Study finds solutions for spacetimes with negative cosmological constant.
problem Existence of spacetimes with negative cosmological constant.
method Proved existence of solutions for Einstein-complex scalar field equations.
result Found large families of solutions with negative cosmological constant.
In this paper we study the supremum of Perelman's λ-functional {λ}_M(g) on Riemannian 4-manifold M by using the Seiberg-Witten equations. We prove among others that, for a compact Kähler-Einstein complex surface (M, J, g_{0}) with negative scalar curvature, (i) If g_{1} is a Riemannian metric on M with λ_{M}(g_{1})= λ_…
Research examines properties of special metrics on complex manifolds.
problem Properties of Hermitian metrics with proportional curvature traces.
method Study of basic properties and examples of Chern-Einstein metrics.
result Examples of Chern-Einstein metrics are provided.
Construct SU(3) structures on bundles over complex manifolds.
problem Creating globally-defined SU(3) structures on bundles. method Construction on smooth compact toric varieties and extensions to Kähler-Einstein manifolds.
result Extends SU(3) structures to non-toric Kähler-Einstein manifolds. The Einstein-scalar field theory can be used to model gravitational physics with scalar field matter sources. We discuss the initial value formulation of this field theory, and show that the ideas of Leray can be used to show that the Einstein-scalar field system of partial differential equations is well-posed as an ev…
We use the conformal method to obtain solutions of the Einstein-scalar field gravitational constraint equations. Handling scalar fields is a bit more challenging than handling matter fields such as fluids, Maxwell fields or Yang-Mills fields, because the scalar field introduces three extra terms into the Lichnerowicz e…
Proves global existence and uniqueness of solutions for Einstein-scalar-field equations.
problem Global existence and uniqueness of solutions for specific Einstein-scalar-field equations.
method Proves global existence and uniqueness of classical solutions with small initial data and wake-like decaying null infinity.
result Global existence and uniqueness of solutions for the equations with wake-like decaying null infinity.
Paper proves trapped surface formation for Einstein-Maxwell-charged scalar field system.
problem Formation of trapped surfaces in Einstein-Maxwell-charged scalar field system.
method Generalized Christodoulou's approach for spherical symmetry and improved for Minkowskian data.
result Improved bound on trapped surface formation for Minkowskian data.
Study of charged scalar fields on Reissner-Nordström spacetimes via energy estimates.
problem Understanding the behavior and stability of charged scalar fields on near-extremal Reissner-Nordström spacetimes.
method Global integrated energy decay and boundedness estimates for solutions to the charged scalar field equation.
result Established global, weighted integrated energy decay and boundedness estimates for solutions on (near-)extremal Reissner-Nordström(--de Sitter) spacetimes.
Study on scalar-flat Kahler 4-manifolds with a continuous symmetry.
problem Understanding scalar-flat Kahler 4-manifolds with a Killing field.
method Analysis of manifolds with a Killing field and asymptotic conditions.
result Rigidity results that restrict the behavior of scalar-flat Kahler manifolds at infinity.
Second part of series studying charged scalar fields on Reissner--Nordström spacetimes.
problem Analyzing late-time behavior and stability of charged scalar fields on black hole backgrounds.
method Purely physical-space based methods, energy estimates, inverse-power laws.
result First pointwise decay estimates for charged scalar fields on black hole backgrounds.
Unique global solutions found for specific initial data.
problem Einstein-scalar-field equations with specific initial conditions.
method Spherically symmetric analysis of small, slowly decaying data.
result Unique global solutions exist for the equations.
Researchers calculate the precise boundary operator for interacting bulk scalar fields in AdS/CFT.
problem Understanding the precise form of boundary operators dual to interacting bulk scalar fields.
method Holographic renormalization coupled with the Caffarelli/Silvestre extension theorem.
result Boundary operator dual to a bulk scalar field is an anti-local operator, the fractional Laplacian.
Study connects Riemann-Finsler geometry to Lorentz-violating scalar fields.
problem Exploring the connection between Riemann-Finsler geometries and Lorentz-violating scalar fields.
method Deriving quadratic actions and classical relativistic point-particle lagrangians in various spacetime dimensions.
result Support for open conjectures about Riemann-Finsler geometries in Lorentz-violating field theories.
Classifies patterns of symmetry breaking and vacuum degeneracy in scalar and gauge fields.
problem Understanding patterns of symmetry breaking and vacuum degeneracy in complex field systems.
method Uses mathematical classification of singular foliations to encode and classify patterns of spontaneous symmetry breaking and vacuum degeneracy.
result Mathematical classification provides a qualitative understanding of possible patterns of vacuum degeneracy.
