arXiv research

Jet bundles as higher-order polarised $k$-contact manifolds

Develops k-contact geometry theory for field theories.

The paper studies Einstein-Hilbert functional and its relation to K-semistability.

Analytic K-semistability connects curvature to metric existence.

The study finds resonance points in polarised curves with polynomial conserved quantities.

We provide a sufficient condition for polarisations of Fano varieties to be K-stable in terms of Tian's alpha invariant, which uses the log canonical threshold to measure singularities of divisors in the linear system associated to the polarisation. This generalises a result of Odaka-Sano in the anti-canonically polari…

For a stratified group $G$, we construct a class of polarised Lie groups, which we call modifications of $G$, that are locally contactomorphic to it. Vice versa, we show that if a polarised group is locally contactomorphic to a stratified group $G$, whose Lie algebra has finite Tanaka prolongation, then it must be a mo…

Introduces valuative stability for polarised varieties, equivalent to K-stability.

We construct a moduli space of polarised manifolds which admit a constant scalar curvature Kähler metric. We show that this space admits a natural Kähler metric.

Abstract: Survey on contact submanifolds.

Discusses the tight versus overtwisted dichotomy in 3D contact geometry.

Spinors help study unique five-dimensional contact structures.

Geodesic rays and chordal distances link algebraic and geometric properties of positive metrics.

Infinite-dimensional contact geometry explored.

New discrete models for constant mean curvature surfaces and tori.

Study contact geometry of symplectic divisors, invariant under specific transformations.

A new method simplifies contact Hamiltonian mechanics.

Characterizes Anosov flows in 3D using symplectic and contact geometry.

The study explores polarized deformations of SKT Calabi-Yau manifolds using Aeppli classes.

Paper proves isoperimetric inequality for Minkowski spacetime.

A dictionary connects symplectic to contact geometry, with applications to complex and G-structures.

Study new $k$-contact distributions and Lie systems.

Study explores weak generalized K-contact structures in contact metric spaces.

Local flatness theorem for paraquaternionic contact structures.

In this paper the notion of the intrinsic geometry of an almost contact metric manifold is introduced. Description of some classes of spaces with almost contact metric structures in terms of the intrinsic geometry is given. A new type of almost contact metric spaces, more precisely, Hermitian almost contact metric spac…

Introduces stability conditions for polarized varieties, linking to K-stability.

Defines generalized Sasakian structures in contact geometry.

Established a generalized Boothby-Wang theorem in contact geometry.

These notes are an expanded version of an introductory lecture on contact geometry given at the 2001 Georgia Topology Conference. They are intended to present some of the "topological" aspects of three dimensional contact geometry.

On décrit ici des relations entre la géométrie globale des variétés de contact closes et celle de certaines variétés symplectiques, à savoir les variétés de Stein compactes. L'origine de ces relations est l'existence de livres ouverts adaptés aux structures de contact. We discuss relations between the global geometry o…

We identify the difference between the CM polarisation and the Chow polarisation on the ``Hilbert scheme''. As a consequence, we give a numerical criterion for the CM stability as in Mumfords' G.I.T.. Also, we write down an explicit formula for the generalised futaki invariant interms of weights and multiplicities of t…

This article sketches various ideas in contact geometry that have become useful in low-dimensional topology. Specifically we (1) outline the proof of Eliashberg and Thurston's results concerning perturbations of foliatoins into contact structures, (2) discuss Eliashberg and Weinstein's symplectic handle attachments, an…

Lie contact structures generalize the classical Lie sphere geometry of oriented hyperspheres in the standard sphere. They can be equivalently described as parabolic geometries corresponding to the contact grading of orthogonal real Lie algebra. It follows the underlying geometric structure can be interpreted in several…

We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope $μ$ for a projective manifold and for each of its subschemes, and show that if $X$ is cscK then $μ(Z)\leμ(X)$ for all subschemes $Z$. This gives man…

Proves a conjecture for Calabi-Yau manifolds.

In this paper, the notion of an almost contact Kählerian structure is introduced. The interior geometry of almost contact Kählerian spaces is investigated. On the zero-curvature distribution of an almost contact metric structure, as on the total space of a vector bundle, an almost contact Kählerian structure is obtaine…

It is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group. We prove partial results towards this conjecture. We also show that it would give a new proof of the K-polystability of constant…

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,ξ) admits a …

Equivalence proven between divisorial stability and quotient log divisorial stability.

In this thesis we study the relationship between the existence of canonical metrics on a complex manifold and stability in the sense of geometric invariant theory. We introduce a modification of K-stability of a polarised variety which we conjecture to be equivalent to the existence of an extremal metric in the polaris…

Characterizes Anosov flows via contact geometry.

A taut contact sphere on a 3-manifold is a linear 2-sphere of contact forms, all defining the same volume form. In the present paper we completely determine the moduli of taut contact spheres on compact left-quotients of SU(2) (the only closed manifolds admitting such structures). We also show that the moduli space of …

New surgery method preserves Anosov flow properties using bi-contact geometry.

Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of…

Study on null submanifolds in indefinite complex contact geometry.

Study new warped product metric in contact geometry.

Lecture notes introduce contact complete integrability for odd-dimensional manifolds.

The paper studies deformations of Calabi-Yau manifolds using Gauduchon metrics.