Unified positive mass theorem and Dirac operator study on weighted manifolds.
problem Establishing a unified positive mass theorem for weighted manifolds and smooth metric measure spaces.
method Analyzing Dirac operators on warped product manifolds and applying results to the positive mass theorem.
result Equivalence of weighted positive mass theorem to usual positive mass theorem.
The k-Dirac operator is a differential operator which is natural to geometric structure of a parabolic type. We will give a set of initial conditions for this operator. In the proof of the claim we will need to adapt some parts from the theory of exterior differential systems to the setting of weighted differential ope…
This paper develops a weighted L2-method for the (half) Dirac equation. For Dirac bundles over closed Riemann surfaces, we give a sufficient condition for the solvability of the (half) Dirac equation in terms of a curvature integral. Applying this to the Dolbeault-Dirac operator, we establish an automatic transversa…
Uniform elliptic theory for Dirac operators on orbifold resolutions.
problem Analyzing Dirac operators on orbifold resolutions.
method Viewing orbifolds as conically fibred singular spaces and resolving them by gluing asymptotically conical fibrations.
result Uniform index formula for Dirac operators on orbifold resolutions.
Paper introduces new fractional Dirac operator and Q-curvature.
problem Fractional Dirac operator and Q-curvature in spinors.
method Caffarelli-Silvestre extension, energy inequalities, weighted Sobolev inequality.
result Introduction of conformal fractional Dirac operator and Q-curvature.
Paper finds solutions to a complex equation on surfaces with boundary conditions.
problem Existence of solutions to a super-Liouville equation on compact Riemannian surfaces with boundary.
method Introduced a weighted Dirac operator and constructed a Nehari manifold to show existence of non-trivial solutions.
result Existence of non-trivial solutions to the super-Liouville equation.
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.
Characterizes low energy behavior of fibered Dirac operators.
problem Understanding the behavior of fibered Dirac operators near zero energy.
method Pseudodifferential characterization of the resolvent's low energy limit.
result Pseudodifferential characterization of the inverse of a suspended Dirac operator.
The paper extends spin geometry to weighted manifolds and defines a new mass for Ricci flow.
problem Generalizing spin geometry to weighted manifolds and defining a new mass.
method Investigates spectral properties of the weighted Dirac operator and defines a new mass.
result Defines a new mass for weighted asymptotically Euclidean manifolds and shows its monotonicity under Ricci flow.
This is the first paper in a series of two papers. In this paper we construct complexes of invariant differential operators which live on homogeneous spaces of ∣2∣-graded parabolic geometries of some particular type. We call them k-Dirac complexes. More explicitly, we will show that each k-Dirac complex arises as…
Lower bounds for Dirac eigenvalues on manifolds with boundary.
problem Finding lower bounds for eigenvalues of the Dirac operator on manifolds with boundary.
method Using the relative Yamabe constant to derive a conformal lower bound.
result Equality in the lower bound holds if and only if the manifold is a hemisphere and the eigenfunction is a Killing spinor.
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.
Study on symplectic Dirac operators on foliations, estimating eigenvalues.
problem Estimating eigenvalues of transversely symplectic Dirac operators.
method Analysis of transversely symplectic structures and use of Weitzenbock formula.
result Estimation of lower bounds for eigenvalues of transversely symplectic Dirac operators.
Study on magnetic Dirac operators and their spectrum.
problem Understanding the spectrum of magnetic Dirac operators.
method Analysis of magnetic Dirac operators over complete Riemannian manifolds.
result Find sufficient conditions for maximal or discrete spectrum.
In this paper we introduce the Dirac and spin-Dirac operators associated to a connection on Riemann-Cartan space(time) and standard Dirac and spin-Dirac operators associated with a Levi-Civita connection on a Riemannian (Lorentzian) space(time) and calculate the square of these operators, which play an important role i…
Mathematical proof of index equality for lattice Dirac operators and continuum operators.
problem Equality of indices for lattice and continuum Dirac operators.
method Using K-theory, we prove equivalence of one-parameter families of continuum and lattice Dirac operators.
result Indices of continuum and lattice Dirac operators are equal.
We derive a weighted L2-estimate of the Witten spinor in a complete Riemannian spin manifold (Mn,g) of non-negative scalar curvature which is asymptotically Schwarzschild. The interior geometry of M enters this estimate only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactifi…
Study perturbs APS boundary conditions for Lorentzian Dirac operators.
problem Maintaining Fredholmness of Dirac operators under perturbations of APS boundary conditions.
method Develop criteria for perturbing compact pairs of projections to remain Fredholm.
result Criteria for perturbing APS boundary conditions without losing Fredholmness.
