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Normal generators for mapping class groups are abundant Comment. Math. Helv. (IF 0.9) Pub Date : 2022-04-14 Justin Lanier, Dan Margalit
We provide a simple criterion for an element of the mapping class group of a closed surface to be a normal generator for the mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator for the mapping class group when the genus is at least 3. We also give many examples of pseudo-Anosov normal generators, answering
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Slow manifolds for infinite-dimensional evolution equations Comment. Math. Helv. (IF 0.9) Pub Date : 2022-04-14 Felix Hummel, Christian Kuehn
We extend classical finite-dimensional Fenichel theory in two directions to infinite dimensions. Under comparably weak assumptions we show that the solution of an infinite-dimensional fast-slow system is approximated well by the corresponding slow flow. After that we construct a two-parameter family of slow manifolds $S_{\epsilon,\zeta}$ under more restrictive assumptions on the linear part of the
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Number fields with prescribed norms (with an appendix by Yonatan Harpaz and Olivier Wittenberg) Comment. Math. Helv. (IF 0.9) Pub Date : 2022-04-14 Christopher Frei, Daniel Loughran, Rachel Newton
We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for 100% of $G$-extensions of $k$, when ordered by conductor. The appendix contains an alternative purely geometric proof of our
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Rationality of even-dimensional intersections of two real quadrics Comment. Math. Helv. (IF 0.9) Pub Date : 2022-04-14 Brendan Hassett, János Kollár, Yuri Tschinkel
We study rationality constructions for smooth complete intersections of two quadrics over nonclosed fields. Over the real numbers, we establish a criterion for rationality in dimension four.
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Bounds on the Lagrangian spectral metric in cotangent bundles Comment. Math. Helv. (IF 0.9) Pub Date : 2022-01-18 Paul Biran, Octav Cornea
Let $N$ be a closed manifold and $U \subset T^*(N)$ a bounded domain in the cotangent bundle of $N$, containing the zero-section. A conjecture due to Viterbo asserts that the spectral metric for Lagrangian submanifolds in $U$ that are exact-isotopic to the zero-section is bounded. In this paper we establish an upper bound on the spectral distance between two such Lagrangians $L_0, L_1$, which depends
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Quantitative estimates for the Bakry–Ledoux isoperimetric inequality Comment. Math. Helv. (IF 0.9) Pub Date : 2022-01-18 Cong Hung Mai, Shin-ichi Ohta
We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with $\operatorname{Ric}_{\infty} \ge 1$. Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle decomposition), in terms of the deficit in Bakry–Ledoux’s Gaussian isoperimetric
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Opening nodes in the DPW method: Co-planar case Comment. Math. Helv. (IF 0.9) Pub Date : 2022-01-18 Martin Traizet
We combine the DPW method and opening nodes to construct embedded surfaces of positive constant mean curvature with Delaunay ends in euclidean space, with no limitation to the genus or number of ends.
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Skew-amenability of topological groups Comment. Math. Helv. (IF 0.9) Pub Date : 2022-01-18 Kate Juschenko, Friedrich Martin Schneider
We study skew-amenable topological groups, i.e., those admitting a left-invariant mean on the space of bounded real-valued functions left-uniformly continuous in the sense of Bourbaki. We prove characterizations of skew-amenability for topological groups of isometries and automorphisms, clarify the connection with extensive amenability of group actions, establish a Følner-type characterization, and
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You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces Comment. Math. Helv. (IF 0.9) Pub Date : 2021-11-22 Moon Duchin, Viveka Erlandsson, Christopher J. Leininger, Chandrika Sadanand
We give a complete characterization of the relationship between the shape of a Euclidean polygon and the symbolic dynamics of its billiard flow. We prove that the only pairs of tables that can have the same bounce spectrum are right-angled tables that differ by an affine map. The main tool is a new theorem that establishes that a flat cone metric is completely determined by the support of its Liouville
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Subadditivity of Kodaira dimension does not hold in positive characteristic Comment. Math. Helv. (IF 0.9) Pub Date : 2021-11-22 Paolo Cascini, Sho Ejiri, János Kollár, Lei Zhang
Over any algebraically closed field of positive characteristic, we construct examples of fibrations violating subadditivity of Kodaira dimension.
