arXiv research

This paper finds a global surface of section in dynamically convex L(p,p-1) using ECH.

Classifies surfaces of section for Seifert fibrations.

We exhibit sufficient conditions for a finite collection of periodic orbits of a Reeb flow on a closed $3$-manifold to bound a positive global surface of section with genus zero. These conditions turn out to be $C^\infty$-generically necessary. Moreover, they involve linking assumptions on periodic orbits with Conley-Z…

We survey some recent developments in the quest for global surfaces of section for Reeb flows in dimension three using methods from Symplectic Topology. We focus on applications to geometry, including existence of closed geodesics and sharp systolic inequalities. Applications to topology and celestial mechanics are als…

The restricted planar three-body problem has a rich history, yet many unanswered questions still remain. In the present paper we prove the existence of a global surface of section near the smaller body in a new range of energies and mass ratios for which the Hill's region still has three connected components. The appro…

A Killing submersion is a Riemannian submersion from a 3-manifold to a surface, both connected and orientable, whose fibres are the integral curves of a Killing vector field, not necessarily unitary. The first part of this paper deals with the classification of all Killing submersions in terms of two geometric function…

We consider minimal compact complex surfaces S with Betti numbers b_1=1 and n=b_2>0. A theorem of Donaldson gives n exceptional line bundles. We prove that if in a deformation, these line bundles have sections, S is a degeneration of blown-up Hopf surfaces. Besides, if there exists an integer m>0 and a flat line bundle…

This thesis predicts the distribution of smoothed zeros of random sections on line bundles.

Solutions of Hitchin's self-duality equations corresponds to special real sections in the Deligne-Hitchin moduli space -- twistor lines. A question posed by Simpson in 1997 asks whether all real sections give rise to global solutions of the self-duality equations. An affirmative answer would allow for complex analytic …

Let $S$ be a compact hyperbolic Riemann surface of genus $g \geq 2$. We call a systole a shortest simple closed geodesic in $S$ and denote by $\mathop{sys}(S)$ its length. Let $\mathop{msys(g)}$ be the maximal value that $\mathop{sys}(\cdot)$ can attain among the compact Riemann surfaces of genus $g$. We call a (global…

The correspondence between 2-parameter families of oriented lines in ${\Bbb{R}}^3$ and surfaces in $T{\Bbb{P}}^1$ is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences generated by global sections of $T{\Bbb{P}}^1$ are investigated and a number of theorems…

New invariants defined for volume-preserving flows on 3-manifolds.

Study calculates global sections on complex curves.

New global section found for geodesic flows on convex hypersurfaces.

The theory of complete surfaces of (nonzero) constant mean curvature in $\RR^3$ has progressed markedly in the last decade. This paper surveys a number of these developments in the setting of Alexandrov embedded surfaces; the focus is on gluing constructions and moduli space theory, and the analytic techniques on which…

Study describes global sections of chiral de Rham complex on compact Ricci-flat Kähler manifolds.

A family of holomorphic vector bundles is constructed on a complex manifold $X$. The space of the holomorphic sections of these bundles are calculated in certain cases. As an application, if $X$ is an $N$-dimensional compact Kähler manifold with holonomy group $SU(N)$, the space of holomorphic vector fields on its jet …

The paper proves a special case of Yau's conjecture for Kähler surfaces.

We give necessary and sufficient conditions for a closed connected co-orientable contact $3$-manifold $(M,ξ)$ to be a standard lens space based on assumptions on the Reeb flow associated to a defining contact form. Our methods also provide rational global surfaces of section for nondegenerate Reeb flows on $(L(p,q),ξ_{…

The paper proves the existence of surfaces of section for geodesic flows on closed surfaces.

In this paper we study lightlike surfaces of Minkowski 3- space such that they have degenerate or non-degenerate planar normal sections. We first show that every lightlike surface of Minkowski $3-$ space has degenerate planar normal sections. Then we study lightlike surfaces with non-degenerate planar normal sections a…

Geodesic spheres are the only quasicomplete surfaces in 3-space-forms.

