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Commentarii Mathematici Helvetici
Comment. Math. Helv.
CMH
0010-2571
1420-8946
General
10.4171/CMH
http://www.ems-ph.org/doi/10.4171/CMH
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
85
2010
1
Topological rigidity and Gromov simplicial volume
Pierre
Derbez
Université de Provence, MARSEILLE CEDEX 13, FRANCE
Haken manifold, Seifert fibered space, hyperbolic 3-manifold, Gromov simplicial volume, non-zero degree maps, Dehn filling, subgroup separability
A natural problem in the theory of 3-manifolds is the question of whether two 3-manifolds are homeomorphic or not. The aim of this paper is to study this problem for the class of closed Haken manifolds using degree one maps. To this purpose we introduce an invariant τ(N) = (Vol(N),||N||), where ||N|| denotes the Gromov simplicial volume of N and Vol(N) is a 2-dimensional simplicial volume which measures the volume of the base 2-orbifolds of the Seifert pieces of N. After studying the behavior of τ(N) under the action of non-zero degree maps, we prove that if M and N are closed Haken manifolds such that ||M|| = |deg(f)| ||N|| and Vol(M) = Vol(N) then any non-zero degree map f: M → N is homotopic to a covering map. As a corollary we prove that if M and N are closed Haken manifolds such that τ(N) is sufficiently close to τ(M) then any degree one map f: M → N is homotopic to a homeomorphism.
Manifolds and cell complexes
Geometry
General
1
37
10.4171/CMH/186
http://www.ems-ph.org/doi/10.4171/CMH/186
On minimal surfaces bounded by two convex curves in parallel planes
Martin
Traizet
Université François Rabelais, TOURS, FRANCE
Minimal surface with boundary, Robin function, concentration of curvature
We prove that a compact minimal surface bounded by two closed convex curves in parallel planes close enough to each other must be topologically an annulus.
Differential geometry
General
39
71
10.4171/CMH/187
http://www.ems-ph.org/doi/10.4171/CMH/187
Foliations and global inversion
Eduardo
Cabral Balreira
Trinity University, SAN ANTONIO, UNITED STATES
Jacobian conjecture, Hadamard’s theorem, global inversion theorem
We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism f: M → ℝn is bijective if and only if Hn − 1(M) = 0 and the pre-image of every affine hyperplane is non-empty and acyclic. The proof is based on some geometric constructions involving foliations and tools from intersection theory. This topological result generalizes in finite dimensions the classical analytic theorem of Hadamard–Plastock, including its recent improvement by Nollet–Xavier. The main theorem also relates to a conjecture of the aforementioned authors, involving the well-known Jacobian conjecture in algebraic geometry.
Global analysis, analysis on manifolds
Manifolds and cell complexes
General
73
93
10.4171/CMH/188
http://www.ems-ph.org/doi/10.4171/CMH/188
Monodromy in Hamiltonian Floer theory
Dusa
McDuff
Columbia University, NEW YORK, UNITED STATES
Hamiltonian group, Floer theory, Seidel representation, spectral invariant, Hofer norm, Calabi quasimorphism
Schwarz showed that when a closed symplectic manifold (M,ω) is symplectically aspherical (i.e. the symplectic form and the first Chern class vanish on π2(M)) then the spectral invariants, which are initially defined on the universal cover of the Hamiltonian group, descend to the Hamiltonian group Ham(M,ω). In this note we describe less stringent conditions on the Chern class and quantum homology of M under which the (asymptotic) spectral invariants descend to Ham(M,ω). For example, they descend if the quantum multiplication of M is undeformed and H2(M) has rank > 1, or if the minimal Chern number is at least n + 1 (where dim M = 2n) and the even cohomology of M is generated by divisors. The proofs are based on certain calculations of genus zero Gromov–Witten invariants. As an application, we show that the Hamiltonian group of the one point blow up of T4 admits a Calabi quasimorphism. Moreover, whenever the (asymptotic) spectral invariants descend it is easy to see that Ham(M,ω) has infinite diameter in the Hofer norm. Hence our results establish the infinite diameter of Ham in many new cases. We also show that the area pseudonorm – a geometric version of the Hofer norm – is nontrivial on the (compactly supported) Hamiltonian group for all noncompact manifolds as well as for a large class of closed manifolds.