Study of Bondi-Sachs formalism for massless scalar field with zero cosmological constant.
problem Analyzing the Bondi-Sachs formalism for Einstein's massless scalar field equations.
method Asymptotic expansions and peeling property for Bondi-Sachs metrics and scalar fields.
result Positivity of Bondi energy-momentum under specific conditions.
We study the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new conformal invariant, which is sensitive to the presence of the initial data for …
Study on future stability of FLRW spacetime solutions with decelerated expansion.
problem Stability of solutions to Einstein equations coupled with a nonlinear scalar field.
method Decomposition of metric and scalar field perturbations into spatial averages and oscillatory remainders.
result Future-stability of FLRW spacetime solutions for 1/3<p<1. Paper proves trapped surface formation for EMCSF system without symmetry assumptions.
problem Formation of trapped surfaces for the Einstein--Maxwell--charged scalar field system.
method Scale-critical trapped surface formation result established from past null infinity.
result Focusing of gravitational waves, concentration of electromagnetic fields, or condensation of scalar fields can lead to trapped surface formation.
Stability of extremal Reissner-Nordström black holes proven in spherical symmetry.
problem Stability of extremal Reissner-Nordström black holes in spherical symmetry.
method Proved nonlinear asymptotic stability through spherically symmetric characteristic data.
result Existence of a submanifold Mstab leading to stable solutions. Study proves stability of big bang singularity in complex system.
problem Stability of Kasner solutions in Einstein-Maxwell-scalar field-Vlasov system.
method Detailed mathematical structures and new delicate arguments.
result Nonlinear stability with Kasner exponents in full strong sub-critical regime.
The paper proves non-existence of torqued and anti-torqued vector fields on hyperbolic spaces.
problem Existence of torqued and anti-torqued vector fields on hyperbolic spaces.
method Analyzing the properties of conformal scalar functions and their impact on the existence of vector fields.
result Non-existence of proper torqued and anti-torqued vector fields on hyperbolic spaces.
The paper finds conditions for Yamabe solitons' metrics to be constant scalar curvature.
problem Finding conditions for Yamabe solitons' metrics to be constant scalar curvature.
method Using properties of conformal vector fields to find sufficient conditions.
result Sufficient conditions on soliton vector fields under which their metrics are of Yamabe metrics.
We establish new existence and non-existence results for positive solutions of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. Our analysis introduces variational techniques, i…
The presence of a conformal gradient vector field on a Riemannian manifold implies it is isometric to a hemisphere.
problem Characterizing Riemannian manifolds with boundary using conformal gradient vector fields.
method Analyzing the properties of conformal gradient vector fields on Riemannian manifolds with boundary and applying Ricci curvature and scalar curvature conditions.
result Riemannian manifolds with boundary are isometric to a hemisphere under certain conditions on conformal gradient vector fields and curvature.
Proves weak cosmic censorship for a specific Einstein-scalar field system in 2+1 dimensions.
problem Proves the absence of naked singularities in a specific Einstein-scalar field system.
method Establishes a mass gap and shows the presence of infinite blueshift to prove the absence of naked singularities.
result Proves the weak cosmic censorship conjecture for the circularly symmetric Einstein-scalar field system in 2+1 dimensions.
Extends K-energy to complexified Kähler classes for scalar curvature study.
problem Scalar curvature equation with B-field on complexified Kähler classes.
method Extended K-energy functional, convex along geodesics.
result Uniqueness of solutions in some cases.
Massless scalar and vector fields are coupled to Lyra geometry by means of Duffin-Kemmer-Petiau (DKP) theory. Using Schwinger Variational Principle, equations of motion, conservation laws and gauge symmetry are implemented. We find that the scalar field couples to the anholonomic part of the torsion tensor, and the gau…
Unique solutions found for wave-like decaying null infinity equations.
problem Wave-like decaying null infinity equations with spherically symmetric Einstein-scalar-field.
method Local and global unique solutions for small initial data.
result Sharp decaying condition for unique solutions.
New conditions found for Riemannian foliations with specific conformal fields.
problem Conditions for Riemannian foliations with transversal conformal fields.
method Analyzes conditions for transversal isometry of foliations.
result Found conditions for F to be isometric to the sphere. The paper proves rigidity theorems on holomorphic isometries into homogeneous domains.
problem Characterizing and comparing holomorphic isometries into homogeneous bounded domains.
method Two rigidity theorems on holomorphic isometries into homogeneous bounded domains.
result The flat (definite or indefinite) complex Euclidean space is not a relative of a homogeneous bounded domain.
