arXiv research

Using zippered rectangle coordinates we parametrize a Poincaré section for horocycle flow on the space of genus 2 translation surfaces with one singular cone point of angle $6π$. In addition, we bound the return time under horocycle flow to this Poincaré section by examining a subset of surfaces where a certain sum of …

This paper is a continuation of the paper F. A. Arias and M. Malakhaltsev "A generalization of the Gauss-Bonnet and Hopf-Poincaré theorems", ArXiv:1510.01395 [MathDG] 5 Oct 2015. Let $π: E \to M$ be a locally trivial fiber bundle over a two-dimensional manifold $M$, and $Σ\subset M$ be a discrete subset. A subset $Q \s…

We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, $n$-dimensional Riemannian manifold with pinched negative sectional curvature follows …

Expands Euler-Poincare characteristic to supergeometry.

We construct a Poincaré section for the horocycle flow on the modular surface $SL(2, \R)/SL(2, \Z)$, and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of pe…

Let $(M,g)$ be a compact Kähler-Einstein manifold with $c_1 > 0$. Denote by $K\to M$ the canonical line-bundle, with total space $X$, and $X_0$ the singular space obtained by blowing down $X$ along its zero section. We employ a construction by Page and Pope and discuss an interesting multi-parameter family of Poincaré-…

The paper defines and analyzes Kähler metrics near a compact manifold, showing their deviation from Poincaré-type metrics.

The article proves a Poincaré inequality for hypersurfaces and applies it to rigidity results.

New proof shows certain 4D metrics are non-degenerate if curvature is negative definite.

We consider a locally trivial fiber bundle $π: E \to M$ over a compact oriented two-dimensional manifold $M$, and a section $s$ of this bundle defined over $M \setminus Σ$, where $Σ$ is a discrete subset of $M$. We call the set $Σ$ the set of singularities of the section $s : M \setminus Σ\to E$. We assume that the beh…

The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids. From a variational principle we derive the discrete Euler-Lagrange equations and we introduce a symplectic 2-section, which is preserved by the Lagrange evolution operator. In terms of the discrete Leg…

We develop a general theory for the existence of extremal Kähler metrics of Poincaré type in the sense of Auvray, defined on the complement of a toric divisor of a polarized toric variety. In the case when the divisor is smooth, we obtain a list of necessary conditions which must be satisfied for such a metric to exist…

Study Bergman kernels on complex hyperbolic cusps, generalizing previous results.

Automorphic forms on a bounded symmetric domain D=G/K can be viewed as holomorphic sections of $L^{\otimes k}$, where L is a quantizing line bundle on a compact quotient of D and k is a positive integer. Let $Γ$ be a cocompact discrete subgroup of SU(n,1) which acts freely on SU(n,1)/U(n). We suggest a construction of …

Paper analyzes origami slope gaps and their distribution, finding a unique pattern.

We prove the existence of a $C^{1,1}$ conformally compact Einstein metric on the ball that has asymptotic sectional curvature decay to $-1$ plus terms of order $e^{-2r}$ where $r$ is the distance from any fixed compact set. This metric has no $C^2$ conformal compactification.

Following Bryant, Ferry, Mio and Weinberger we construct generalized manifolds as limits of controlled sequences p_i: X_i --> X_{i-1} : i = 1,2,... of controlled Poincaré spaces. The basic ingredient is the epsilon-delta-surgery sequence recently proved by Pedersen, Quinn and Ranicki. Since one has to apply it not only…

Extends transversality to supergeometry, proving stability and genericity.

We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler--Poincaré equations defined on the Virasoro-Bott group, by using the inverse map (also called `back-to-labels' map). This family contains as special cases the well-known Korteweg-de Vries, Ca…

In this paper we shall study smooth submanifolds immersed in a k-step Carnot group G of homogeneous dimension Q. Among other results, we shall prove an isoperimetric inequality for the case of a $C^2$-smooth compact hypersurface S with - or without - boundary $\partial S$; S and $\partial S$ are endowed with their homo…

In the first part of this paper we prove some new Poincaré inequalities, with explicit constants, for domains of any hypersurface of a Riemannian manifold with sectional curvatures bounded from above. This inequalities involve the first and the second symmetric functions of the eigenvalues of the second fundamental for…

The paper proves inequalities for varifolds on Riemannian manifolds.