Differential geometry
Manifolds and cell complexes
General
95
133
10.4171/CMH/189
http://www.ems-ph.org/doi/10.4171/CMH/189
Virtually soluble groups of type FP∞
Conchita
Martínez-Pérez
Universidad de Zaragoza, ZARAGOZA, SPAIN
Brita E. A.
Nucinkis
Royal Holloway University of London, EGHAM, SURREY, UNITED KINGDOM
Cohomological finiteness conditions, soluble groups
We prove that a virtually soluble group G of type FP∞ admits a finitely dominated model for EG of dimension the Hirsch length of G. This implies in particular that the Brown conjecture is satisfied for virtually torsion-free elementary amenable groups.
Group theory and generalizations
General
135
150
10.4171/CMH/190
http://www.ems-ph.org/doi/10.4171/CMH/190
SLn(ℤ[t]) is not FPn − 1
Kai-Uwe
Bux
Universität Bielefeld, BIELEFELD, GERMANY
Amir
Mohammadi
The University of Texas at Austin, AUSTIN, UNITED STATES
Kevin
Wortman
University of Utah, SALT LAKE CITY, UNITED STATES
Euclidean buildings, finiteness properties, geometric group theory
We prove the result from the title using the geometry of Euclidean buildings.
Group theory and generalizations
General
151
164
10.4171/CMH/191
http://www.ems-ph.org/doi/10.4171/CMH/191
Reconstructing p-divisible groups from their truncations of small level
Adrian
Vasiu
Binghamton University, BINGHAMTON, UNITED STATES
p-divisible groups, F-crystals, algebras, affine group schemes
Let k be an algebraically closed field of characteristic p > 0. Let D be a p-divisible group over k. Let nD be the smallest non-negative integer for which the following statement holds: if C is a p-divisible group over k of the same codimension and dimension as D and such that C[pnD] is isomorphic to D[pnD], then C is isomorphic to D. To the Dieudonné module of D we associate a non-negative integer ℓD which is a computable upper bound of nD. If D is a product ∏i ∈ I Di of isoclinic p-divisible groups, we show that nD = ℓD; if the set I has at least two elements we also show that nD ≤ max{1, nDi, nDi + nDj − 1 | i, j ∈ I, j ≠ i}. We show that we have nD ≤ 1 if and only if ℓD ≤ 1; this recovers the classification of minimal p-divisible groups obtained by Oort. If D is quasi-special, we prove the Traverso truncation conjecture for D. If D is F-cyclic, we explicitly compute nD. Many results are proved in the general context of latticed F-isocrystals with a (certain) group over k.
Number theory
Algebraic geometry
Group theory and generalizations
General
165
202
10.4171/CMH/192
http://www.ems-ph.org/doi/10.4171/CMH/192
Contact homology of Hamiltonian mapping tori
Oliver
Fabert
Universität München, MÜNCHEN, GERMANY
Symplectic field theory, Floer homology, J-holomorphic curve
In the general geometric setup for symplectic field theory the contact manifolds can be replaced by mapping tori MΦ of symplectic manifolds (M,ω) with symplectomorphisms Φ. While the cylindrical contact homology of MΦ is given by the Floer homologies of powers of Φ, the other algebraic invariants of symplectic field theory for MΦ provide natural generalizations of symplectic Floer homology. For symplectically aspherical M and Hamiltonian Φ we study the moduli spaces of rational curves and prove a transversality result, which does not need the polyfold theory by Hofer, Wysocki and Zehnder. We use our result to compute the full contact homology of MΦ ≅ S1 × M.
Differential geometry
General
203
241
10.4171/CMH/193
http://www.ems-ph.org/doi/10.4171/CMH/193
2
Geometric approach to the Weil–Petersson symplectic form
Reynir
Axelsson
University of Iceland, REYKJAVIK, ICELAND
Georg
Schumacher
Philipps-Universität, MARBURG, GERMANY
Weil–Petersson form, Fenchel–Nielsen coordinates, symplectic geometry Publishers Note. Due to an error the submission date as printed in the article is not correct. The article was received by the editors on February 27, 2008.
In a family of compact, canonically polarized, complex manifolds the first variation of the lengths of closed geodesics is computed. As an application, we show the coincidence of the Fenchel–Nielsen and Weil–Petersson symplectic forms on the Teichmüller spaces of compact Riemann surfaces in a purely geometric way. The method can also be applied to situations like moduli spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available.
