Scattering theory developed for linearised gravity near Schwarzschild black hole.
problem Linear stability of Schwarzschild spacetime and scattering of gravitational waves.
method Physical-space Chandrasekhar transformation and Teukolsky-Starobinsky correspondence.
result Construction of scattering theory for spin 2 Teukolsky equations.
The Teukolsky connection is linked to a complex structure on Einstein spacetimes.
problem Understanding the symmetries and structures in Einstein spacetimes.
method Analyzing the Teukolsky connection and its relation to conformal and GHP covariant connections.
result The Teukolsky connection is a manifestation of a complex structure on the conformal class of the spacetime.
2D twistor manifolds explain Teukolsky equations for rotating black holes.
problem Understanding the geometry of Teukolsky equations for rotating black holes.
method Using 2D twistor manifolds to explain the Teukolsky equations.
result All geometric structures can be understood by considering a 2D twistor manifold.
Based on operator identities and their formal adjoints, we derive two symmetry operators for the linearized Einstein operator on vacuum backgrounds of Petrov type D and in particular the Kerr spacetime. One of them is of differential order four and coincides with a result of Cohen and Kegeles. The other one is a new op…
The paper proves boundedness and decay of Teukolsky equations on Kerr backgrounds.
problem Analyzing boundedness and decay of Teukolsky equations on Kerr backgrounds.
method Adapting techniques from scalar waves, uniform-in-frequency estimates for Teukolsky PDEs were obtained.
result Solutions of Teukolsky equation on subextremal Kerr backgrounds remain bounded and decay in time.
The paper analyzes Teukolsky equations on Kerr backgrounds, proving boundedness and decay of solutions.
problem Analyzing boundedness and decay of solutions to Teukolsky equations on Kerr backgrounds.
method Frequency space analysis of transformed Teukolsky equations on Kerr backgrounds.
result Fixed frequency solutions remain bounded and decay in time for subextremal Kerr backgrounds.
Proves boundedness and decay for spin 1 Teukolsky equation on Reissner-Nordström spacetime.
problem Analyzing stability of charged black holes in perturbations.
method Gauge-invariant quantities and Fackerell-Ipser-type equations.
result Proves boundedness and polynomial decay for ℓ=1 spherical mode. Proves boundedness and decay for spin 2 Teukolsky system on Reissner-Nordström spacetime.
problem Analyzing stability of Reissner-Nordström spacetime for small charge.
method Derived quantities and generalized coupled Regge-Wheeler system.
result Proves boundedness and polynomial decay for spin 2 solutions.
Study on mode stability of gravitational instantons of type D.
problem Proving mode stability of gravitational instantons of type D.
method Analogous to Lorentzian case, analyze Weyl curvature scalars satisfying a separable Teukolsky equation.
result Prove mode stability, showing no solutions compatible with regularity and asymptotic flatness.
Kahler geometry explains decoupling of Kerr perturbations.
problem Decoupling of curvature scalars in Kerr spacetime.
method Hidden Kahler structure in Kerr spacetime, showing decoupling as a consequence of Kahler geometry.
result Decoupling of Teukolsky equations on Kahler background.
Paper develops a new geometric framework for Kerr stability.
problem Uniform decay properties of Kerr solutions.
method Geometric framework for Teukolsky equation in nonlinear Kerr spacetime.
result First nonlinear version of Chandrasekhar transformation.
In this paper we derive a differential identity for linearized gravity on the Kerr spacetime and more generally on vacuum spacetimes of Petrov type D. We show that a linear combination of second derivatives of the linearized Weyl tensor can be formed into a complex symmetric 2-tensor Mab which solves the…
Scattering theory for linearised gravity on Schwarzschild black hole exterior.
problem Constructing a scattering theory for linearised gravity equations on Schwarzschild background.
method Building on previous work, constructing Hilbert space-isomorphisms for finite energy initial data and scattering states.
result Past and future linear memories are related by an antipodal map for Bondi-normalised solutions.
Proves stability of charged black holes with small charge.
problem Linear stability of Reissner-Nordström spacetime for small charge.
method Proves stability of Reissner-Nordström family of charged black holes using linearized Einstein-Maxwell equations and geodesic null foliations.
result Solutions decay to a linearized Kerr-Newman metric, proving stability.
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.
Study peels tensor equations on Schwarzschild spacetime.
problem Analyzing the asymptotic behavior of tensorial wave equations on Schwarzschild spacetime.
method Combining conformal compactification and vector field techniques to estimate tensorial field energies.
result Obtains optimal initial data for peeling at all orders.