New spectral functionals for Dirac operators with inner fluctuations computed.
problem Spectral functionals and Dirac operators with inner fluctuations.
method Extension of spectral functionals for Dirac operators with inner fluctuations.
result Computed spectral Einstein functional for Dirac operator with inner fluctuations on even-dimensional spin manifolds.
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 calculates a functional for a specific Dirac operator.
problem Computing a spectral Einstein functional for a Dirac operator with torsion.
method Computing the spectral Einstein functional for even-dimensional spin manifolds without boundary.
result The spectral Einstein functional for the Dirac operator with torsion is computed.
We determine the structure of conformal powers of the Dirac operator on Einstein {\it Spin}-manifolds in terms of the product formula for shifted Dirac operators. The result is based on the techniques of higher variations for the Dirac operator on Einstein manifolds and spectral analysis of the Dirac operator on the as…
Defines spectral Einstein functionals for sub-Dirac operators on manifolds with boundary.
problem Calculating Einstein-like functionals for sub-Dirac operators.
method Introduced spectral Einstein functional for sub-Dirac operators on manifolds with boundary.
result Proved a theorem for spectral Einstein functions on four-dimensional manifolds.
Study on Dirac operators on lightlike hypersurfaces in 4D Lorentzian manifolds.
problem Investigating Dirac operators on hypersurfaces with degenerate metrics.
method Spinorial Gauss formula, investigation of Dirac operator, relation with Riemannian curvatures.
result Established relation between Dirac operators and curvatures of the manifold and hypersurface.
The study examines conditions for scalar curvature and Dirac operators on singular spaces.
problem Existence of scalar curvature measures and Dirac operators on singular spaces.
method Investigation of smooth manifolds with singular Riemannian metrics.
result Sufficient conditions for the existence of scalar curvature measures and Dirac operators.
We use the spectra of Dirac type operators on the sphere Sn to produce sharp L2 inequalities on the sphere. These operators include the Dirac operator on Sn, the conformal Laplacian and Paenitz operator. We use the Cayley transform, or stereographic projection, to obtain similar inequalities for powers o…
The paper proves formulas and theorems for Dirac-Witten operators on manifolds with or without boundaries.
problem Analyzing Dirac-Witten operators on manifolds with boundaries.
method Obtained Lichnerowicz type formulas and proved Kastler-Kalau-Walze type theorems.
result Validated Kastler-Kalau-Walze type theorems for 4D and 6D compact manifolds with or without boundaries.
The paper introduces spectral Einstein functionals for Dirac operators on manifolds with boundary.
problem Analyzing perturbations of Dirac operators on manifolds with boundary.
method Introduction of spectral Einstein functionals and proof of Dabrowski-Sitarz-Zalecki type theorems.
result Proof of theorems associated with spectral Einstein functionals for perturbations of Dirac operators.
The paper calculates spectral torsion for rescaled Dirac operators on manifolds.
problem Computing spectral torsion for rescaled Dirac operators.
method Using trilinear Clifford multiplication and functional of differential one-forms.
result Computed spectral torsion for four types of rescaled Dirac operators.
Dirac operator invertibility proven for specific manifolds.
problem Invertibility of twisted Dirac operator on manifolds.
method Closed connected spin manifold with non-negative scalar curvature, flat Hilbert module bundle.
result Dirac operator is invertible under given conditions.
We derive an inequality that relates nodal set and eigenvalues of a class of twisted Dirac operators on closed surfaces and point out how this inequality naturally arises as an eigenvalue estimate for the Spinc Dirac operator. This allows us to obtain eigenvalue estimates for the twisted Dirac operator appearing…
Study perturbs Dirac operators in any dimension, focusing on Majorana fermions.
problem Understanding perturbations of Dirac operators in various dimensions.
method Analyzes canonical perturbations of Dirac operators on Hermitian Clifford modules.
result Characterizes the low-energy spectrum of these operators on complete surfaces.
Teaches Dirac operators for geometry and topology.
problem Interactions between geometry and topology.
method Families of Dirac operators and index theorems.
result Applications to metrics of positive scalar curvature and the three-dimensional Weinstein conjecture.
Geometric approach to Dirac operator evolution on spacetimes.
problem Constructing the Cauchy evolution operator for Lorentzian Dirac operators.
method Realizing the operator as a sum of oscillatory integrals, relating to Feynman propagator.
result Relating Cauchy evolution operators to Feynman propagators and constructing Hadamard states.
