Recursos de colección
Project Euclid (Hosted at Cornell University Library) (203.669 recursos)
Geometry & Topology
Geometry & Topology
Di Cerbo, Luca F; Stover, Matthew
We classify the minimum-volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces, and they are all arithmetic; ie they are associated with quotients of the ball by an arithmetic lattice. Moreover, the associated lattices are all commensurable. The first compactification, originally discovered by Hirzebruch, is the blowup of an abelian surface at one point. The others are bielliptic surfaces blown up at one point. The bielliptic examples are new and are the first known examples of smooth toroidal compactifications birational to bielliptic surfaces.
van Limbeek, Wouter
Let [math] be a closed manifold that admits a self-cover [math] of degree [math] . We say [math] is strongly regular if all iterates [math] are regular covers. In this case, we establish an algebraic structure theorem for the fundamental group of [math] : We prove that [math] surjects onto a nontrivial free abelian group [math] , and the self-cover is induced by a linear endomorphism of [math] . Under further hypotheses we show that a finite cover of [math] admits the structure of a principal torus bundle. We show that this applies when [math] is Kähler and [math] is...
Kojima, Sadayoshi; McShane, Greg
Thanks to a recent result by Jean-Marc Schlenker, we establish an explicit linear inequality between the normalized entropies of pseudo-Anosov automorphisms and the hyperbolic volumes of their mapping tori. As corollaries, we give an improved lower bound for values of entropies of pseudo-Anosovs on a surface with fixed topology, and a proof of a slightly weaker version of the result by Farb, Leininger and Margalit first, and by Agol later, on finiteness of cusped manifolds generating surface automorphisms with small normalized entropies. Also, we present an analogous linear inequality between the Weil–Petersson translation distance of a pseudo-Anosov map (normalized by...
Murphy, Emmy; Siegel, Kyler
We introduce a class of Weinstein domains which are sublevel sets of flexible Weinstein manifolds but are not themselves flexible. These manifolds exhibit rather subtle behavior with respect to both holomorphic curve invariants and symplectic flexibility. We construct a large class of examples and prove that every flexible Weinstein manifold can be Weinstein homotoped to have a nonflexible sublevel set.
Lou, Beicheng; Tan, Ser Peow; Vo, Anh Duc
We show that there are infinitely many commensurability classes of pseudomodular groups, thus answering a question raised by Long and Reid. These are Fuchsian groups whose cusp set is all of the rationals but which are not commensurable to the modular group. We do this by introducing a general construction for the fundamental domains of Fuchsian groups obtained by gluing together marked ideal triangular tiles, which we call hyperbolic jigsaw groups.
Eskin, Alex; Kontsevich, Maxim; Möller, Martin; Zorich, Anton
Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top [math] Lyapunov exponents of the flat bundle is always greater than or equal to the degree of any rank- [math] holomorphic subbundle. We generalize the original context from Teichmüller curves to any local system over a curve with nonexpanding cusp monodromies. As an application we obtain the large-genus limits of individual Lyapunov exponents in hyperelliptic strata of abelian differentials, which Fei Yu proved conditionally on his conjecture. ¶ Understanding the case of equality with the degrees of subbundle coming from...
Inspired by the Katz–Mazur theorem on crystalline cohomology and by the numerical experiments of Eskin, Kontsevich and Zorich, we conjecture that the polygon of the Lyapunov spectrum lies above (or on) the Harder–Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons and the integral of eigenvalues of the curvature of the Hodge bundle by using the works of Atiyah and Bott, Forni, and Möller. We obtain several applications to Teichmüller dynamics conditional on the conjecture.
We suggest a method to construct new examples of partially hyperbolic diffeomorphisms. We begin with a partially hyperbolic diffeomorphism [math] which leaves invariant a submanifold [math] . We assume that [math] is an Anosov submanifold for [math] , that is, the restriction [math] is an Anosov diffeomorphism and the center distribution is transverse to [math] . By replacing each point in [math] with the projective space (real or complex) of lines normal to [math] , we obtain the blow-up [math] . Replacing [math] with [math] amounts to a surgery on the neighborhood of [math] which alters the topology of the...
Heller, Jeremiah; Ormsby, Kyle M
Let [math] be a field of characteristic different from [math] . We establish surjectivity of Balmer’s comparison map ¶ ¶ from the tensor triangular spectrum of the homotopy category of compact motivic spectra to the homogeneous Zariski spectrum of Milnor–Witt [math] –theory. We also comment on the tensor triangular geometry of compact cellular motivic spectra, producing in particular novel field spectra in this category. We conclude with a list of questions about the structure of the tensor triangular spectrum of...
