111 research outputs found
The flux homomorphism on closed hyperbolic surfaces and Anti-de Sitter three-dimensional geometry
Given a smooth spacelike surface of negative curvature in Anti-de
Sitter space of dimension 3, invariant by a representation
where
is a closed oriented surface of genus , a canonical construction
associates to a diffeomorphism of . It turns out that
is a symplectomorphism for the area forms of the two hyperbolic
metrics and on induced by the action of on
. Using an algebraic construction related to
the flux homomorphism, we give a new proof of the fact that is
the composition of a Hamiltonian symplectomorphism of and the unique
minimal Lagrangian diffeomorphism from to .Comment: 20 page
On the maximal dilatation of quasiconformal minimal Lagrangian extensions
Given a quasisymmetric homeomorphism of the circle, Bonsante and
Schlenker proved the existence and uniqueness of the minimal Lagrangian
extension to the hyperbolic plane. By
previous work of the author, its maximal dilatation satisfies , where denotes the cross-ratio
norm. We give constraints on the value of an optimal such constant , and
discuss possible lower inequalities, by studying two one-parameter families of
minimal Lagrangian extensions in terms of maximal dilatation and cross-ratio
norm.Comment: 25 pages. Results of Theorem A improved. Several mistakes corrected,
Remark 4.9 added, general exposition improve
Spacelike convex surfaces with prescribed curvature in (2+1)-Minkowski space
We prove existence and uniqueness of solutions to the Minkowski problem in
any domain of dependence in -dimensional Minkowski space, provided
is contained in the future cone over a point. Namely, it is possible to
find a smooth convex Cauchy surface with prescribed curvature function on the
image of the Gauss map. This is related to solutions of the Monge-Amp\`ere
equation on the unit disc, with the
boundary condition , for a smooth
positive function and a bounded lower semicontinuous function.
We then prove that a domain of dependence contains a convex Cauchy
surface with principal curvatures bounded from below by a positive constant if
and only if the corresponding function is in the Zygmund class.
Moreover in this case the surface of constant curvature contained in
has bounded principal curvatures, for every . In this way we get a full
classification of isometric immersions of the hyperbolic plane in Minkowski
space with bounded shape operator in terms of Zygmund functions of .
Finally, we prove that every domain of dependence as in the hypothesis of the
Minkowski problem is foliated by the surfaces of constant curvature , as
varies in .Comment: 45 pages, 17 figures. Final version, improved presentation and
details of some proof
On Codazzi tensors on a hyperbolic surface and flat Lorentzian geometry
Using global considerations, Mess proved that the moduli space of globally
hyperbolic flat Lorentzian structures on is the tangent
bundle of the Teichm\"uller space of , if is a closed surface. One of
the goals of this paper is to deepen this surprising occurrence and to make
explicit the relation between the Mess parameters and the embedding data of any
Cauchy surface. This relation is pointed out by using some specific properties
of Codazzi tensors on hyperbolic surfaces. As a by-product we get a new
Lorentzian proof of Goldman's celebrated result about the coincidence of the
Weil-Petersson symplectic form and the Goldman pairing.
In the second part of the paper we use this machinery to get a classification
of globally hyperbolic flat space-times with particles of angles in
containing a uniformly convex Cauchy surface. The analogue of Mess' result is
achieved showing that the corresponding moduli space is the tangent bundle of
the Teichm\"uller space of a punctured surface. To generalize the theory in the
case of particles, we deepen the study of Codazzi tensors on hyperbolic
surfaces with cone singularities, proving that the well-known decomposition of
a Codazzi tensor in a harmonic part and a trivial part can be generalized in
the context of hyperbolic metrics with cone singularities.Comment: 49 pages, 4 figure
Fibered spherical 3-orbifolds
In early 1930s Seifert and Threlfall classified up to conjugacy the finite
subgroups of , this gives an algebraic classification of
orientable spherical 3-orbifolds. For the most part, spherical 3-orbifolds are
Seifert fibered. The underlying topological space and singular set of
non-fibered spherical 3-orbifolds were described by Dunbar. In this paper we
deal with the fibered case and in particular we give explicit formulae relating
the finite subgroups of with the invariants of the
corresponding fibered 3-orbifolds. This allows to deduce directly from the
algebraic classification topological properties of spherical 3-orbifolds.Comment: 27 pages, 6 figures. Several misprint corrected, improved expositio
Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfaces
A well-known theorem of Wolpert shows that the Weil-Petersson symplectic form
on Teichm\"uller space, computed on two infinitesimal twists along simple
closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines
of the intersection angles. We define an infinitesimal deformation starting
from a more general object, namely a balanced geodesic graph, by which any
tangent vector to Teichm\"uller space can be represented. We then prove a
generalization of Wolpert's formula for these deformations. In the case of
simple closed curves, we recover the theorem of Wolpert.Comment: 21 pages, 11 figure
Geometric transition from hyperbolic to Anti-de Sitter structures in dimension four
We provide the first examples of geometric transition from hyperbolic to
Anti-de Sitter structures in dimension four, in a fashion similar to Danciger's
three-dimensional examples. The main ingredient is a deformation of hyperbolic
4-polytopes, discovered by Kerckhoff and Storm, eventually collapsing to a
3-dimensional ideal cuboctahedron. We show the existence of a similar family of
collapsing Anti-de Sitter polytopes, and join the two deformations by means of
an opportune half-pipe orbifold structure. The desired examples of geometric
transition are then obtained by gluing copies of the polytope.Comment: 50 pages, 27 figures. Part 3 of the previous version has been removed
and will be part of a new preprint to appear soo
Area-preserving diffeomorphism of the hyperbolic plane and K-surfaces in Anti-de Sitter space
We prove that any weakly acausal curve in the boundary of Anti-de
Sitter (2+1)-space is the asymptotic boundary of two spacelike -surfaces,
one of which is past-convex and the other future-convex, for every
. The curve is the graph of a quasisymmetric
homeomorphism of the circle if and only if the -surfaces have bounded
principal curvatures. Moreover in this case a uniqueness result holds.
The proofs rely on a well-known correspondence between spacelike surfaces in
Anti-de Sitter space and area-preserving diffeomorphisms of the hyperbolic
plane. In fact, an important ingredient is a representation formula, which
reconstructs a spacelike surface from the associated area-preserving
diffeomorphism.
Using this correspondence we then deduce that, for any fixed
, every quasisymmetric homeomorphism of the circle admits a
unique extension which is a -landslide of the hyperbolic plane. These
extensions are quasiconformal.Comment: 47 pages, 18 figures. More details added to Remark 4.14, Remark 6.2
and Theorem 7.8 Step 2. Several references added and typos corrected. Final
version. To appear in Journal of Topolog
Examples of geometric transition in low dimensions
The purpose of this note is to discuss examples of geometric transition from
hyperbolic structures to half-pipe and Anti-de Sitter structures in dimensions
two, three and four. As a warm-up, explicit examples of transition to Euclidean
and spherical structures are presented. No new results appear here; nor an
exhaustive treatment is aimed. On the other hand, details of some elementary
computations are provided to explain certain techniques involved. This note,
and in particular the last section, can also serve as an introduction to the
ideas behind the four-dimensional construction of [RS19].Comment: 26 pages, 18 figures. To appear in: Actes du s\'eminaire de Th\'eorie
Spectrale et G\'eom\'etri
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