Let $X$ be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer $k\geq 2$, we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle $Ω^{\otimes k}_X$, where $Ø$ is the holomorphic cotangent bundle of $X$. Our first main result estimates the corresponding B…

The paper embeds CR manifolds into twistor spaces and constructs neutral hyperkähler metrics.

We analyze the slope gap distribution of Veech surfaces, finding finite non-analytic points and quadratic tail decay.

New Finsler flow on 2-torus has chaotic dynamics.

The article investigates conditions for isomorphism of singular tangent bundles.

New curvature K(x) measures manifold properties without integrals.

Sharp bounds derived for eigenvalues on specific geometric spaces.

Generalizes double transgression formulas on complex manifolds.

The paper explores how vector fields relate to volume in geometric contexts.

The study finds infinitely many counterexamples to a generalized Double Soul Conjecture.

The behavior of geodesic curves on even seemingly simple surfaces can be surprisingly complex. In this paper we use the Hamiltonian formulation of the geodesic equations to analyze their integrability properties. In particular, we examine the behavior of geodesics on surfaces defined by the spherical harmonics. Using t…

We explicitly compute the limiting gap distribution for slopes of saddle connections on the flat surface associated to the regular octagon with opposite sides identified. This is the first such computation where the Veech group of the translation surface has multiple cusps. We also show how to parametrize a Poincaré se…

This work is a short, self-contained introduction to subriemannian geometry with special emphasis on Chow's Theorem. As an application, a regularity result for the Poincaré Lemma is presented. At the beginning, the definitions of a subriemannian geometry, horizontal vector fields and horizontal curves are given. Then t…

Discrete Lagrange problems solved with Lie group constraints.

The jet formalism for Classical Field theories is extended to the setting of Lie algebroids. We define the analog of the concept of jet of a section of a bundle and we study some of the geometric structures of the jet manifold. When a Lagrangian function is given, we find the equations of motion in terms of a Cartan fo…

In this short Note, we establish that the constant $C_1$ in Lemma $0.4$ of the correction (Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture, arXiv/math/DG:1512.00699 (2015)) by John Morgan and Gang Tian to their Clay Institute Monograph (Ricci Flow and the Poincare Conjecture, vol. 3, Clay Mathemati…

This text presents some basic notions in symplectic geometry, Poisson geometry, Hamiltonian systems, Lie algebras and Lie groups actions on symplectic or Poisson manifolds, momentum maps and their use for the reduction of Hamiltonian systems. It should be accessible to readers with a general knowledge of basic notions …

Proves Poincaré duality for Hopf algebroids with bijective antipode.

This note corrects a mistake in the original book in the evolution equations of total curvature for the curve-shrinking flow in an ambient Ricci Flow. The resulting upper bound for the evolution of total curvature is an exponential bound in time. The change involves the multiplicative constant. Here we show that it dep…

New spaces help connect manifold structures on equivariant Poincaré spaces.

Proves Poincaré surgery theorem using homotopy theory.

Extends Poincaré-Lefschetz duality to pairs of ∞-categories.

Uniform Poincaré inequalities established for various metric spaces.

Proof outlined for 4D smooth Poincaré conjecture.

Galatius, Madsen, Tillmann and Weiss have identified the homotopy type of the classifying space of the cobordism category with objects (d-1)-dimensional manifolds embedded in R^\infty. In this paper we apply the techniques of spaces of manifolds, as developed by the author and Galatius, to identify the homotopy type of…

This survey starts with the historical landmarks leading to the study of principal configurations on surfaces, their structural stability and further generalizations. Here it is pointed out that in the work of Monge, 1796, are found elements of the qualitative theory of differential equations (QTDE), founded by Poincar…