation relation between Lefschetz operator and exterior differential.
method Analyzing almost complex manifolds with compatible metrics.
result Generalizes local Kähler identities to almost Hermitian manifolds.
The paper studies the spectrum of Laplace-Beltrami operators on complex spaces.
problem Analyzing the spectrum of Laplace-Beltrami operators on compact complex spaces.
method Examined the Friedrichs extension of Laplace-Beltrami and Hodge-Kodaira Laplacians, providing estimates for eigenvalues and trace-class properties.
result Discrete spectrum and trace-class properties of Laplace-Beltrami operators on compact complex spaces.
Proves stability of gravitational instantons, proving operator positivity.
problem Stability of gravitational instantons.
method Riemannian analog of black hole mode stability for Hermitian, non-self-dual gravitational instantons.
result Teukolsky equation is a positive definite operator on Hermitian, non-self-dual gravitational instantons.
Paper proves inequalities on Hermitian manifolds with applications to bounded solutions.
problem Establishing mixed Hessian inequalities on Hermitian manifolds.
method Weak convergence theorem of complex Hessian operators and general mixed Hessian inequality.
result Existence of bounded solutions of complex Hessian equations.
Let X=G/H be a symmetric space for a real simple Lie group G, equipped with a G-invariant complex structure. Then, X is a pseudo-Hermitian manifold, and in this geometric setting, higher Laplacians Lm are defined for each positive integer m, which generalize the ordinary Laplace-Beltrami operator. We show …
We study the compact Hermitian spin surfaces with positive conformal scalar curvature on which the first eigenvalue of the Dolbeault operator of the spin structure is the smallest possible. We prove that such a surface is either a ruled surface or a Hopf surface. We give a complete classification of the ruled surfaces …
Essential self-adjointness proved for perturbed quadharmonic operators on Riemannian manifolds.
problem Proving essential self-adjointness for perturbed quadharmonic operators.
method Using bounded geometry assumptions and a non-positive potential function.
result Essential self-adjointness condition established for perturbed quadharmonic operators.
This paper demonstrates the power of the calculus developed in the two previous parts of the series for all real forms of the almost Hermitian symmetric structures on smooth manifolds, including e.g. conformal Riemannian and almost quaternionic geometries. Exploiting some finite dimensional representation theory of sim…
Study weak geodesics in deformed Hermitian-Yang-Mills equation space.
problem Geodesics in the space of potentials for deformed Hermitian-Yang-Mills equation.
method Formulated as degenerate elliptic equation, used nonlinear Dirichlet duality theory, constructed continuous solutions.
result Continuous solutions constructed for Dirichlet problem.
The moduli space NK of infinitesimal deformations of a nearly Kähler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6…
We consider the Dolbeault operator of K1/2 -- the square root of the canonical line bundle which determines the spin structure of a compact Hermitian spin surface (M,g,J). We prove that the Dolbeault cohomology groups of K1/2 vanish if the scalar curvature of g is non-negative and non-identically zero. Moreov…
Study perturbations of submodules in Drury-Arveson space, finding smooth vector bundles with Hermitian connections.
problem Geometry of perturbations in Drury-Arveson space.
method Analysis of smooth vector bundles with Hermitian connections and computation of parallel transport operators.
result Found natural Hermitian connections on perturbed submodules.
This is the first part of a series of papers. The whole series aims to develop the tools for the study of all almost Hermitian symmetric structures in a unified way. In particular, methods for the construction of invariant operators, their classification and the study of their properties will be worked out. In this pap…
In this paper we classify maps from a torus phase space X to Hn∗, the space of n×n, non-singular hermitian operators up to equivariant homotopy. The equivariance is with respect to a time-reversal involution on X and an involution on Hn∗ defining a certain symmetry class. Furthe…
Theory developed for complex Hessian measures on Hermitian manifolds.
problem Defining and analyzing complex Hessian measures on Hermitian manifolds.
method Potential theory for m-subharmonic functions with respect to a Hermitian metric.
result Equivalence between polar sets and negligible sets for m-subharmonic functions.
The paper studies asymptotic expansions of operators related to Bochner-Schrödinger on Riemannian manifolds.
problem Asymptotic expansions of operators related to Bochner-Schrödinger on Riemannian manifolds.
method Analyzes the Bochner-Schrödinger operator Hp and its function φ(Hp) in L2(X,Lp⊗E), providing an asymptotic expansion of its smooth Schwartz kernel. result The trace of the operator φ(Hp) admits a complete asymptotic expansion in powers of p−1/2 as po∞. Study of zonal spherical functions on partial flag manifolds using Jacobi polynomials.
problem Understanding zonal spherical functions on partial flag manifolds.
method Matrix variate version of Koornwinder's method for constructing orthogonal polynomials, using Hermitian Jacobi polynomials and multivariate Schur polynomials.
result Conjecture that Hermitian Jacobi polynomials are elementary zonal spherical functions.
Study on geometrically formal metrics on complex manifolds.
problem Existence and properties of geometrically formal metrics on complex manifolds.
method Topological and cohomological obstructions, detailed analysis for specific manifolds, and metric constructions.
result Existence and non-existence conditions for geometrically formal metrics on various complex manifolds.
Paper localizes ADHM construction for ASD instantons over smooth domains.
problem Characterizing ASD instantons over smooth bounded domains.
method Develops a local analogue of ADHM construction using Hilbert spaces and bounded Hermitian operators.
result Describes degeneration of construction when instantons develop curvature singularity.
We construct almost complex algebraic curvature tensors for pseudo Hermitian inner products whose skew-symmetric curvature operator has constant Jordan normal form on the set of non-degenerate complex lines.
We show that the Weyl structure of an almost-Hermitian Weyl manifold of dimension at least 6 is trivial if the associated curvature operator satisfies the Kaehler identity. Similarly if the curvature of an almost para-Hermitian Weyl manifold of dimension at least 6 satisfies the para-Kaehler identity, then the Weyl str…