New curvature measures for 4D manifolds with corners defined and related to Gauss-Bonnet.
problem Defining curvature measures for 4D manifolds with corners.
method Defined two new extrinsic curvature quantities, one conformal invariant, and a new conformally invariant operator.
result Gauss-Bonnet theorem reformulated in terms of new curvature measures.
New C \mathcal{C} C -connection characterizes 4D spaces conformal to Einstein spaces.
problem Characterizing 4D spaces conformal to Einstein spaces.
method Introducing C \mathcal{C} C -connection, a Weyl connection that preserves conformal invariance. result Characterizes non-degenerate spaces conformal to Einstein spaces.
Study on 4D PDEs with half-flat conformal structure leading to Monge-Ampere equations.
problem Characterizing second-order PDEs in 4D with specific conformal structures.
method Analysis of Monge-Ampere property and dispersionless Lax pairs.
result All known scalar second-order integrable dispersionless PDEs in 4D are of Monge-Ampere type.
Paper studies critical points of curvature energies in 4D.
problem Critical points of conformally invariant extrinsic energies on 4-manifolds.
method Converted Euler-Lagrange equations to a system with favourable structures using invariances and Noether's theorem.
result Generalized Tristan Rivière's work on Willmore energy to 4D.
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.
The paper studies Willmore surfaces in 4D conformal manifolds and finds the Clifford torus is strictly Willmore-stable.
problem Exploring the Willmore functional for surfaces in 4D conformal manifolds.
method Detailed calculation of first and second variations, derivation of Euler-Lagrange equation in a conformally invariant form.
result The Clifford torus in C P 2 \mathbb{C}P^2 C P 2 is strictly Willmore-stable, supporting a conjecture. 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 C m , α C^{m,α} C m , 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.
Study of critical points for 4D conformally invariant curvature energies.
problem Analyzing critical points of conformally invariant curvature energies in 4 dimensions.
method Using Noether's theorem and divergence-free potentials, generating an algebraic structure, and considering Palais-Smale sequences.
result Improved energy estimates for critical points under small-energy hypotheses.
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.
New invariant for 4D hypersurfaces ensures smooth critical points.
problem Understanding smoothness of curvature energies on 4D hypersurfaces.
method Developed a new conformally invariant energy.
result Critical points of new energy are smooth.
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.
Scroll structures on solutions of 4D integrable equations are involutive and governed by a dispersionless hierarchy.
problem Characterizing the geometry of solutions to 4D integrable equations.
method Defining rational normal scrolls and showing their involutivity.
result Involutive scroll structures are governed by a dispersionless integrable hierarchy.
New classification of conformal structures with maximal G 2 G_2 G 2 symmetry.
problem Classifying conformal structures with maximal G 2 G_2 G 2 symmetry. method Complete local classification of homogeneous 4D split-conformal structures.
result Established a complete local classification of conformal structures with maximal G 2 G_2 G 2 symmetry. Random Gaussian fields on 4D Riemannian manifolds with conformal invariance.
problem Characterizing and analyzing Gaussian fields on 4D Riemannian manifolds.
method Constructing and analyzing co-biharmonic Gaussian fields with covariance kernels defined by the Paneitz operator.
result Rigorous derivation of quantum Liouville measure for ∣ γ ∣ < 8 |γ|<\sqrt8 ∣ γ ∣ < 8 . We prove that integrability of a dispersionless Hirota type equation implies the symplectic Monge-Ampere property in any dimension ≥ 4 \geq 4 ≥ 4 . In 4D this yields a complete classification of integrable dispersionless PDEs of Hirota type through a list of heavenly type equations arising in self-dual gravity. As a by-produc…
A theorem connects two Willmore energies in 4D.
problem Understanding the Willmore energy in 4D.
method Proving a duality theorem for a specific Willmore energy.
result The Willmore energy is equal to two conformally invariant energies.
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…
Representations of Dirac-Hestenes and Dirac spinor fields via coordinates of surfaces conformally immersed into 4-dimensional complex space are proposed. A relation between time evolution of spinor fields and integrable deformations of surfaces is discussed.
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.
The study shows rigidity of Kähler-Ricci solitons on a specific 4D manifold.
problem Investigating the rigidity of Kähler-Ricci solitons near a Kähler model.
method Analyzing gradient shrinking Ricci solitons close to a Kähler model, focusing on norms of the self-dual Weyl tensor and scalar curvature.
result Gradient Ricci solitons on S 2 i m e s R 2 \mathbb{S}^2 imes \mathbb{R}^2 S 2 im es R 2 are either half-conformally flat or locally Kähler if certain conditions are met. 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.
We use the conformal invariance and the holographic correspondence to fully specify the dependence of entanglement entropy on the extrinsic geometry of the 2d surface Σ Σ Σ that separates two subsystems of quantum strongly coupled N = 4 {\mathcal{N}}=4 N = 4 SU(N) superconformal gauge theory. We extend this result and calculate en…
We consider compactification of type IIA supergravity on nearly Kaehler manifolds. These represent a simple class of SU(3) structure manifolds which includes S^6 and CP^3. We exhibit for the first time an explicit reduction ansatz in this context, obtaining an N=2 gauged supergravity in 4d with a single vector and hype…
Geometric interpretation of 2d-4d wall-crossing formulas.
problem Understanding wall-crossing phenomena in coupled 2d-4d systems.
method Deformation theory of holomorphic pairs and relation to scattering diagrams.
result Geometric interpretation of wall-crossing formulas.
