Four-dimensional Einstein Dehn filling is impossible.
problem Complex-hyperbolic Einstein Dehn filling in four dimensions.
method Proof of impossibility.
result Complex-hyperbolic Einstein Dehn filling cannot be performed in dimension four.
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48 results for “Dimension four”
Study finds all isometries for specific Lie groups.
problem Identifying isometry groups in nonunimodular Lie groups.
method Examined left-invariant Riemannian metrics on Lie groups of dimension four.
result Determined full group of isometries for each metric.
Researchers found all special metrics in 4D for certain curvature functionals.
problem Identifying special metrics in 4D for quadratic curvature functionals.
method Determined all homogeneous metrics that are critical for quadratic curvature functionals.
result All homogeneous metrics in 4D for some quadratic curvature functionals have been identified.
Smoothness of Sklyanin algebras examined in 3D and 4D cases.
problem Differential smoothness of Sklyanin algebras in different dimensions.
method Analysis of differential smoothness in three and four variables.
result Three-dimensional Sklyanin algebras are differentially smooth, but four-dimensional ones are not.
Recent work by physicists on gravity in two dimensions has a natural generalization to four dimensions, formulated in terms of an analogue of Segal's category [defined for the study of conformal field theory].
Volume renormalization for conformally compact asymptotically hyperbolic manifolds in dimension fourmath.DG
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.
Study on images and singularities of pseudoholomorphic maps.
problem Characterize images and singularities of pseudoholomorphic maps.
method Analyzes pseudoholomorphic maps in domains and targets of dimension four.
result Proves properties of images and singularities of pseudoholomorphic maps.
We determine the Killing superalgebras underpinning field theories with rigid unextended supersymmetry on Lorentzian four-manifolds by re-interpreting them as filtered deformations of -graded subalgebras with maximum odd dimension of the Poincaré superalgebra in four dimensions. Part of this calcula…
New steady Ricci solitons found in even dimensions, including a four-dimensional example.
problem Existence of non-collapsed steady Ricci solitons in even dimensions.
method Construction of a family of non-collapsed, non-Kähler, non-Einstein steady Ricci solitons on complex line bundles over Kähler-Einstein manifolds.
result Existence of new non-collapsed steady Ricci solitons in even dimensions, including a four-dimensional example.
Found a new connected component in symplectic structures.
problem Understanding symplectic structures in higher dimensions.
method Provided an example in dimension four.
result First example of a nontrivial connected component.
Review of gravitational instantons in physics.
problem Understanding solutions to Einstein equations in four dimensions.
method Lecture review of examples.
result Examples of gravitational instantons.
We determine all Chern numbers of smooth complex projective varieties of dimension at least four which are determined up to finite ambiguity by the underlying smooth manifold. We also give an upper bound on the dimension of the space of linear combinations of Chern numbers with that property and prove its optimality in…
In this paper we deal with symplectic Lie algebras. All symplectic structures are determined for dimension four and the corresponding Lie algebras are classified up to equivalence. Symplectic four dimensional Lie algebras are described either as solutions of the cotangent extension problem or as symplectic double exten…
A method, due to Élie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2,2) signature are Einstein of which one is Ricci-flat. If…
The paper extends the study of conformally flat spaces to four dimensions.
problem Understanding conformal transformations and flatness in four-dimensional Finsler spaces.
method Extending the study of three-dimensional conformally flat Landsberg and Berwald spaces to four dimensions.
result Conditions for a four-dimensional conformally flat Landsberg space to become a Berwald space.
We prove that every piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly. This solves a challenge posed by Thurston in dimension three and confirms a conjecture by Kwasik and L…
Study finds all 4D Lie groups with harmonic curvature.
problem Finding Lie groups with harmonic curvature in 4D.
method Analyzing pp-wave Lie groups and determining harmonic curvature conditions.
result Description of 4D pp-wave Lie groups obtained.
This paper introduces two-dimensional diagrams that are slight generalizations of moment map images for toric four-manifolds and catalogs techniques for reading topological and symplectic properties of a symplectic four-manifold from these diagrams. The paper offers a purely topological approach to toric manifolds as w…
Survey on gradient Ricci solitons in 4D, focusing on geometry and classification.
problem Understanding gradient Ricci solitons in four dimensions.
method Geometric analysis and classification of solitons.
result Recent results on classification and rigidity of gradient Ricci solitons in 4D.
Study of Brown--York mass for four-dimensional asymptotically flat manifolds.
problem Calculating mass for hypersurfaces in four-dimensional asymptotically flat manifolds.
method Intrinsic definition of mean curvature, expansion analysis for large uniformly convex hypersurfaces.
result Shape-dependent correction to ADM mass for nearly round surfaces vanishes under certain conditions.
We review the relations between (twisted) supersymmetric gauge theories in four dimensions and moduli problems in four-dimensional topology, and we study in detail the non-abelian monopole equations from this point of view. The relevance of exact results in N=1 and N=2 supersymmetric gauge theories to the computation o…
Classifies almost-toric systems in four dimensions.
problem Classifying compact almost-toric systems in four dimensions.
method Symplectic classification using base, Taylor series, and twisting indices.
result Classification into five invariants, analogous to semitoric systems.
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.
Classifies SNC-algebras in 5D, calculating curvature.
problem Classifying SNC-algebras in higher dimensions.
method Defined SNC-algebras and used Lie group properties.
result Classified SNC-algebras in dimension five.
