Study jets of flat partial connections in foliations.
problem Characterize and understand flat partial connections in foliations.
method Define and apply jets to flat partial connections in smooth foliations and locally free sheaves, focusing on codimension one and arbitrary codimension foliations.
result Define and apply jets to characterize transversely affine and projective structures in foliations.
Bounds on saddle connections on flat spheres with conical singularities.
problem Counting saddle connections on flat spheres with conical singularities.
method Geometry of immersed disks and explicit upper bounds.
result Explicit upper bounds on the number and lengths of saddle connections.
Defines tangent spaces on causal sets using partial derivatives and metrics.
problem Defining geometric structures on causal sets.
method Using partial derivatives and metrics to define tangent spaces, connection, curvature, parallel transport, and geodesics.
result Approaches expected values for a flat spacetime as density increases.
In this paper we study almost complex manifolds admitting a quasi-Kähler Chern-flat metric (Chern-flat means that the holonomy of the Chern connection is trivial). We prove that in the compact case such manifolds are all nilmanifolds. Some partial classification results are established and we prove that a quasi-Kähler …
The purpose of this article is to investigate Bach-flat critical metrics of the volume functional on a compact manifold M with boundary ∂M. Here, we prove that a Bach-flat critical metric of the volume functional on a simply connected 4-dimensional manifold with boundary isometric to a standard sphere must …
We describe two extensions of the notion of a self-dual connection in a vector bundle over a manifold M from dim M=4 to higher dimensions. The first extension, Omega-self-duality, is based on the existence of an appropriate 4-form Omega on the Riemannian manifold M and yields solutions of the Yang-Mills equations. The …
It is known that there is a bijection between the perturbed closed geodesics, below a given energy level, on the moduli space of flat connections M and families of perturbed Yang-Mills connections depending on a small parameter. In this paper we study the heat flow on the loop space on M and the Yang-Mills L^2-flows fo…
Given a complex manifold M equipped with a holomorphic action of a connected complex Lie group G, and a holomorphic principal H--bundle EH over X equipped with a G--connection h, we investigate the connections on the principal H--bundle EH that are (strongly) adapted to h. Examples are provided by…
New polystability theory connects Calabi-Yau varieties to gravitational instantons.
problem Understanding the structure of Calabi-Yau manifolds and their metrics.
method Introducing a new concept of poly-stability and relating it to gravitational instantons.
result Polystability is equivalent to the existence of certain gravitational instantons.
Study solves overdetermined problem on compact surfaces.
problem Overdetermined problem on compact Riemannian surfaces.
method Analyzes Steklov eigenvalues and geometric properties.
result Characterizes domains for which the problem is solvable.
Establishes uniform Hörmander estimates for flat line bundles on Kähler manifolds.
problem Estimating ∂-operators for flat line bundles. method Uniform L2-estimates for ∂-operators on Kähler manifolds. result Recovers Ueda's lemma for compact Kähler manifolds and generalizes to Ricci-flat manifolds.
We introduce a weak concept of Morita equivalence, in the birational context, for Poisson modules on complex normal Poisson projective varieties. We show that Poisson modules, on projective varieties with mild singularities, are either rationally Morita equivalent to a flat partial holomorphic sheaf, or a sheaf with a …
The study proves leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
problem Proving the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
method Using the ∂∂-class, the study proves the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces. result Uniform convergence of the normalized Chern-Ricci flow starting at any Gauduchon metric on all Inoue-Bombieri surfaces, with smooth convergence and bounded curvature for initial metrics in the ∂∂-class of the Tricerri/Vaisman metric. The paper examines SKT and CYT metrics on Lie groups.
problem Existence and properties of SKT and CYT metrics on Lie groups.
method Study left-invariant metrics on compact semi-simple Lie groups.
result Left-invariant metrics that are both CYT and SKT must be Bismut flat.
Two-dimensional Riemannian manifolds uniquely determined by boundary data.
problem Determining a 2D Riemannian manifold from boundary data.
method Calderón problem approach using Dirichlet-to-Neumann operator.
result A 2D compact connected Riemannian manifold is uniquely determined up to conformal equivalence.
