Extends spectral Einstein functionals computation to 4D spin manifolds with boundary.
problem Computing spectral Einstein functionals for 4D spin manifolds with boundary.
method Generalizes Dabrowski's results to 4D spin manifolds with boundary using noncommutative residue.
result Generalized spectral Einstein functionals computation for 4D spin manifolds with boundary.
Study second-Chern-Einstein metrics on 4D manifolds, finding Killing vector fields and examples.
problem Investigate second-Chern-Einstein metrics on 4D almost-Hermitian manifolds.
method Analyze compact and unimodular almost-abelian Lie algebras, use Killing vector fields and parallel non-zero Lee forms.
result Describe 4D compact second-Chern-Einstein locally conformally symplectic manifolds and classify unimodular almost-abelian Lie algebras with second-Chern-Einstein metrics.
Proves biharmonic maps on 4D Einstein manifolds relate to Yamabe-type equations.
problem Constructing biharmonic conformal maps on 4D Einstein manifolds.
method Reduces biharmonic problem to Yamabe-type equations.
result Characterizes all solutions and constructs infinite family of examples.
The paper proves a spin manifold's 4D quasi-Einstein satisfies Hitchin-Thorpe inequality.
problem Proving a specific inequality for a class of 4D manifolds.
method Analyzing properties of gradient m-quasi-Einstein manifolds, focusing on spin structures. result Compact 4D spin gradient m-quasi-Einstein manifolds satisfy the Hitchin-Thorpe Inequality when m≥1. Classifies special types of 4D spaces.
problem Classifying non-reductive 4D spaces.
method Classification based on conformal Einstein manifolds.
result Classification of non-reductive 4D homogeneous conformally Einstein manifolds.
New C-connection characterizes 4D spaces conformal to Einstein spaces.
problem Characterizing 4D spaces conformal to Einstein spaces.
method Introducing C-connection, a Weyl connection that preserves conformal invariance. result Characterizes non-degenerate spaces conformal to Einstein spaces.
New method computes Yamabe invariants for specific 4D manifolds.
problem Computing Yamabe invariants for new 4D manifolds.
method Using twisted Seiberg-Witten equations, Pin(2)-monopole equations.
result Computed Yamabe invariants for a new class of 4D manifolds.
Classifies non-Einstein solutions to Einstein--Maxwell equations on 4D Lie algs.
problem Finding non-Einstein solutions to Einstein--Maxwell equations on 4D Lie algebras.
method Classification of left-invariant solutions up to automorphisms.
result Classification of all left-invariant non-Einstein solutions.
3d gauge theories encode 4d simplicial geometries, with quantum blocks approximating gravity.
problem Encoding 4d quantum gravity in 3d gauge theories.
method Applying Dimofte-Gaiotto-Gukov construction to graph complements in 3-manifolds.
result Holomorphic blocks approximate quantum 4d simplicial geometries.
Study compactifies Einstein metrics on 4D manifolds.
problem Compactification of conformally compact Einstein metrics.
method Assumptions on local and non-local conformal invariants, boundary metrics, and manifold topology.
result Established compactness results for 4D manifolds.
Desingularizes Einstein metrics with A1 singularities in 4D.
problem Desingularizing Einstein metrics with specific singularities.
method Recursive procedure to desingularize Fuchsian singularities.
result Desingularizations of non degenerate Poincaré-Einstein metrics with A1 singularities remain non degenerate.
Study on 4D Einstein manifolds with improved boundary regularity.
problem Boundary regularity of conformally compact Einstein manifolds.
method Proves Hölder continuity of scalar curvature and Cm,α boundary metric, studies new structure and defining function. result Improved compactification and regularity of 4D Einstein manifolds.
Study on 4D Einstein manifolds with Kähler conformal geometry.
problem Exploring 4D Poincaré-Einstein manifolds with Kähler metrics.
method Formulated a Dirichlet boundary value problem and established existence and uniqueness theory.
result Existence and uniqueness of new Poincaré-Einstein metrics.
Derives Bochner formulas for the Weyl tensor on 4D Einstein manifolds.
problem Understanding the geometry of Einstein manifolds through the Weyl tensor.
method Derives higher-order Bochner type formulas for the Weyl tensor on 4D Einstein manifolds.
result Proves a second Bochner type formula for the Weyl tensor.