Several complex variables and analytic spaces
Differential geometry
General
243
257
10.4171/CMH/194
http://www.ems-ph.org/doi/10.4171/CMH/194
A characterization of round spheres in terms of blocking light
Benjamin
Schmidt
Michigan State University, EAST LANSING, UNITED STATES
Juan
Souto
Université de Rennes 1, RENNES CEDEX, FRANCE
Riemannian manifold, cross blocking, sphere blocking, blocking number
A closed Riemannian manifold M is said to have cross (compact rank one symmetric space) blocking if whenever p ≠ q are less than the diameter apart, all light rays from p can be shaded away from q with at most two point shades. Similarly, a closed Riemannian manifold is said to have sphere blocking if for each p ∈ M all the light rays from p are shaded away from p by a single point shade. We prove that Riemannian manifolds with cross and sphere blocking are isometric to round spheres.
Differential geometry
Dynamical systems and ergodic theory
General
259
271
10.4171/CMH/195
http://www.ems-ph.org/doi/10.4171/CMH/195
Conformal arc-length as ½-dimensional length of the set of osculating circles
Rémi
Langevin
Université de Bourgogne, DIJON CEDEX, FRANCE
Jun
O'Hara
Tokyo Metropolitan University, TOKYO, JAPAN
Conformal arc-length, osculating circles, pseudo-Riemannian manifolds
The set of osculating circles of a given curve in S3 forms a lightlike curve in the set of oriented circles in S3. We show that its “½-dimensional measure” with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a conformally invariant local quantity discovered in the first half of the last century.
Differential geometry
General
273
312
10.4171/CMH/196
http://www.ems-ph.org/doi/10.4171/CMH/196
Finiteness results for flat surfaces: large cusps and short geodesics
John
Smillie
Cornell University, ITHACA, UNITED STATES
Barak
Weiss
Tel Aviv University, TEL AVIV, ISRAEL
Flat surface, Veech group, Fuchsian group, Markov partition
For fixed g and T we show the finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech group contains a cusp of hyperbolic co-area less than T. We obtain new restrictions on Veech groups: any non-elementary Fuchsian group can appear only finitely many times in a fixed stratum; any non-elementary Veech group is of finite index in its normalizer; and the quotient of ℍ by a non-lattice Veech group admits arbitrarily large embedded disks. A key ingredient of the proof is the finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech group contains a hyperbolic element with eigenvalue less than T.
Manifolds and cell complexes
Dynamical systems and ergodic theory
General
313
336
10.4171/CMH/197
http://www.ems-ph.org/doi/10.4171/CMH/197
Bounding the symbol length in the Galois cohomology of function fields of p-adic curves
Venapally
Suresh
Emory University, ATLANTA, UNITED STATES
Galois cohomology, central simple algebras, cyclic algebras, symbols
Let K be a function field of a p-adic curve and l a prime not equal to p. Assume that K contains a primitive l th root of unity. We show that every element in the l-torsion subgroup of the Brauer group of K is a tensor product of two cyclic algebras over K.
Number theory
General
337
346
10.4171/CMH/198
http://www.ems-ph.org/doi/10.4171/CMH/198
Automatic transversality and orbifolds of punctured holomorphic curves in dimension four
Chris
Wendl
Humboldt-Universität zu Berlin, BERLIN, GERMANY
Pseudoholomorphic curves, transversality, symplectic cobordisms, symplectic 4-manifolds, contact 3-manifolds
We derive a numerical criterion for J-holomorphic curves in 4-dimensional symplectic cobordisms to achieve transversality without any genericity assumption. This generalizes results of Hofer–Lizan–Sikorav [HLS97] and Ivashkovich–Shevchishin [IS99] to allow punctured curves with boundary that generally need not be somewhere injective or immersed. As an application, we combine this with the intersection theory of punctured holomorphic curves to prove that certain geometrically natural moduli spaces are globally smooth orbifolds, consisting generically of embedded curves, plus unbranched multiple covers that form isolated orbifold singularities.
Several complex variables and analytic spaces
Manifolds and cell complexes
General
347
407
10.4171/CMH/199
http://www.ems-ph.org/doi/10.4171/CMH/199
The K(π,1) conjecture for a class of Artin groups
Graham
Ellis
National University of Ireland, GALWAY, IRELAND
Emil
Sköldberg
National University of Ireland, GALWAY, IRELAND
Artin group, Eilenberg–Mac Lane space, cohomology groups
Salvetti constructed a cellular space BD for any Artin group AD defined by a Coxeter graph D. We show that BD is an Eilenberg–Mac Lane space if BD' is an Eilenberg–Mac Lane space for every subgraph D' of D involving no ∞-edges.