Optimal lower bounds for eigenvalues of Dirac-Witten operator on certain submanifolds.
problem Estimating eigenvalues of the Dirac-Witten operator on specific submanifolds.
method Optimal lower bounds derived using intrinsic and extrinsic expressions.
result Limiting-cases of eigenvalues studied and optimal bounds obtained.
The paper proves a theorem for a twisted Dirac operator on specific manifolds.
problem Analyzing the Dirac operator with torsion on spin manifolds.
method Develops a Lichnerowicz type formula and proves a Kastler-Kalau-Walze type theorem.
result Proves a Kastler-Kalau-Walze type theorem for the J-twist of the Dirac operator with torsion on 4D and 6D almost product Riemannian spin manifolds. Under various elliptic boundary conditions, we obtain lower eigenvalue estimates for Dirac operators by using Hormander's weighted L2-technique. Lower bounds in terms of the volume of the underlying manifolds are also deduced from the sharp Sobolev inequality due to Li and Zhu(\cite{LZ}).
The paper proves a theorem for a twisted Dirac operator on specific manifolds.
problem Analyzing the J-twist of the Dirac operator on spin manifolds.
method Lichnerowicz type formula and Kastler-Kalau-Walze type theorem for the J-twist of the Dirac operator.
result Proves a Kastler-Kalau-Walze type theorem for the J-twist of the Dirac operator on 3D and 4D almost product Riemannian spin manifolds with boundary.
This article is one of a series of papers. For this decade, the Dirac operator on a submanifold has been studied as a restriction of the Dirac operator in n-dimensional euclidean space $\EE^n$ to a surface or a space curve as physical models. These Dirac operators are identified with operators of the Frenet-Serret re…
It is postulated that quantum gravity is a sum over causal structures coupled to matter via scale evolution. Quantized causal structures can be described by studying simple matrix models where matrices are replaced by an algebra of quantum mechanical observables. In particular, previous studies constructed quantum grav…
Optimizes eigenvalue bounds for submanifold Dirac operators.
problem Estimating eigenvalues of submanifold Dirac operators.
method Optimal lower bounds derived using intrinsic and extrinsic expressions.
result Optimal eigenvalue bounds established for submanifold Dirac operators.
This is the second part in a series of two papers. The k-Dirac complex is a complex of differential operators which are natural to a particular ∣2∣-graded parabolic geometry. In this paper we will consider the k-Dirac complex over a homogeneous space of the parabolic geometry and as a first result, we will prove …
The paper calculates the noncommutative residue for a rescaled Dirac operator on 6D manifolds.
problem Computing the noncommutative residue for a specific Dirac operator on 6D manifolds.
method Calculations and proofs for the rescaled Dirac operator fDh on 6D compact manifolds.
result Proof of the Kastler-Kalau-Walze type theorem for the rescaled Dirac operator on 6D compact manifolds with boundary.
We consider an elliptic self-adjoint first order differential operator L acting on pairs (2-columns) of complex-valued half-densities over a connected compact 3-dimensional manifold without boundary. The principal symbol of the operator L is assumed to be trace-free and the subprincipal symbol is assumed to be zero. Gi…
The paper proves new theorems for Dirac operators on even-dimensional manifolds with boundary.
problem Proving theorems for Dirac operators on manifolds with boundary.
method Establishing general Kastler-Kalau-Walze type theorems for conformal perturbations of Dirac operators.
result Proof of new theorems for Dirac operators on even-dimensional manifolds with boundary.
We extend our earlier work in [TZ1], where an analytic approach to the Guillemin-Sternberg conjecture [GS] was developed, to cases where the Spinc-complex under consideration is allowed to be further twisted by certain natural exterior power bundles. The main result is a weighted quantization formula in the presence…
Study examines metrics and functionals for Hodge-Dirac operator on manifolds.
problem Examining metrics and functionals for Hodge-Dirac operator on manifolds.
method Analyzing the metric and Einstein bilinear functionals of differential forms for Hodge-Dirac operator d+δ. result The functionals reproduce those for the canonical Dirac operator on a spin manifold up to a numerical factor.
The article studies eigenvalues and spectrum of magnetic Dirac operators.
problem Fundamental mathematical properties of magnetic Dirac operators remain unexplored.
method Eigenvalue estimates and explicit spectrum computation for specific cases.
result Explicit computation of spectrum for magnetic fields on specific manifolds.