M-theory compactified on G 2 G_2 G 2 -holonomy manifolds results in 4d N = 1 \mathcal{N}=1 N = 1 supersymmetric gauge theories coupled to gravity. In this paper we focus on the gauge sector of such compactifications by studying the Higgs bundle obtained from a partially twisted 7d super Yang-Mills theory on a supersymmetric three-cycle…
Generalized Weierstrass representations for generic surfaces conformally immersed into four-dimensional Euclidean and pseudo-Euclidean spaces of different signatures are presented. Integrable deformations of surfaces in these spaces generated by the Davey-Stewartson hierarchy of integrable equations are proposed. Willm…
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.
Initial data for p p pp pp -wave spacetimes constructed in 4D.
problem Characterizing initial data for p p pp pp -wave spacetimes. method Constructs a vacuum initial data set with extra conditions related to CKID.
result Data development is a subset of a vacuum p p pp pp -wave. In this paper we address several aspects of flat Bogomolnyi-Prasad-Sommerfeld (BPS) domain walls together with their Lorentz invariant vacua of 4d N=1 supergravity coupled to a chiral multiplet. The scalar field spans a one-parameter family of 2d Kähler manifolds satisfying a Kähler-Ricci flow equation. We find that BP…
The paper classifies 4D Ricci-flat ALE manifolds and their properties.
problem Characterizing 4D Ricci-flat ALE manifolds and their properties.
method Analyzing the correspondence between ALE gravitational instantons and Kähler orbifolds, and proving properties of these structures.
result There is a one-to-one correspondence between Hermitian non-Kähler ALE gravitational instantons and Bach-flat Kähler orbifolds in 2D.
Symmetrizes 4d and 3d BPS quivers for Argyres-Douglas theories.
problem Understanding the relationship between 4d and 3d BPS quivers.
method Analyzes geometric backgrounds and uses skein modules to derive quiver partition functions.
result Proves isomorphism between 4d wall-crossing and unlinking of symmetric quivers.
A class of 3d N = 2 \mathcal{N}=2 N = 2 supersymmetric gauge theories are constructed and shown to encode the simplicial geometries in 4-dimensions. The gauge theories are defined by applying the Dimofte-Gaiotto-Gukov construction in 3d/3d correspondence to certain graph complement 3-manifolds. Given a gauge theory in this class…
Study on flat metrics on 3D and 4D manifolds, focusing on topology and algebra.
problem Topology and algebraic structure of flat metrics on manifolds.
method Algebraic and topological descriptions of moduli spaces.
result Algebraic description and topology of moduli spaces for 4D manifolds with a single holonomy generator.
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 on curvature flow in 4D ball, proving existence and convergence.
problem Existence of metrics with prescribed T T T -curvature on the 4D unit ball. method Using T T T -curvature flow and Morse-theoretic approach, combining Ache-Chang's inequality. result Existence results and exponential convergence to extremal metric.
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…
A regularization procedure developed in [1] for the integral curvature invariants on manifolds with conical singularities is generalized to the case of squashed cones. In general, the squashed conical singularities do not have rotational O(2) symmetry in a subspace orthogonal to a singular surface Σ Σ Σ so that the surfa…
Study on 4D Lie groups and related almost hypercomplex manifolds.
problem Characterizing almost hypercomplex manifolds with specific metrics.
method Construction and classification of manifolds based on Lie algebras.
result Established a connection between Lie algebra classes and manifold classifications.
The paper studies 4D Ricci flow manifolds with curvature constraints.
problem Investigating 4D Ricci flow manifolds with specific curvature conditions.
method Analyzing 4D manifolds with curvature constraints via Ricci flow.
result Proves topological and geometric gap theorems for maximal volume growth.
Perelman's proof confirmed, new method uses 4D topology.
problem Confirming the classical Poincaré conjecture.
method 4D topology, spun torus-knots, ribbonness, disk-chord system, Bing's result.
result Homotopy 3-sphere is diffeomorphic to the 3-sphere.
Study finds all 4D neutral manifolds.
problem Classifying neutral manifolds in four dimensions.
method Examined homogeneous semi-symmetric neutral manifolds.
result Identified all four-dimensional neutral manifolds.
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 m m -quasi-Einstein manifolds, focusing on spin structures. result Compact 4D spin gradient m m m -quasi-Einstein manifolds satisfy the Hitchin-Thorpe Inequality when m ≥ 1 m\ge 1 m ≥ 1 . The study proves properties of 4D projective manifolds and builds non-hyperbolic examples.
problem Characterizing and understanding geometric properties of 4D projective manifolds.
method Analyzing geometric decompositions and using properties of locally symmetric spaces.
result Closed, indecomposable 4D projective manifolds are either real hyperbolic or have real hyperbolic pieces.
Study uses superalgebra homology to classify 4D Engel-like Lie algebras.
problem Classifying 4-dimensional Engel-like Lie algebras.
method Applied homology groups of Lie superalgebras.
result Distinguished and classified 4D Engel-like Lie algebras.
Paper connects hypersurfaces in 4D to surfaces in 3-sphere.
problem Understanding conformally flat hypersurfaces in 4D space forms.
method Characterizes conformal structures and relates to surfaces in 3-sphere.
result Relates 2-metrics in 3-sphere to surfaces giving rise to conformally flat hypersurfaces.
This paper confirms volumes of geodesic balls can identify 4D space forms.
problem Determining if a 4D manifold is a space form using geodesic ball volumes.
method Tensor calculus and classical theorems, not topological characterizations.
result Similar results for 4D manifold space forms confirmed.