Almost complex Hodge theorymath.DG
Study on Hodge theory for almost complex manifolds.
problem Determining Hodge numbers for almost complex manifolds.
method Review and analysis of recent developments in Hodge theory for almost complex manifolds.
result Hodge numbers are almost complex, almost Kähler, or birational invariants in dimension four.
We construct an example of Ricci-flat almost-Kähler non-Kähler structure in four dimensions.
Four-dimensional GL(2) structures are flat with a specific subgroup transformation.
problem Flatness of GL(2) structures in four dimensions.
method Showed flatness through a coframe transformation involving a subgroup isomorphic to a semidirect product of Heisenberg group and Abelian group.
result The relevant PDE system is integrable with a dispersionless Lax-pair.
The paper constructs Ricci flow solutions for non-smooth metrics in four dimensions.
problem Constructing Ricci flow solutions for non-smooth metrics in four dimensions.
method Ricci DeTurck flow on closed manifolds with initial values in .
result Constructs solutions to Ricci flow for non-smooth metrics in four dimensions.
It is shown how solutions to the Tzitzéica equation can be used to construct a family of (pseudo) hyper-complex metrics in four dimensions.
The paper proves smoothness of weakly biharmonic almost complex structures in dimension four.
problem Existence and regularity of weakly polyharmonic almost complex structures.
method Elliptic system analysis and critical growth nonlinearities.
result Weakly biharmonic almost complex structures are smooth in dimension four.
Fukaya-Yamaguchi conjecture holds in 4D manifolds with nonnegative curvature.
problem Fundamental group of nonnegative curvature manifolds.
method Observation in dimension 4.
result Fukaya-Yamaguchi conjecture holds in 4D.
We study curvature properties of four-dimensional Lorentzian manifold with two-symmetry property. We then consider Einstein-like metrics, Ricci solitons and homogeneity over these spaces.
In this note, the geography problem in dimension four is reviewed and then its extension to dimension six for the symplectic case is explained. Finally some examples in dimension six are provided.
Geometrically transitions hyperbolic to anti-de Sitter structures in 4D.
problem Creating geometric transitions between hyperbolic and anti-de Sitter structures in 4D.
method Deformation of hyperbolic 4-polytopes and joining with anti-de Sitter polytopes.
result Existence of geometric transition examples in 4D.
We consdier in dimension four weakly convergent sequences of approximate biharmonic maos into sphere with bi-tension fields bounded in for some . We prove an energy identity that accounts for the loss of Hessian energies by the sum of Hessian energies over finitely many nontrivial biharmonic maps on $\mathbb…
We construct a family of split signature Einstein metrics in four dimensions, corresponding to particular classes of third order ODEs considered modulo fiber preserving transformations of variables.
Following an approach of the second author for conformally invariant variational problems in two dimensions, we show in four dimensions the existence of a conservation law for fourth order systems, which includes both intrinsic and extrinsic biharmonic maps. With the help of this conservation law we prove the continuit…
Topological Obstructions to the Existence of Compact Shrinking Ricci Solitons in Dimension Fourmath.DG
Undergraduate thesis explores topological barriers to compact Ricci solitons in 4D.
problem Finding topological obstructions to compact gradient shrinking Ricci solitons in dimension four.
method Discussion of background material, introduction of new problem, exploration of limitations of current results.
result Introduction of new problem and limitations of current results in extending Hitchin-Thorpe inequality.
A strong KT (SKT) manifold consists of a Hermitian structure whose torsion three-form is closed. We classify the invariant SKT structures on four-dimensional solvable Lie groups. The classification includes solutions on groups that do not admit compact four-dimensional quotients. It also shows that there are solvable g…
This work concerns the non-flat metrics on the Heisenberg Lie group of dimension three $\Heis_3(\RR)$ and the bi-invariant metrics on the solvable Lie groups of dimension four. On $\Heis_3(\RR)$ we prove that the property of the metric being naturally reductive is equivalent to the property of the center being non-dege…
In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic curvature. As a corollary we generalize a result of Perelman on three dimensional…
It is well known that the classification of the Weyl tensor in Lorentzian manifolds of dimension four, the so called Petrov classification, was a great tool to the development of general relativity. Using the bivector approach it is shown in this article a classification for the Weyl tensor in all four-dimensional mani…
Four dimensional simply connected Lie groups admitting a pseudo Kähler metric are determined. The corresponding Lie algebras are modelized and the compatible pairs are parametrized up to complex isomorphism (where is a complex structure and is a symplectic structure). Such structure gives rise to a pseu…
In this note we show that the Lagrangian Luttinger surgery preserves the symplectic Kodaira dimension. Some constraints on Lagrangian tori in symplectic four manifolds with non-positive Kodaira dimension are also derived.
In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in \cite{K} for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result Böhm and Wi…
Algorithm classifies five-dimensional spacetimes, generalizing Karlhede's for four dimensions.
problem Classifying five-dimensional spacetimes for general relativity.
method Introduces an algorithm to determine spacetime equivalence using alignment classification of the Weyl tensor.
result Illustrates the algorithm with three examples and discusses its applications.
Recent developments in the understanding of supersymmetric Yang-Mills theory in four dimensions suggest a new point of view about Donaldson theory of four manifolds: instead of defining four-manifold invariants by counting instantons, one can define equivalent four-manifold invariants by counting solution…
Upper bounds found for Seiberg-Witten moduli spaces under specific conditions.
problem Bounding the virtual dimension of Seiberg-Witten moduli spaces.
method Analyzing mod p^r basic classes and conditions on r and b2+.
result Virtual dimension is bounded by 2r(p-1)-2 under given conditions.