Algebraic topology
Group theory and generalizations
General
409
415
10.4171/CMH/200
http://www.ems-ph.org/doi/10.4171/CMH/200
Local signature defect of fibered complex surfaces via monodromy and stable reduction
Tadashi
Ashikaga
Tohoku Gakuin University, Tagajo, JAPAN
Signature, fibered surface, monodromy, eta invariant, stable reduction, signature defect
We consider the localization problem of signature of a fibered complex surface. In this paper, we analyze the contribution of topological monodromy to the local signature of a fiber germ. In order to realize it, we describe explicitly the behavior of Atiyah–Patodi–Singer’s eta invariant under the stable reduction of a fiber germ in terms of Nielsen’s data.
Algebraic geometry
Several complex variables and analytic spaces
Manifolds and cell complexes
General
417
461
10.4171/CMH/201
http://www.ems-ph.org/doi/10.4171/CMH/201
A converse theorem for Dirichlet L-functions
Jerzy
Kaczorowski
Adam Mickiewicz University, POZNAN, POLAND
Giuseppe
Molteni
Università di Milano, MILANO, ITALY
Alberto
Perelli
Università di Genova, GENOVA, ITALY
Dirichlet L-functions, converse theorems, Hamburger theorem, Selberg class
It is known that the space of solutions (in a suitable class of Dirichlet series with continuation over ℂ) of the functional equation of a Dirichlet L-function L(s,χ) has dimension ≥ 2 as soon as the conductor q of χ is at least 4. Hence the Dirichlet L-functions are not characterized by their functional equation for q ≥ 4. Here we characterize the conductors q such that for every primitive character χ (mod q), L(s,χ) is the only solution with an Euler product in the above space. It turns out that such conductors are of the form q = 2a3bm with any square-free m coprime to 6 and finitely many a and b.
Number theory
General
463
483
10.4171/CMH/202
http://www.ems-ph.org/doi/10.4171/CMH/202
3
Dimensional reduction and the long-time behavior of Ricci flow
John
Lott
University of California, BERKELEY, UNITED STATES
Ricci flow, geometrization
If g(t) is a three-dimensional Ricci flow solution, with sectional curvatures that are O(t−1) and diameter that is O(t1/2), then the pullback Ricci flow solution on the universal cover approaches a homogeneous expanding soliton.
Differential geometry
Manifolds and cell complexes
General
485
534
10.4171/CMH/203
http://www.ems-ph.org/doi/10.4171/CMH/203
On the integro-differential equation satisfied by the p-adic log Γ-function
Eduardo
Friedman
Universidad de Chile, SANTIAGO DE CHILE, CHILE
p-adic gamma function, p-adic integral equations, p-adic differential equations
Diamond’s p-adic analogue log ΓD(x) of the classical function log Γ(x) has recently been shown to satisfy the integro-differential equation (∗) ∫ℤp f(x + t) dt = (x −1) f′(x) − x + 1/2 (x ∈ ℚp − ℤp), where ∫ℤp is a Volkenborn integral and f′ is the derivative of f. We show that this equation characterizes log ΓD(x) up to a function with everywhere vanishing second derivative. Namely, every solution f of (∗) is infinitely differentiable and satisfies f′′ = log ΓD′′. We show that the set of solutions of the homogeneous equation ∫ℤp y(x + t) dt = (x − 1) y′(x) associated to (∗) is an infinite-dimensional commutative and associative p-adic algebra under the product law (y1 ◊ y2)(x) : = y2′(x)y1(x) + y1′(x)y2(x) − (x − 1/2) y1′(x)y2′(x), the unit being y(x) = x − 1/2. We also study Morita’s alternate p-adic analogue log ΓM of log Γ(x) and prove similar results.
Number theory
Field theory and polynomials
Integral equations
General
535
549
10.4171/CMH/204
http://www.ems-ph.org/doi/10.4171/CMH/204
Compactly supported cohomology of buildings
Michael
Davis
Ohio State University, COLUMBUS, UNITED STATES
Jan
Dymara
Uniwersytet Wrocławski, WROCŁAW, POLAND
Tadeusz
Januszkiewicz
Polskiej Akademii Nauk, WARSZAWA, POLAND
John
Meier
Lafayette College, EASTON, UNITED STATES
Boris
Okun
University of Wisconsin at Milwaukee, MILWAUKEE, UNITED STATES
Building, cohomology of groups, Coxeter group
We compute the compactly supported cohomology of the standard realization of any locally finite building.