Passeggi, Alejandro; Potrie, Rafael; Sambarino, Martín
We show that if [math] is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of [math] is positive. Further, the entropy is shown to be associated to a [math] –robust rotational horseshoe. On the other hand, we construct examples of annular homeomorphisms with such attractors for which the rotation interval is uniformly large but the entropy approaches zero as much as desired. ¶ The developed techniques allow us to obtain similar results in the context of Birkhoff attractors.
Qin, Lizhen; Wang, Botong
We construct a family of [math] –dimensional compact manifolds [math] which are simultaneously diffeomorphic to complex Calabi–Yau manifolds and symplectic Calabi–Yau manifolds. They have fundamental groups [math] , their odd-degree Betti numbers are even, they satisfy the hard Lefschetz property, and their real homotopy types are formal. However, [math] is never homotopy equivalent to a compact Kähler manifold for any topological space [math] . The main ingredient to show the non-Kählerness is a structure theorem of cohomology jump loci due to the second author.
Khandhawit, Tirasan; Lin, Jianfeng; Sasahira, Hirofumi
Let [math] be a closed and oriented [math] –manifold. We define different versions of unfolded Seiberg–Witten Floer spectra for [math] . These invariants generalize Manolescu’s Seiberg–Witten Floer spectrum for rational homology [math] –spheres. We also compute some examples when [math] is a Seifert space.
We construct bi-Lipschitz embeddings into Euclidean space for bounded-diameter subsets of manifolds and orbifolds of bounded curvature. The distortion and dimension of such embeddings is bounded by diameter, curvature and dimension alone. We also construct global bi-Lipschitz embeddings for spaces of the form [math] , where [math] is a discrete group acting properly discontinuously and by isometries on [math] . This generalizes results of Naor and Khot. Our approach is based on analyzing the structure of a bounded-curvature manifold at various scales by specializing methods from collapsing theory to a certain class of model spaces. In the process, we develop...
We introduce the notion of an operator category and two different models for homotopy theory of [math] –operads over an operator category — one of which extends Lurie’s theory of [math] –operads, the other of which is completely new, even in the commutative setting. We define perfect operator categories, and we describe a category [math] attached to a perfect operator category [math] that provides Segal maps. We define a wreath product of operator categories and a form of the Boardman–Vogt tensor product that lies over it. We then give examples of operator categories that provide universal properties for the operads...
In this paper, we prove that if an asymptotically Euclidean manifold with nonnegative scalar curvature has long-time existence of Ricci flow, the ADM mass is nonnegative. We also give an independent proof of the positive mass theorem in dimension three.
Mann, Etienne; Robalo, Marco
Let [math] be a smooth projective variety. Using the idea of brane actions discovered by Toën, we construct a lax associative action of the operad of stable curves of genus zero on the variety [math] seen as an object in correspondences in derived stacks. This action encodes the Gromov–Witten theory of [math] in purely geometrical terms and induces an action on the derived category [math] which allows us to recover the quantum K–theory of Givental and Lee.
Studenmund, Daniel; Wortman, Kevin
Let [math] be a global function field of characteristic [math] , and let [math] be a finite-index subgroup of an arithmetic group defined with respect to [math] and such that any torsion element of [math] is a [math] –torsion element. We define semiduality groups, and we show that [math] is a [math] –semiduality group if [math] acts as a lattice on a product of trees. We also give other examples of semiduality groups, including lamplighter groups, Diestel–Leader groups, and countable sums of finite groups.
Martelli, Bruno; Riolo, Stefano
We introduce and study some deformations of complete finite-volume hyperbolic four-manifolds that may be interpreted as four-dimensional analogues of Thurston’s hyperbolic Dehn filling. ¶ We construct in particular an analytic path of complete, finite-volume cone four-manifolds [math] that interpolates between two hyperbolic four-manifolds [math] and [math] with the same volume [math] . The deformation looks like the familiar hyperbolic Dehn filling paths that occur in dimension three, where the cone angle of a core simple closed geodesic varies monotonically from [math] to [math] . Here, the singularity of [math] is an immersed geodesic surface whose cone angles also vary monotonically from...
Ballas, Samuel A; Danciger, Jeffrey; Lee, Gye-Seon
Y Benoist proved that if a closed three-manifold [math] admits an indecomposable convex real projective structure, then [math] is topologically the union along tori and Klein bottles of finitely many submanifolds each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial results in the direction of a potential converse to Benoist’s theorem. We show that a cusped hyperbolic three-manifold may, under certain assumptions, be deformed to convex projective structures with totally geodesic torus boundary. Such structures may be convexly glued together whenever the geometry at the boundary matches up. In particular, we prove...
Let [math] be a finite group. We show that the rational equivariant homotopy groups of symmetric products of the [math] –equivariant sphere spectrum are naturally isomorphic to the rational homology groups of certain subcomplexes of the subgroup lattice of [math] .