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Self-referential discs and the light bulb lemma Comment. Math. Helv. (IF 0.9) Pub Date : 2021-11-22 David Gabai
We show how self-referential discs in 4-manifolds lead to the construction of pairs of discs with a common geometrically dual sphere which are homotopic rel $\partial$, concordant and coincide near their boundaries, yet are not properly isotopic. This occurs in manifolds without 2-torsion in their fundamental group, e.g. the boundary connect sum of $S^2\times D^2$ and $S^1\times B^3$, thereby exhibiting
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Isometric immersions of RCD spaces Comment. Math. Helv. (IF 0.9) Pub Date : 2021-11-22 Shouhei Honda
We prove that if an RCD space has a regular isometric immersion in a Euclidean space, then the immersion is a locally bi-Lipschitz embedding map. This result leads us to prove that if a compact non-collapsed RCD space has an isometric immersion in a Euclidean space via an eigenmap, then the eigenmap is a locally bi-Lipschitz embedding map to a sphere, which generalizes a fundamental theorem of Takahashi
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Filling random cycles Comment. Math. Helv. (IF 0.9) Pub Date : 2021-11-22 Fedor Manin
We compute the asymptotic behavior of the average-case filling volume for certain models of random Lipschitz cycles in the unit cube and sphere. For example, we estimate the minimal area of a Seifert surface for a model of random knots first studied by Millett. This is a generalization of the classical Ajtai–Komlós–Tusnády optimal matching theorem from combinatorial probability. The author hopes for
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Topological dynamics beyond Polish groups Comment. Math. Helv. (IF 0.9) Pub Date : 2021-11-22 Gianluca Basso, Andy Zucker
When $G$ is a Polish group, metrizability of the universal minimal flow has been shown to be a robust dividing line in the complexity of the topological dynamics of $G$. We introduce a class of groups, the CAP groups, which provides a neat generalization of this to all topological groups. We prove a number of characterizations of this class, having very different flavors, and use these to prove that
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Linear independence in linear systems on elliptic curves Comment. Math. Helv. (IF 0.9) Pub Date : 2021-06-23 Bradley W. Brock, Bruce W. Jordan, Bjorn Poonen, Anthony J. Scholl, Joseph L. Wetherell
Let $E$ be an elliptic curve, with identity $O$, and let $C$ be a cyclic subgroup of odd order $N$, over an algebraically closed field $k$ with char $k \nmid N$. For $P \in C$, let $s_P$ be a rational function with divisor $N \cdot P - N \cdot O$. We ask whether the $N$ functions $s_P$ are linearly independent. For generic $(E,C)$, we prove that the answer is yes. We bound the number of exceptional
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The Nirenberg problem of prescribed Gauss curvature on $S^2$ Comment. Math. Helv. (IF 0.9) Pub Date : 2021-06-23 Michael T. Anderson
We introduce a new perspective on the classical Nirenberg problem of understanding the possible Gauss curvatures of metrics on $S^{2}$ conformal to the round metric. A key tool is to employ the smooth Cheeger–Gromov compactness theorem to obtain general and essentially sharp a priori estimates for Gauss curvatures $K$ contained in naturally defined stable regions. We prove that in such stable regions
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On the rate of equidistribution of expanding translates of horospheres in $\Gamma\backslash G$ Comment. Math. Helv. (IF 0.9) Pub Date : 2021-06-23 Samuel Edwards
Let $G$ be a semisimple Lie group and $\Gamma$ a lattice in $G$. We generalize a method of Burger to prove precise effective equidistribution results for translates of pieces of horospheres in the homogeneous space $\Gamma\backslash G$.