In this paper we construct a Hitchin connection in a setting, which significantly generalizes the setting covered by the first author previously, which in turn was a generalisation of the moduli space case covered by Hitchin in his original work on the Hitchin connection. In fact, our construction provides a Hitchin co…
The purpose of this article is to give an interpretation of real projective structures and associated cohomology classes in terms of connections, sections, etc. satisfying elliptic partial differential equations in the spirit of Hodge theory. We shall also give an application of these results as the uniqueness of a min…
Geodesic sprays on Finsler manifolds studied with covariant coefficients.
problem Understanding geometric properties of Finsler metrics through covariant coefficients.
method Introduced F-covariant coefficients Hi and studied their geometric consequences. result Existence and uniqueness of spray scalar H for projectively flat metrics. The paper studies Kähler manifolds with partially semi-positive curvature and rational connectedness.
problem Analyzing compact Kähler manifolds with partially semi-positive curvature and rational connectedness.
method Proving rational connectedness for manifolds with BC-p positive tangent bundles, and applying these results to curvature conditions. result Confirming a conjecture and generalizing results on rational connectedness and curvature conditions.
Let M be a compact connected surface with boundary. We prove that the signal condition given by the Gauss-Bonnet theorem is necessary and sufficient for a given smooth function f on ∂M (resp. on M) to be geodesic curvature of the boundary (resp. the Gauss curvature) of some flat metric on M (resp. met…
A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) if its fundamental 2-form ω is ∂∂ˉ-closed. We review some properties of strong KT metrics also in relation with symplectic forms taming complex structures. Starting from a 2n-dimensional SKT Lie algebra $\mathfr…
We construct a Fourier--Mukai transform for smooth complex vector bundles E over a torus bundle π:M→B, the vector bundles being endowed with various structures of increasing complexity. At a minimum, we consider vector bundles E with a flat partial unitary connection, that is families or deformations of flat …
A partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere is given. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Biquard connection is zero and this occurs precisely on 3-Sasakian manifolods. All con…
Solves a problem in Riemannian geometry for scalar-flat metrics with boundary conditions.
problem Finding a conformal metric with zero scalar curvature and prescribed boundary mean curvature.
method Construction of local test functions to resolve open cases and establish new solvability conditions.
result Established new solvability conditions for the problem.
In this paper, we consider an equivalence problem of second order partially differential equations (PDE) and a duality of the flat differential equation. For the equivalence problem, explicit form of invariants (curvatures) are given. We also investigate a duality associated with the flat equation using double fibratio…
Let (M,g) be a smooth compact Riemannian manifold of dimension n with smooth boundary ∂M. Suppose that (M,g) admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric quotient over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric…
Study flat connections on Courant algebroids using Lie groups.
problem Flatness conditions on Courant algebroids.
method Metric generalized connections, flatness condition analysis.
result Existence of compact simple Lie groups as building blocks for flat transitive Courant algebroids.
Proves Simpson's conjecture about Higgs bundles and moduli spaces.
problem Proving Simpson's conjecture about Higgs bundles and moduli spaces.
method Analyzes Higgs bundles and their associated moduli spaces, proving Simpson's conjecture.
result Proves Simpson's conjecture about holomorphic Lagrangian submanifolds in de Rham moduli space.
Study of symplectically flat connections and their functionals on smooth manifolds.
problem Understanding symplectically flat connections and their functionals on smooth manifolds.
method Extend symplectically flat connections to ζ-flat connections, introduce functionals with zeroes as symplectically flat connections, study critical points of these functionals, describe characteristic classes of ζ-flat bundles. result Novel geometric flows and characteristic classes of ζ-flat bundles are described. One of the main aims of this article is to give the complete classification of critical metrics of the volume functional on a compact manifold M with boundary ∂M and with harmonic Weyl tensor, which improves the corresponding classification for complete locally conformally flat case, due to Miao and Tam [18…
Connected sum affects crossing numbers of flat virtual knots.
problem Understanding how connected sum impacts the crossing numbers of flat virtual knots.
method Analyzing minimal crossing diagrams and using super-additivity properties.
result Crossing number of flat virtual knots is super-additive under connected sum.
Classifies hypersurfaces with specific curvature properties in 4D space.
problem Classifying hypersurfaces with three distinct principal curvatures in 4D space.
method Used classification results for hypersurfaces in R4, S3imesR, and H3imesR to derive new classifications. result Alternative classification of cyclic conformally flat hypersurfaces in R4. New gravitational solitons and infinite topological manifolds found.
problem Finding new gravitational solitons and Riemannian manifolds.
method Space-periodic solutions of Einstein equations, quotients, Wick rotation.
result Complete Ricci flat Riemannian manifolds of infinite topological type.