Study on instability of extreme Reissner-Nordström spacetime perturbations.
problem Linear stability of gravitational and electromagnetic perturbations in extreme Reissner-Nordström spacetime.
method Extends Giorgi's framework to prove instability results for a set of gauge invariant quantities along the event horizon.
result Proves decay, non-decay, and polynomial blow-up estimates for certain quantities along the event horizon, depending on the number of derivatives.
New proof of Schwarzschild stability using geometric gauge.
problem Linear stability of Schwarzschild spacetime under gravitational perturbations.
method Employing a new geometric gauge and exploiting the structure of transport equations.
result Established both orbital and asymptotic stability for linearised quantities.
We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to…
Scalar-tensor gravitation theories, such as the Brans-Dicke family of theories, are commonly partly described by a modified Einstein equation in which the Ricci tensor is replaced by the Bakry-Émery-Ricci tensor of a Lorentzian metric and scalar field. In physics this formulation is sometimes referred to as the "Jordan…
A new connection in Finsler geometry unifies various types of connections.
problem Introducing a unified connection in Finsler geometry.
method Using the pullback formalism, a new linear connection is introduced and investigated.
result The existence and uniqueness of the new connection are proved intrinsically.
Study finds six homogeneous surfaces with multiple invariant connections.
problem Characterizing homogeneous surfaces with invariant connections.
method Computed all simply connected homogeneous and infinitesimally homogeneous surfaces.
result Found six non-equivalent surfaces with multiple invariant connections.
A manifold's canonical involution defines a projection of connections.
problem Defining a projection of connections on (J2=±1)-metric manifolds. method Introducing a canonical involution to project connections.
result The projection sends Levi Civita to first canonical connection.
Study non-integrable distributions with various affine connections.
problem Characterize non-integrable distributions in Riemannian manifolds with different connections.
method Obtain Gauss, Codazzi, and Ricci equations for non-integrable distributions with semi-symmetric metric, non-metric, and statistical connections.
result Find new examples of Einstein and distributions with constant scalar curvature.
Paper explores connection cochain in abelian extensions and its relation to connection forms.
problem Understanding the connection cochain in abelian extensions.
method Apply Moriyoshi's connection cochain concept to abelian extensions and relate it to connection 1-forms.
result Established the relationship between connection cochain and connection 1-forms in abelian extensions.
The paper classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying algebraic Ricci solitons on three-dimensional Lorentzian Lie groups.
method Computed canonical and Kobayashi-Nomizu connections and their curvatures; defined algebraic Ricci solitons.
result Classified algebraic Ricci solitons on specific Lorentzian Lie groups.
New normalization condition for sub-Riemannian connections.
problem Normalizing connections on sub-Riemannian manifolds.
method Formulated in terms of Cartan connections, depends on curvature's first degree of homogeneity.
result A compatible partial affine connection can be uniquely extended to a full affine connection and a grading of the tangent bundle.
Paper proves unique Finsler connections for scalar forms.
problem Existence and uniqueness of Finsler connections.
method Pullback approach to global Finsler geometry, study of horizontally recurrent connections.
result Existence and uniqueness of horizontally recurrent Finsler connections for scalar forms.
Odd connections on supermanifolds are defined and their properties studied.
problem Defining and understanding odd quasi-connections on supermanifolds.
method Examined odd quasi-connections, defined torsion and curvature, and identified special classes.
result Odd connections on supermanifolds are shown to have torsion and curvature tensors.
The paper classifies Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures.
problem Classifying Lorentzian Lie groups based on specific tensor properties.
method Classification of three-dimensional Lorentzian Lie groups based on Ricci tensors and quasi-statistical structures associated with different affine connections.
result The paper classifies three-dimensional Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures associated with Bott, canonical, and Kobayashi-Nomizu connections.
The paper proves monotonicity formulas for minimal connections and their applications.
problem Understanding critical points of volume functionals in Riemannian geometry.
method Developed monotonicity formulas for minimal connections under specific conditions.
result Established vanishing theorems for minimal connections on Euclidean spaces and dDT connections on G2-manifolds.
Knots connected via a trivial band sum to connected sum.
problem Conditions for band-connected sum to equal connected sum.
method Analyzing knots and bands to determine conditions for equality.
result A band is trivial if and only if a band-connected sum equals a connected sum.