Group theory and generalizations
Manifolds and cell complexes
General
551
582
10.4171/CMH/205
http://www.ems-ph.org/doi/10.4171/CMH/205
Local–global principles for embedding of fields with involution into simple algebras with involution
Gopal
Prasad
University of Michigan, ANN ARBOR, UNITED STATES
Andrei
Rapinchuk
University of Virginia, CHARLOTTESVILLE, UNITED STATES
Local–global principles, central simple algebras, involutions, arithmetic groups, locally symmetric spaces
In this paper we prove local–global principles for the existence of an embedding (E, σ) ↪ (A, τ) of a given global field E endowed with an involutive automorphism σ into a simple algebra A given with an involution τ in all situations except where A is a matrix algebra of even degree over a quaternion division algebra and τ is orthogonal (Theorem A of the introduction). Rather surprisingly, in the latter case we have a result which in some sense is opposite to the local–global principle, viz. algebras with involution locally isomorphic to (A, τ) are distinguished by their maximal subfields invariant under the involution (Theorem B of the introduction). These results can be used in the study of classical groups over global fields. In particular, we use Theorem B to complete the analysis of weakly commensurable Zariski-dense S-arithmetic groups in all absolutely simple algebraic groups of type different from D4 which was initiated in our paper [23]. More precisely, we prove that in a group of type Dn, n even > 4, two weakly commensurable Zariski-dense S-arithmetic subgroups are actually commensurable. As indicated in [23], this fact leads to results about length-commensurable and isospectral compact arithmetic hyperbolic manifolds of dimension 4n + 7, with n ≥ 1. The appendix contains a Galois-cohomological interpretation of our embedding theorems.
Number theory
Algebraic geometry
Group theory and generalizations
Topological groups, Lie groups
583
645
10.4171/CMH/206
http://www.ems-ph.org/doi/10.4171/CMH/206
The large sieve and random walks on left cosets of arithmetic groups
Florent
Jouve
Université Paris-Sud 11, ORSAY CEDEX, FRANCE
Random walks on arithmetic groups, Property (τ), large sieve, polynomials and orthogonal matrices over finite fields
Building on Kowalski’s work on random walks on the groups SL(n,ℤ) and Sp(2g,ℤ), we consider similar problems (we try to estimate the probability with which, after k steps, the matrix obtained has a characteristic polynomial with maximal Galois group or has no nonzero squares among its entries) for more general classes of sets: in GL(n,A), where A is a subring of ℚ containing ℤ that we specify, we perform a random walk on the set of matrices with fixed determinant D ∈ A×. We also investigate the case where the set involved is any of the two left cosets of the special orthogonal group SO(n,m)(ℤ) with respect to the spinorial kernel Ω(n,m)(ℤ).
Linear and multilinear algebra; matrix theory
Number theory
Field theory and polynomials
General
647
704
10.4171/CMH/207
http://www.ems-ph.org/doi/10.4171/CMH/207
Small exotic Stein manifolds
Selman
Akbulut
Michigan State University, EAST LANSING, UNITED STATES
Kouichi
Yasui
Kyoto University, KYOTO, JAPAN
4-manifold, Stein manifold, handlebody, cork, plug
It is known that the only Stein filling of the standard contact structure on S3 is B4. In this paper, we construct pairs of homeomorphic but not diffeomorphic simply connected compact Stein 4-manifolds, for any Betti number b2 ≥ 1; we do this by enlarging corks and plugs.
Manifolds and cell complexes
General
705
721
10.4171/CMH/208
http://www.ems-ph.org/doi/10.4171/CMH/208
4
Dynamics on the unit disk: Short geodesics and simple cycles
Curtis
McMullen
Harvard University, CAMBRIDGE, UNITED STATES
Blaschke products, complex dynamics, rotation numbers
Functions of a complex variable
Dynamical systems and ergodic theory
General
723
749
10.4171/CMH/209
http://www.ems-ph.org/doi/10.4171/CMH/209
The non-triviality of the Grope filtrations of the knot and link concordance groups
Peter
Horn
Columbia University, NEW YORK, UNITED STATES
Concordance, Grope filtration, solvable filtration
We consider the Grope filtration of the classical knot concordance group that was introduced by Cochran, Orr and Teichner. Our main result is that each successive quotient in this filtration has infinite rank. We also establish the analogous result for the Grope filtration of the concordance group of string links consisting of more than one component.