The paper classifies 4D quasi-Einstein manifolds with harmonic Weyl curvature.
problem Classifying 4D (m,ρ)-quasi-Einstein manifolds with harmonic Weyl curvature. method Local isometric classification of non-trivial metrics.
result Non-trivial (m,ρ)-quasi-Einstein metrics are locally isometric to specific types of manifolds. Study on filling 3D metrics with 4D Poincaré-Einstein structures.
problem Finding a conformal filling by a Poincaré-Einstein metric in 4D.
method Compactness result for conformally compact Einstein 4-manifolds under invariant conditions, with a rigidity result for hyperbolic metrics.
result Established compactness results and derived existence results for conformal fillings.
Study proves rigidity and gap theorems for specific metrics.
problem Existence and properties of self-dual and even Poincaré-Einstein metrics in 4D.
method Rigorous mathematical proofs, including gap theorems and rigidity results.
result Obtained new scalar conformal invariants and identified obstructions to metric existence.
It is formulated a new 'anholonomic frame' method of constructing exact solutions of Einstein equations with off--diagonal metrics in 4D and 5D gravity. The previous approaches and results are summarized and generalized as three theorems which state the conditions when two types of ansatz result in integrable gravitati…
Classifies weakly Einstein curvature tensors in 4D Euclidean space.
problem Classifying algebraic curvature tensors in 4D Euclidean space.
method Algebraic formulation and geometric interpretation of weakly Einstein manifolds.
result Complete classification of non-Einstein weakly Einstein curvature tensors in dimension four.
New formula for volume in 4D hyperbolic manifolds.
problem Volume calculation for specific geometric manifolds.
method Derive new renormalized volume formula.
result Generalizes existing formulas for Poincare-Einstein manifolds.
The paper proves that certain 4D metrics are isometric to a sphere.
problem Investigating critical points of the total scalar curvature functional.
method Analyzing CPE metrics with constant scalar curvature and unitary volume.
result A 4D CPE metric with harmonic tensor W+ is isometric to a round sphere S4. New proof shows certain 4D metrics are non-degenerate if curvature is negative definite.
problem Proving non-degeneracy of Poincaré-Einstein metrics.
method Proved non-degeneracy for 4D metrics satisfying a chiral curvature inequality.
result 4D Poincaré-Einstein metrics are non-degenerate if curvature is negative definite.
The paper classifies 4D Einstein manifolds with specific curvature bounds.
problem Classifying 4D Einstein manifolds with sectional curvature bounded from above.
method Analyzing sectional curvature and applying rigidity theorems.
result Einstein four-manifolds with nonnegative sectional curvature are isometric to S4, RP4, or CP2. The study finds static solutions in symplectic curvature flow in 4D.
problem Finding static solutions in symplectic curvature flow in 4D.
method Derived a local normal form for static solutions and used Cartan-Kahler theorem for solitons.
result Every complete static solution to symplectic curvature flow in 4D is Kahler-Einstein.
The paper finds the explicit expression of Graham-Witten's invariant for 4D submanifolds.
problem Finding conformal invariants of submanifolds.
method Volume renormalization of minimal surfaces in conformally compact Einstein manifolds.
result Explicit expression of Graham-Witten's conformal invariant for 4D submanifolds.
The paper connects web theory to heavenly PDEs and Einstein metrics.
problem Understanding the correspondence between self-dual metrics and vector fields.
method Using Nijenhuis operators and web theory to construct new heavenly PDEs.
result New integrable heavenly PDEs derived from Nijenhuis operators.
Proves equivalence of Lax pairs and Einstein-Weyl/self-dual structures in 3D/4D.
problem Equivalence of Lax pairs and geometric structures in 3D and 4D.
method Characteristic property of dispersionless Lax pairs, projective behavior of Lax pair.
result Equivalence of Lax pairs and Einstein-Weyl/self-dual structures.
The 'anholonomic frame' method (see gr-qc/0005025, gr-qc/0001060 and hep-th/0110250) is applied for constructing new classes of exact solutions of vacuum Einstein equations with off-diagonal metrics in 4D and 5D gravity. We examine several black tori solutions generated by anholonomic transforms with non-trivial topolo…
Study classifies Einstein-Yang-Mills spaces in 4D symmetric spaces.
problem Classifying Lorentzian symmetric spaces with Einstein-Yang-Mills properties.
method Classification based on invariant metric connections and diagonal metrics.
result Four-dimensional symmetric spaces with nontrivial isotropy groups are classified.
Establishes 4D regularity for certain metric spaces.
problem Noncollapsed sequences of metrics with Ricci tensor bounds.
method A priori L2 curvature estimates.
result Diffeomorphism finiteness and rigidity theorems.
Weak harmonic Weyl metrics found on all 4D closed manifolds.
problem Finding canonical metrics on 4D closed manifolds.
method Critical points of a quadratic functional involving the divergence of the Weyl tensor.
result Every 4D closed manifold admits a unique weak harmonic Weyl metric.