The paper solves the Integration Problem for principal connections.
problem Describing discrete connections associated with a principal connection.
method Using the Lie or derivative functor to induce connections on the principal bundle.
result For flat principal connections, the Integration Problem has a unique solution among flat discrete connections.
This article is a continuation of my former article "On Connectivity Spaces". After some brief historical references relating to the subject, separation spaces and then adjoint notions of connective representation and connective foliation are developed. The connectivity order previously defined only in the finite case …
Study of multiplicative connections in Lie groupoids.
problem Defining and understanding multiplicative connections in Lie groupoids.
method Definition and study of multiplicative connections satisfying compatibility with the groupoid structure.
result Identification of the obstruction to the existence of a multiplicative connection.
Study on submanifolds in generalized Sasakian-space-forms with various connections.
problem Analyzing submanifolds in generalized Sasakian-space-forms with different connections.
method Examines submanifolds in generalized Sasakian-space-forms with semisymmetric metric, non-metric, Schouten-van Kampen, and Tanaka-webster connections.
result Provides results on submanifolds in generalized Sasakian-space-forms with respect to various connections.
Develops torsion dual connections for statistical manifolds.
problem Defining statistical manifolds using dual connections.
method Introduces torsion dual connections and proves their properties.
result Curvature tensor of torsion dual connections has specific divergence.
Extends connections on Lie groupoids, proving completeness conditions.
problem Existence and completeness of multiplicative connections on Lie groupoid fibrations.
method Introduces and investigates multiplicative Ehresmann connections on Lie groupoid fibrations.
result Conditions for completeness of multiplicative connections on Lie groupoid fibrations.
The study connects conic connections and torsion-free principal connections on G-structures.
problem Relating torsion tensors of principal connections to characteristic conic connections.
method Formulating and verifying conditions for the existence of characteristic conic connections implying torsion-free principal connections.
result Conditions for the existence of characteristic conic connections imply the existence of torsion-free principal connections, verified for adjoint varieties of simple Lie algebras.
Study on solvable Lie groups with specific Weyl connections.
problem Characterizing solvable Lie groups with invariant stretched non-positive Weyl connections.
method Analyzing structure and classification of solvable Lie groups.
result Classification of solvable Lie groups and compact solvmanifolds with invariant SNP connections.
Defines semi-symmetric metric connections on differential forms.
problem Analyzing connections on differential forms.
method Defined and studied semi-symmetric metric connections, computed their curvature and Ricci tensors, and analyzed Lie derivatives.
result Derived Gauss-Codazzi-Ricci equations and properties of canonical, Schouten, and Vrancreanu connections.
Sprays on Frechet manifolds connect connections and tangent structures.
problem Characterizing linear symmetric connections on Frechet manifolds.
method Constructing connection maps and linear symmetric connections on tangent and second-order tangent bundles using sprays.
result A bijective correspondence exists between linear symmetric connections on tangent bundles and sprays.
Defines a new natural connection on Riemannian Π-manifolds.
problem Characterizing natural connections on Riemannian Π-manifolds.
method Introducing and analyzing the first natural connection with torsion.
result Relations between the first natural connection and Levi-Civita connection are established.
New connections found with specific torsion properties.
problem Understanding metric connections with specific torsion properties.
method Described Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
result Found new Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
This paper explores how local behavior of meromorphic connections on the projective line determines the global connection.
problem Determining the global meromorphic connection based on specified local behavior at singular points.
method Expository discussion of various problems related to meromorphic connections with specified local behavior, including Deligne-Simpson and rigidity problems.
result The existence and nonemptiness of moduli spaces of meromorphic connections with specified local behavior.
Paper extends Simons theorem to F-Yang-Mills connections for instability.
problem Tackles instability of F-Yang-Mills connections. method Extends Simons theorem to F-Yang-Mills connections using Kobayashi-Ohnita-Takeuchi's method. result Derives a sufficient condition for instability of non-flat F-Yang-Mills connections. In this paper, we prove a local index theorem for the DeRham Hodge-laplacian which is defined by the connection compatible with metric. This connection need not be the Levi-Civita connection. When the connection is Levi-Civita connection, this is the classical local Gauss-Bonnet-Chern theorem.
Recently the present authors introduced a general class of Finsler connections which leads to a smart representation of connection theory in Finsler geometry and yields to a classification of Finsler connections into the three classes. Here the properties of one of these classes namely the Berwald-type connections whic…