We prove a convergence result for a family of Yang-Mills connections over an elliptic K3 surface M as the fibers collapse. In particular, assume M is projective, admits a section, and has singular fibers of Kodaira type I1 and type II. Let Ξtk be a sequence of SU(n) connections on a principal SU(n)…
Develops a method to prove Penrose inequality for half-spaces.
problem Proving the Riemannian Penrose inequality for asymptotically flat half-spaces.
method Doubling procedure for asymptotically flat half-spaces with non-negative scalar curvature and mean-convex boundary.
result Obtains the Penrose-type inequality for dimensions 3 to 7.
Solves wave equation on non-flat harmonic manifolds using Abel transform and Fourier analysis.
problem Wave equation on non-flat harmonic manifolds with specific curvature conditions.
method Explicit representation using inverse dual Abel transform and Fourier transform.
result Shows asymptotic Huygens principle and equidistribution of energy.
Study connects G2-structures to flat connections on compact 3-manifolds.
problem Understanding moduli spaces of G2-structures and flat connections. method Equivalence between moduli spaces of G2-structures and flat connections on compact 3-manifolds. result Equivalence between moduli spaces of G2-structures and flat connections on compact 3-manifolds. Bi-flat F-structures link to differential bicomplexes and Gauss-Manin connections.
problem Understanding the geometric structure of bi-flat F-structures.
method Showed bi-flat F-structures define a differential bicomplex and relate to Gauss-Manin connections.
result Flat connections ablaGM associated with bi-flat structures can be identified with Levi-Civita connections of flat metrics. Study partially hyperbolic flows on flat bundles, proving equivalence for complete affine manifolds.
problem Characterize partially hyperbolic representations of fundamental groups of manifolds.
method Representation theory techniques, focusing on holonomy representations and their properties.
result Show equivalence between partially hyperbolic representations and P-Anosov representations for complete affine manifolds. Let (M,g) be a smooth compact Riemannian manifold of dimension n with smooth boundary ∂M, admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Eu…
New method constructs geometric flat outputs for robotic systems using symmetry.
problem Finding flat outputs for arbitrary robotic systems remains an open question.
method Employing symmetry directly to construct a flat output.
result Demonstrated geometric flat outputs for various robotic systems.
We prove an abstract compactness theorem for a family of generalized Seiberg-Witten equations in dimension three. This result recovers Taubes' compactness theorem for stable flat PSL2(C)-connections as well as the compactness theorem for Seiberg-Witten equations with multiple spinors. Furt…
New rigidity result for CAT(0) spaces of higher rank.
problem Rigidity of CAT(0) spaces of rank at least 2.
method Study of Morse flats and their Tits boundaries.
result CAT(0) spaces of rank at least n contain a regular point in their Tits boundary.
We show a correspondence between the set of all G-invariant projectively flat connections on a homogeneous apace M=G/K, and the one of all {G}^~-invariant flat connections on a homogeneous space {M}^~={G}^~/K, where {G}^~ is a central extension of G.
Flat Yang-Mills connections on pinched manifolds.
problem Stability of Yang-Mills connections on compact manifolds.
method Pinching conditions and weak stability criteria.
result No non-flat weakly stable Yang-Mills connections on δ(n)-pinched compact simply-connected Riemannian manifolds.
Complete classification of Hermitian manifolds with flat Gauduchon connections.
problem Classifying compact Hermitian manifolds with flat Gauduchon connections.
method Analyzing properties of Hermitian manifolds and using Gauduchon connections.
result Established a conjecture about Kähler-like conditions and flatness.
The paper explores Hermitian structures on tangent bundles of affine manifolds with Riemannian metrics.
problem Understanding Hermitian structures on tangent bundles of affine manifolds.
method Analyzes the conditions for various generalized Kähler conditions on (TM,J1,g1) and (TkM,Jk,gk). result Develops a machinery to construct generalized Kähler manifolds.
New invariant from non-acyclic flat connections.
problem Constructing a higher-loop perturbative invariant.
method Integral of a Chern-Simons volume form over moduli space of flat connections.
result Generalization of Chern-Simons invariant to non-acyclic connections.