Geometriae Dedicata最新文献

Coarse geometry studies metric spaces on the large scale. The recently introduced notion of coarse entropy is a tool to study dynamics from the coarse point of view. We prove that all isometries of a given metric space have the same coarse entropy and that this value is a coarse invariant. We call this value the coarse entropy of the space and investigate its connections with other properties of the space. We prove that it can only be either zero or infinity, and although for many spaces this dichotomy coincides with the subexponential–exponential growth dichotomy, there is no relation between coarse entropy and volume growth more generally. We completely characterise this dichotomy for spaces with bounded geometry and for quasi-geodesic spaces. As an application, we provide an example where coarse entropy yields an obstruction for a coarse embedding, where such an embedding is not precluded by considerations of volume growth.

In this paper, we provide new and simpler proofs of two theorems of Gluck and Harrison on contact structures induced by great circle or line fibrations. Furthermore, we prove that a geodesic vector field whose Jacobi tensor is parallel along flow lines (e.g. if the underlying manifold is locally symmetric) induces a contact structure if the ‘mixed’ sectional curvatures are nonnegative, and if a certain nondegeneracy condition holds. Additionally, we prove that in dimension three, contact structures admitting a Reeb flow which is either periodic, isometric, or free and proper, must be universally tight. In particular, we generalise an earlier result of Geiges and the author, by showing that every contact form on ({mathbb {R}}^3) whose Reeb vector field spans a line fibration is necessarily tight. Furthermore, we provide a characterisation of isometric Reeb vector fields. As an application, we recover a result of Kegel and Lange on Seifert fibrations spanned by Reeb vector fields, and we classify closed contact 3-manifolds with isometric Reeb flows (also known as R-contact manifolds) up to diffeomorphism.

In this paper, we are concerned with the classification of complex prime (mathbb {Q})-Fano 3-folds of anti-canonical codimension 4 which are produced, as weighted complete intersections of appropriate weighted projectivizations of certain affine varieties related with (mathbb {P}^{1}times mathbb {P}^{1}times mathbb {P}^{1})-fibrations. Such affine varieties or their appropriate weighted projectivizations are called key varieties for prime (mathbb {Q})-Fano 3-folds. We realize that the equations of the key varieties can be described conceptually by Freudenthal triple systems (FTS, for short). The paper consists of two parts. In Part 1, we revisit the general theory of FTS; the main purpose of Part 1 is to derive the conditions of so called strictly regular elements in FTS so as to fit with our description of key varieties. Then, in Part 2, we define several key varieties for prime (mathbb {Q})-Fano 3-folds from the conditions of strictly regular elements in FTS. Among other things obtained in Part 2, we show that there exists a 14-dimensional factorial affine variety (mathfrak {U}_{mathbb {A}}^{14}) of codimension 4 in an affine 18-space with only Gorenstein terminal singularities, and we construct examples of prime (mathbb {Q})-Fano 3-folds of No.20544 in as reported by Altınok et al. (The graded ring database, http://www.grdb.co.uk/forms/fano3) as weighted complete intersections of the weighted projectivization of (mathfrak {U}_{mathbb {A}}^{14}) in the weighted projective space (mathbb {P}(1^{15},2^{2},3)). We also clarify in Part 2 a relation between (mathfrak {U}_{mathbb {A}}^{14}) and the (G_{2}^{(4)})-cluster variety, which is a key variety for prime (mathbb {Q})-Fano 3-folds constructed in Coughlan and Ducat (Compos. Math. 156:1873-1914, 2020).

Let X be a smooth projective threefold and let H be an ample line bundle on X. We investigate the existence of vector bundles on X which are (mu )-stable with respect to an ample divisor (H_{epsilon }=H+epsilon D) for sufficiiently small (epsilon >0) where D is a divisor with (Dcdot H^2=0). In particular, when X is a Fano conic bundle over a rational surface, we show that there exists a family ({E_n}) of (H_{epsilon })-stable vector bundles with (c_1(E_n)=0) and (c_2(E_n)cdot H) becomes arbitrarily large as n goes to infinity.

We generalise results about isometric group actions on metric spaces and their fundamental regions to the context of merely continuous group actions. In particular, we obtain results that yield the relative compactness of a fundamental region for a cocompact group action. As a consequence, we obtain a criterion for a cocompact cyclic group of semi-Riemannian homotheties to be inessential.

We introduce the concept of boundary rigidity for Gromov hyperbolic spaces. We show that a proper geodesic Gromov hyperbolic space with a pole is boundary rigid if and only if its Gromov boundary is uniformly perfect. As an application, we show that for a non-compact Gromov hyperbolic complete Riemannian manifold or a Gromov hyperbolic uniform graph, boundary rigidity is equivalent to having positive Cheeger isoperimetric constant and also to being nonamenable. Moreover, several hyperbolic fillings of compact metric spaces are proved to be boundary rigid if and only if the metric spaces are uniformly perfect. Also, boundary rigidity is shown to be equivalent to being geodesically rich, a concept introduced by Shchur (J Funct Anal 264(3):815–836, 2013).

We provide minimal constructions of meanders with particular combinatorics. Using these meanders, we give minimal constructions of hyperelliptic pillowcase covers with a single horizontal cylinder and simultaneously a single vertical cylinder so that one or both of the core curves are separating curves on the underlying surface. In the case where both of the core curves are separating, we use these surfaces in a construction of Aougab–Taylor in order to prove that for any hyperelliptic connected component of the moduli space of quadratic differentials with no poles there exist ratio-optimising pseudo-Anosovs lying arbitrarily deep in the Johnson filtration and stabilising the Teichmüller disk of a quadratic differential lying in this connected component.

We study the closure of horocycles on rank 1 nonpositively curved surfaces with finitely generated fundamental group. Each horocycle is closed or dense on a certain subset of the unit tangent bundle. In fact, we classify each half-horocycle in terms of the associated geodesic rays. We also determine the nonwandering set of the horocyclic flow and characterize the surfaces admitting a minimal set for this flow.

We prove nonemptyness of domains of proper discontinuity of Anosov groups of affine Lorentzian transformations of ({mathbb R}^n).

We will show that in any characteristic every nonsingular cubic surface is projectively isomorphic to the surface given by the octanomial normal form. This normal form is discovered in Panizzut et al. (LeMatematiche 75(2), 2020) only in characteristic 0 by exhaustive computer search. We offer a conceptual explanation that has the added benefit of being characteristic free. As an application, we give octanomial normal forms of the strata of the coarse moduli space of cubic surfaces defined in Dolgachev and Duncan (Compos Math 25(1):1–59, 1972) which preserve most specialization with respect to automorphisms.