arXiv research

Four-dimensional Einstein Dehn filling is impossible.

Study finds all isometries for specific Lie groups.

Researchers found all special metrics in 4D for certain curvature functionals.

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].

Smoothness of Sklyanin algebras examined in 3D and 4D cases.

Study on 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 $\mathbb{Z}$-graded subalgebras with maximum odd dimension of the $N{=}1$ Poincaré superalgebra in four dimensions. Part of this calcula…

Found a new connected component in symplectic structures.

Review of gravitational instantons in physics.

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…

Study finds all 4D Lie groups with harmonic curvature.

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…

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.

Study of Brown--York mass for four-dimensional asymptotically flat manifolds.

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.

The study finds static solutions in symplectic curvature flow in 4D.

Study on Hodge theory for almost complex manifolds.

We construct an example of Ricci-flat almost-Kähler non-Kähler structure in four dimensions.

Classifies SNC-algebras in 5D, calculating curvature.

The paper constructs Ricci flow solutions for non-smooth metrics in four dimensions.

We derive a new renormalized volume formula for conformally compact asymptotically hyperbolic manifolds in dimension four. The formula generalizes the ones given by Anderson, Albin, and Chang-Qing-Yang for the case of Poincare-Einstein manifolds. We also derive variational formulas for the renormalized seen as a functi…

It is shown how solutions to the Tzitzéica equation can be used to construct a family of (pseudo) hyper-complex metrics in four dimensions.

Fukaya-Yamaguchi conjecture holds in 4D manifolds with nonnegative curvature.

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.

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.

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.

We construct a family of non-collapsed, non-Kähler, non-Einstein steady Ricci solitons in even dimensions greater or equal to four. These solitons exist on complex line bundles over Kähler-Einstein manifolds of positive scalar curvature. They include a four-dimensional $U(2)$-invariant, non-collapsed Riemannian steady …

We consdier in dimension four weakly convergent sequences of approximate biharmonic maos into sphere with bi-tension fields bounded in $L^p$ for some $p>1$. 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…

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…

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…

Undergraduate thesis explores topological barriers to compact Ricci solitons in 4D.

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…

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 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.

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 $(J,ω)$ are parametrized up to complex isomorphism (where $J$ is a complex structure and $ω$ is a symplectic structure). Such structure gives rise to a pseu…

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…

Recent developments in the understanding of $N=2$ 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 $SU(2)$ instantons, one can define equivalent four-manifold invariants by counting solution…

Upper bounds found for Seiberg-Witten moduli spaces under specific conditions.

Rigidity for 4D Willmore submanifolds with boundary.

We prove that there is no nontrivial homogeneous order 2 solutions of fully nonlinear uniformly elliptic equations in dimension 4.

This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing result for the invariants of symplectic four-manifolds, which leads to new proo…

For a given quantum field theory, provided the area of the entangling surface is fixed, what surface maximizes entanglement entropy? We analyze the answer to this question in four and higher dimensions. Surprisingly, in four dimensions the answer is related to a mathematical problem of finding surfaces which minimize t…

Study shows strict inequality for Dirac-type equation on spin manifolds, leading to existence results.