Manifolds and cell complexes
General
751
773
10.4171/CMH/210
http://www.ems-ph.org/doi/10.4171/CMH/210
Une minoration du minimum essentiel sur les variétés abéliennes
Aurélien
Galateau
Université Paris-Sud, ORSAY CEDEX, FRANCE
Bogomolov, abelian variety, lower bound, height
On étend à la codimension générale la minoration du minimum essentiel sur les variétés abéliennes obtenue dans [Gal08], sous une conjecture concernant leurs idéaux premiers ordinaires. Cette minoration est optimale “à ε près” en le degré de la sous-variété et rend inconditionnelle la démonstration, par Viada, de la conjecture de Zilber–Pink pour une courbe plongée dans une puissance de courbe elliptique. We extend to the general codimension the lower bound for the essential minimum on abelian varieties found in [Gal08], under a conjecture about ordinary primes in abelian varieties. This lower bound is the best expected, “up to an ε”, in the degree of the subvariety and completes Viada’s proof of the Zilber–Pink conjecture for a curve embedded in a power of an elliptic curve.
Number theory
Algebraic geometry
General
775
812
10.4171/CMH/211
http://www.ems-ph.org/doi/10.4171/CMH/211
Uniformly hyperbolic finite-valued SL(2,ℝ)-cocycles
Artur
Avila
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Jairo
Bochi
PUC-Rio, RIO DE JANEIRO, BRAZIL
Jean-Christophe
Yoccoz
Collège de France (Annexe), PARIS CEDEX O5, FRANCE
Linear cocycles, uniform hyperbolicity, bifurcations, iterated function systems
We consider finite families of SL(2,ℝ) matrices whose products display uniform exponential growth. These form open subsets of (SL(2,ℝ))N, and we study their components, boundary, and complement. We also consider the more general situation where the allowed products of matrices satisfy a Markovian rule.
Dynamical systems and ergodic theory
Linear and multilinear algebra; matrix theory
General
813
884
10.4171/CMH/212
http://www.ems-ph.org/doi/10.4171/CMH/212
A norm compatible system of Galois cohomology classes for GSp(4)
Francesco
Lemma
, VILLAZ, FRANCE
Elliptic polylogarithm pro-sheaf, p-adic L-function, Siegel modular variety
.ionsss { line-height: 1.2; } .ionsss subbb { margin-left: -2.1ex; vertical-align:-1ex;} For a given de Rham p-adic Galois representation M, a conjecture of Perrin–Riou associates a p-adic L-function for M to a norm compatible system of Galois cohomology classes in the projective lim←nH1(ℚ(ζpn), M). We construct such a norm compatible system for the symplectic group GSp4. Our classes are cup-products of torsion sections of the large elliptic polylogarithm pro-sheaf; we rely on its norm compatibility and on some computations of weights in the cohomology of Siegel threefolds of our previous work.
Number theory
General
885
905
10.4171/CMH/213
http://www.ems-ph.org/doi/10.4171/CMH/213
Maslov class rigidity for Lagrangian submanifolds via Hofer’s geometry
Ely
Kerman
University of Illinois at Urbana-Champaign, URBANA, UNITED STATES
Nil
Şirikçi
University of Illinois at Urbana-Champaign, URBANA, UNITED STATES
Lagrangian submanifold, Maslov class, Floer theory, Hofer’s geometry
In this work, we establish new rigidity results for the Maslov class of Lagrangian submanifolds in large classes of closed and convex symplectic manifolds. Our main result establishes upper bounds for the minimal Maslov number of displaceable Lagrangian submanifolds which are product manifolds whose factors each admit a metric of negative sectional curvature. Such Lagrangian submanifolds exist in every symplectic manifold of dimension greater than six or equal to four. The proof utilizes the relations between closed geodesics on the Lagrangian, the periodic orbits of geometric Hamiltonian flows supported near the Lagrangian, and the length minimizing properties of these flows with respect to the negative Hofer length functional.
Differential geometry
Dynamical systems and ergodic theory
General
907
949
10.4171/CMH/214
http://www.ems-ph.org/doi/10.4171/CMH/214