New cohomology η for deformed Sasaki-Einstein manifolds derived from Dolbeault cohomology.
problem Developing a new cohomology structure for deformed Sasaki-Einstein manifolds.
method Introducing η-cohomology defined by a CR structure and a holomorphic function f with non-vanishing η≡df. result Established a direct relation between the cyclic homologies of the Calabi-Yau algebra and the η-cohomology groups. The paper finds exact solutions to a complex Einstein-Dirac-Maxwell system on 4D Sasakian spacetimes.
problem Finding exact solutions to an Einstein-Dirac-Maxwell system with Sasakian quasi-Killing spinors.
method Constructing a family of exact solutions on four-dimensional static Sasakian spacetimes using the Sasakian frame.
result Closed and open universe models are found with specific energy conditions.
We construct new classes of exact solutions of the 4D vacuum Einstein equations which describe ellipsoidal black holes, black tori and combined black hole -- black tori configurations. The solutions can be static or with anisotropic polarizations and running constants. They are defined by off--diagonal metric ansatz wh…
Study of Einstein equations in warped spaces to determine cosmological constants.
problem Understanding the origin of the cosmological constant in higher dimensions.
method Classification and analysis of (m+n)D and (1+n)D warped spaces satisfying Einstein equations. result Warping function can determine both (m+n)D and 4D cosmological constants. For several classes of second order dispersionless PDEs, we show that the symbols of their formal linearizations define conformal structures which must be Einstein-Weyl in 3D (or self-dual in 4D) if and only if the PDE is integrable by the method of hydrodynamic reductions. This demonstrates that the integrability of t…
New 4D gravity theory yields G2-holonomy metrics.
problem Construct G2-holonomy metrics from 4D gravity theory.
method Formulate a 4D gravity theory and lift its solutions to G2-holonomy metrics.
result Every solution of the 4D gravity theory lifts to a G2-holonomy metric.
In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It is outlined a new geometric techniques of constructing Ricci flow solutions. Some conceptual issues on spacetimes provided with generic off-diagonal metrics and ass…
New classification of conformal structures with maximal G2 symmetry.
problem Classifying conformal structures with maximal G2 symmetry. method Complete local classification of homogeneous 4D split-conformal structures.
result Established a complete local classification of conformal structures with maximal G2 symmetry. On a five dimensional simply connected Sasaki-Einstein manifold, one can construct Yang-Mills theories coupled to matter with at least two supersymmetries. The partition function of these theories localises on the contact instantons, however the contact instanton equations are not elliptic. It turns out that these equa…
Knots in 4D manifolds are trivial based on Wedderburn's Theorem.
problem Understanding knots in 4-dimensional manifolds.
method Application of Wedderburn's Theorem.
result All knots and links in smooth 4D manifolds are trivial.
The paper studies projectively equivalent para-Kaehler metrics in 4D.
problem Characterizing para-Kaehler metrics with specific properties.
method Developed c-projective geometry for para-Kaehler metrics, focusing on 4D case.
result Local description and characterization of 4D pc-projectively equivalent metrics, including Einstein type.
We show that the smooth geometry of a hyperbolic 3-manifold emerges from a classical spin system defined on a 2d discrete lattice, and moreover show that the process of this "dimensional oxidation" is equivalent with the dimensional reduction of a supersymmetric gauge theory from 4d to 3d. More concretely, we propose a…
Study classifies special 4D shapes with certain curvature.
problem Classifying specific types of 4D shapes.
method Classifying compact almost-Kähler four manifolds with nonnegative biorthogonal curvature.
result Classified compact almost-Kähler four manifolds with nonnegative biorthogonal curvature.
In this article we consider nonholonomic deformations of disk solutions in general relativity to generic off-diagonal metrics defining knew classes of exact solutions in 4D and 5D gravity. These solutions possess Lie algebroid symmetries and local anisotropy and define certain generalizations of manifolds with Killing …
The study calculates harmonic functions and 1-forms on specific 4D spaces.
problem Computing harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
method Computed the expansion of harmonic functions and 1-forms.
result Computed the expansion of harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
Study describes flat metric moduli spaces on 4D manifolds.
problem Understanding flat metrics on 4D closed manifolds.
method Algebraic and topological description of moduli spaces.
result Algebraic and topological description of moduli spaces of flat metrics.
4D gradient solitons with constant curvature are rigid.
problem Characterizing 4D gradient Ricci solitons with constant scalar curvature.
method Proving rigidity using constant scalar curvature and quotient structures.
result 4D gradient solitons with constant curvature are rigid.