We study the geodesic flow on the global holomorphic sections of the bundle $π:{TS}^2\to {S}^2$ induced by the neutral Kähler metric on the space of oriented lines of ${\Bbb{R}}^3$, which we identify with ${TS}^2$. This flow is shown to be completely integrable when the sections are symplectic and the behaviour of the …

A new method for clearing liability networks using sheaves on directed hypergraphs.

Let $V$ be a maximal globally hyperbolic flat $n+1$--dimensional space--time with compact Cauchy surface of hyperbolic type. We prove that $V$ is globally foliated by constant mean curvature hypersurfaces $M_τ$, with mean curvature $τ$ taking all values in $(-\infty, 0)$. For $n \geq 3$, define the rescaled volume of $…

Lorentzian Ptolemy inequality linked to curvature bounds.

The space of the global sections of chiral de Rham complex on a compact Ricci-flat Kähler manifold is calculated and it is expressed as an invariant subspace of a $βγ-bc$ system under the action of certain Lie algebra.

In this paper, we study the global behaviour of contact structures on oriented manifolds V which are circle bundles over a closed orientable surface S of genus g>0. We establish in particular contact analogs of a number of classical results about foliations due to Milnor, Wood, Thurston, Matsumoto, and Ghys. In Section…

The paper classifies time-like surfaces in a static space-time.

In this note, we study submanifold geometry of the Atiyah-Hitchin manifold, the double cover of the $2$-monopole moduli space. When the manifold is naturally identified as the total space of a line bundle over $S^2$, the zero section is a distinguished minimal $2$-sphere of considerable interest. In particular, there h…

The Weil-Petersson metric for the moduli space of Riemann surfaces has negative sectional curvature. Surfaces represented in the complement of a compact set in the moduli space have short geodesics. At such surfaces the Weil-Petersson metric is approximately a product metric. An almost product metric has sections with …

Study proves properties of compact Hermitian surfaces with specific curvature conditions.

The paper addresses how to add points to existing configurations on surfaces without disrupting continuity.

Study on surfaces with constant curvature under a specific connection.

Study on harmonic metrics for rank 3 Higgs bundles in Hitchin section.

The paper classifies translation surfaces with constant curvature in a specific connection.

New maps show some surfaces can't be sections of 4D spheres.

Proves section conjecture for curves and surface bundles over various fields.

The paper shows how different geodesic flows on surfaces can be mapped to each other.

These are notes based on a mini-course at the conference RIEMain in Contact, held in Cagliari, Sardinia, in June 2018. The main theme is the connection between Reeb dynamics and topology. Topics discussed include traps for Reeb flows, plugs for Hamiltonian flows, the Weinstein conjecture, Reeb flows with finite numbers…

Study embeds ruled surfaces into symplectic manifolds, finds Stein fillability results.

For orthonormal normal sections of two-dimensional immersions in R^4 we define torsion coefficients and a functional for the total torsion. We discuss normal sections which are critical for this functional. In particular, a global estimate for the torsion coefficients of a critical normal section in terms of the curvat…

The paper introduces a new concept of frame vorticity and uses it to find optimal sections in specific geometric settings.

Unified view of integrable systems linking CMC, isothermic, and Willmore surfaces.

We prove that a compact Hermitian manifold with semi-positive but not identically zero holomorphic sectional curvature has Kodaira dimension $-\infty$. As applications, we show that Kodaira surfaces and hyperelliptic surfaces can not admit Hermitian metrics with semi-positive holomorphic sectional curvature although th…

Study Higgs sections and flat sections for nonlinear harmonic bundles.

In a recent paper, Eichmair, Galloway and Pollack have proved a Gannon-Lee-type singularity theorem based on the existence of marginally outer trapped surfaces (MOTS) on noncompact initial data sets for globally hyperbolic spacetimes. However, one might wonder whether the corresponding incomplete geodesics could still …

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler. We also give examples of complete non-Kähler metrics with pointwise negative c…