Second paper in series solves Einstein vacuum equations for three impulsive waves.
problem Solving local Cauchy problem for impulsive gravitational waves.
method Geometric commutators for energy estimates, fractional-derivative regularity, anisotropic Sobolev embedding.
result Scalar field becomes everywhere Lipschitz and C1,θ away from singular region. Formulates a new class of gravity theories coupled to scalar and gauge fields.
problem Global formulation of theories of gravity coupled to scalar and gauge fields.
method Global mathematical formulation of a class of generalized four-dimensional theories of gravity.
result Global solutions can be interpreted as classical U-folds.
The study finds polynomial upper bounds for singularities in Einstein-scalar field system.
problem Understanding the strength of singularities in gravitational collapse.
method Analyzing geometric quantities, focusing on the Kretschmann scalar.
result Polynomial blow-up upper bounds O(1/rN) for the Kretschmann scalar, improving previous bounds. Paper examines conditions making Riemann solitons trivial and estimates their scalar curvature.
problem Conditions making Riemann solitons trivial and scalar curvature estimates.
method Analyzes compactness and behavior at infinity of gradient fields.
result Obtains scalar curvature estimates for certain Riemann solitons.
In this paper, we prove a far-from-CMC result similar to the ones obtained by Holst, Nagy, Tsogtgerel and Maxwell for the conformal Einstein-scalar field constraint equations on compact Riemannian manifolds with positive (modified) Yamabe invariant.
The paper characterizes solitons and estimates scalar curvature.
problem Characterizing and estimating scalar curvature of generalized Ricci-Yamabe solitons.
method Characterization and estimation of scalar curvature through soliton properties.
result Conditions for scalar curvature to be constant and estimation of Ricci curvature.
The study classifies gradient Ricci solitons with specific vector fields.
problem Characterizing gradient Ricci solitons with closed conformal vector fields.
method Analyzing properties of gradient Ricci solitons with constant scalar curvature and closed conformal vector fields.
result Gradient Ricci solitons with these properties are isometric to specific spaces.
The study finds energy gaps for Yang-Mills fields on Kähler surfaces.
problem Finding energy gaps for Yang-Mills fields on Kähler surfaces.
method Proving an L2 energy gap result for Yang-Mills connections on Kähler surfaces with positive scalar curvature. result Proves energy gap results for Yang-Mills fields on Kähler surfaces.
Study shows instability of naked singularities in scalar field models.
problem Stability of naked singularities in spherically symmetric Einstein-Scalar field systems.
method Analysis of a family of incoming null cones becoming increasingly singular.
result Naked singularities are unstable to black hole formation under certain perturbations.
We follow the approach employed by Y. Choquet-Bruhat, J. Isenberg and D. Pollack in the case of closed manifolds and establish existence and non-existence results for the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds.
In this paper we consider the field equations for linearized gravity and other integer spin fields on the Kerr spacetime, and more generally on spacetimes of Petrov type D. We give a derivation, using the GHP formalism, of decoupled field equations for the linearized Weyl scalars for all spin weights and identify the g…
Functional determinant for mixed signature sphere products depends on sphere dimensions and parity.
problem Determining the functional determinant for scalar fields on mixed signature sphere products.
method Analyzing the GJMS operator on SqimesSp to derive the functional determinant. result The functional determinant depends only on the total dimension and parity of the sphere dimensions.
We consider the signed density of the extremal points of (two-dimensional) scalar fields with a Gaussian distribution. We assign a positive unit charge to the maxima and minima of the function and a negative one to its saddles. At first, we compute the average density for a field in half-space with Dirichlet boundary c…
The aim of this note is to prove that any compact non-trivial almost Ricci soliton (Mn,g,X,λ) with constant scalar curvature is isometric to a Euclidean sphere Sn. As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreo…
Investigates how adding a scalar potential affects Dirac-harmonic maps.
problem Analyzing the impact of scalar potential on Dirac-harmonic maps.
method Examines various geometric and analytic properties with different potentials.
result Cannot achieve certain properties with the potential term in general.
Study of spacelike singularities in spherical spacetimes with scalar matter.
problem Characterize spacelike singularities in spherically symmetric spacetimes with scalar matter.
method Analyzes the properties of spacelike singularities in spherically symmetric spacetimes with scalar matter, proving inverse polynomial blow-up rates and providing a BKL-type expansion.
result Provides a rigorous description of Kasner-like singularities in spherically symmetric gravitational collapse.
New definitions and properties of harmonic vector fields on Finsler manifolds.
problem Defining and understanding harmonic vector fields in Finsler geometry.
method Natural definitions of differential, divergence, and p-harmonic form; proving Hodge theorem; Bochner-Yano classification theorem. result A closed orientable Finsler manifold with a positive harmonic Ricci scalar has a zero Betti number.