Geometriae Dedicata最新文献

Gromov introduced two distance functions, the box distance and the observable distance, on the space of isomorphism classes of metric measure spaces and developed the convergence theory of metric measure spaces. We investigate several topological properties on the space equipped with these distance functions toward a deep understanding of convergence theory.

Let M be a smooth 4-manifold underlying some del Pezzo surface of degree (d ge 6). We consider the smooth Nielsen realization problem for M: which finite subgroups of ({{,textrm{Mod},}}(M) = pi _0({{,textrm{Homeo},}}^+(M))) have lifts to ({{,textrm{Diff},}}^+(M) le {{,textrm{Homeo},}}^+(M)) under the quotient map (pi : {{,textrm{Homeo},}}^+(M) rightarrow {{,textrm{Mod},}}(M))? We give a complete classification of such finite subgroups of ({{,textrm{Mod},}}(M)) for (d ge 7) and a partial answer for (d = 6). For the cases (d ge 8), the quotient map (pi ) admits a section with image contained in ({{,textrm{Diff},}}^+(M)). For the case (d = 7), we show that all finite order elements of ({{,textrm{Mod},}}(M)) have lifts to ({{,textrm{Diff},}}^+(M)), but there are finite subgroups of ({{,textrm{Mod},}}(M)) that do not lift to ({{,textrm{Diff},}}^+(M)). We prove that the condition of whether a finite subgroup (G le {{,textrm{Mod},}}(M)) lifts to ({{,textrm{Diff},}}^+(M)) is equivalent to the existence of a certain equivariant connected sum realizing G. For the case (d = 6), we show this equivalence for all maximal finite subgroups (G le {{,textrm{Mod},}}(M)).

This article aims to pursue and generalize, by using the global point of view offered by the stacks, the local study made by Ghys (J für die reine und angewandte Mathematik 468:113–138, 1995) concerning the deformations of complex structures of compact quotients of ({text {SL}}_2({mathbb {C}})). In his article, Ghys showed that the analytic germ of the representation variety ({mathcal {R}}(varGamma ):={text {Hom}}(varGamma ,{text {SL}}_2({mathbb {C}}))) of (varGamma ) in ({text {SL}}_2({mathbb {C}})), pointed at the trivial morphism, determines the Kuranishi space of ({text {SL}}_2({mathbb {C}})/varGamma ). In this note, we show that the tautological family above a Zariski analytic open subset V in ({mathcal {R}}(varGamma )) remains complete. Moreover, the computation of the isotropy group of a complex structure in Teichmüller space, allows us to affirm that the quotient stack ([V/{text {SL}}_2({mathbb {C}})]) is an open substack of the Teichmüller stack of ({text {SL}}_2({mathbb {C}})/varGamma ).

The Epstein-Penner convex hull construction associates to every decorated punctured hyperbolic surface a convex set in the Minkowski space. It works in the de Sitter and anti-de Sitter spaces as well. In these three spaces, the quotient of the spacelike boundary part of the convex set has an induced Euclidean, spherical and hyperbolic metric, respectively, with conical singularities. We show that this gives a bijection from the decorated Teichmüller space to a moduli space of such metrics in the Euclidean and hyperbolic cases, as well as a bijection between specific subspaces of them in the spherical case. Moreover, varying the decoration of a fixed hyperbolic surface corresponds to a discrete conformal change of the metric. This gives a new 3-dimensional interpretation of discrete conformality which is in a sense inverse to the Bobenko-Pinkall-Springborn interpretation.

We bound the derivative of complex length of a geodesic under variation of the projective structure on a closed surface in terms of the norm of the Schwarzian in a neighborhood of the geodesic. One application is to cone-manifold deformations of acylindrical hyperbolic 3-manifolds.

We construct a halfspace in the bidisk, whose boundary acts like a bisector. As an application, we build a fundamental domain consisting of such halfspaces for the action of groups that project to Schottky groups in both factors.

We study notions of asymptotic regularity for a class of minimal submanifolds of complex hyperbolic space that includes minimal Lagrangian submanifolds. As an application, we show a relationship between an appropriate formulation of Colding-Minicozzi entropy and a quantity we call the CR-volume that is computed from the asymptotic geometry of such submanifolds.

Abstract

In 1979, for each signature for Fuchsian groups of the first kind, Bowen and Series constructed an explicit fundamental domain for one group of the signature, and from this a function on ({mathbb {S}}^1) tightly associated with this group. In general, their fundamental domain enjoys what has since been called both the ‘extension property’ and the ‘even corners property’. We determine the exact set of signatures for cocompact triangle groups for which this property can hold for any convex fundamental domain, and verify that for this restricted set, the Bowen-Series fundamental domain does have the property. To each Bowen-Series function in this corrected setting, we naturally associate four continuous deformation families of circle functions. We show that each of these functions is aperiodic if and only if it is surjective; and, is finite Markov if and only if its natural parameter is a hyperbolic fixed point of the triangle group at hand.

Abstract

Let S be an oriented, closed surface of genus g. The mapping class group of S is the group of orientation preserving homeomorphisms of S modulo isotopy. In 1997, Looijenga introduced the Prym representations, which are virtual representations of the mapping class group that depend on a finite, abelian group. Let V be a genus g handlebody with boundary S. The handlebody group is the subgroup of those mapping classes of S that extend over V. The twist group is the subgroup of the handlebody group generated by twists about meridians. Here, we restrict the Prym representations to the handlebody group and further to the twist group. We determine the image of the representations in the cyclic case.

For each lattice complex hyperbolic triangle group, we study the Fuchsian stabilizers of (reprentatives of each group orbit of) mirrors of complex reflections. We give explicit generators for the stabilizers, and compute their signature in the sense of Fuchsian groups. For some groups, we also find explicit pairs of complex lines such that the union of their stabilizers generate the ambient lattice.