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Moments and interpretations of the Cohen–Lenstra–Martinet heuristics Comment. Math. Helv. (IF 0.9) Pub Date : 2021-06-23 Weitong Wang, Melanie Matchett Wood
The goal of this paper is to prove theorems that elucidate the Cohen–Lenstra–Martinet conjectures for the distributions of class groups of number fields, and further the understanding of their implications. We start by giving a simpler statement of the conjectures. We show that the probabilities that arise are inversely proportional to the number of automorphisms of structures slightly larger than
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Finite entropy vs finite energy Comment. Math. Helv. (IF 0.9) Pub Date : 2021-06-23 Eleonora Di Nezza, Vincent Guedj, Chinh H. Lu
Probability measures with either finite Monge–Ampère energy or finite entropy have played a central role in recent developments in Kähler geometry. In this note we make a systematic study of quasi-plurisubharmonic potentials whose Monge–Ampère measures have finite entropy. We show that these potentials belong to the finite energy class $\mathcal{E}^{\frac{n}{n-1}}$, where $n$ denotes the complex dimension
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Geometric cycles and characteristic classes of manifold bundles Comment. Math. Helv. (IF 0.9) Pub Date : 2021-03-12 Bena Tshishiku
We produce new cohomology for non-uniform arithmetic lattices $\Gamma < \mathrm {SO}(p,q)$ using a technique of Millson–Raghunathan. From this, we obtain new characteristic classes of manifold bundles with fiber a closed $4k$-dimensional manifold $M$ with indefinite intersection form of signature $(p,q)$. These classes are defined on finite covers of $B$ Diff $(M)$ and are shown to be nontrivial for
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Irreducibility of a free group endomorphism is a mapping torus invariant Comment. Math. Helv. (IF 0.9) Pub Date : 2021-03-12 Jean Pierre Mutanguha
We prove that the property of a free group endomorphism being irreducible is a group invariant of the ascending HNN extension it defines. This answers a question posed by Dowdall–Kapovich–Leininger. We further prove that being irreducible and atoroidal is a commensurability invariant. The invariance follows from an algebraic characterization of ascending HNN extensions that determines exactly when
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A Lie algebra action on the Chow ring of the Hilbert scheme of points of a K3 surface Comment. Math. Helv. (IF 0.9) Pub Date : 2021-03-12 Georg Oberdieck
We construct an action of the Neron–Severi part of the Looijenga–Lunts–Verbitsky Lie algebra on the Chow ring of the Hilbert scheme of points on a K3 surface. This yields a simplification of Maulik and Negut’s proof that the cycle class map is injective on the subring generated by divisor classes as conjectured by Beauville. The key step in the construction is an explicit formula for Lefschetz duals
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Uniqueness of the measure of maximal entropy for the standard map Comment. Math. Helv. (IF 0.9) Pub Date : 2021-03-12 Davi Obata
In this paper we prove that for sufficiently large parameters the standard map has a unique measure of maximal entropy (m.m.e.). Moreover, we prove: the m.m.e. is Bernoulli, and the periodic points with Lyapunov exponents bounded away from zero equidistribute with respect to the m.m.e.We prove some estimates regarding the Hausdorff dimension of the m.m.e. and about the density of the support of the
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Schauder estimates on products of cones Comment. Math. Helv. (IF 0.9) Pub Date : 2021-03-12 Martin de Borbon, Gregory Edwards
We prove an interior Schauder estimate for the Laplacian on metric products of two dimensional cones with a Euclidean factor, generalizing the work of Donaldson and reproving the Schauder estimate of Guo–Song. We characterize the space of homogeneous subquadratic harmonic functions on products of cones, and identify scales at which geodesic balls can be well approximated by balls centered at the apex
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Sets with constant normal in Carnot groups: properties and examples Comment. Math. Helv. (IF 0.9) Pub Date : 2021-03-12 Costante Bellettini, Enrico Le Donne
We analyze subsets of Carnot groups that have intrinsic constant normal, as they appear in the blowup study of sets that have finite subRiemannian perimeter. The purpose of this paper is threefold. First, we prove some mild regularity and structural results in arbitrary Carnot groups. Namely, we show that for every constant-normal set in a Carnot group its subRiemannian-Lebesgue representative is regularly
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Free rational points on smooth hypersurfaces Comment. Math. Helv. (IF 0.9) Pub Date : 2020-12-07 Tim Browning, Will Sawin
Motivated by a recent question of Peyre, we apply the Hardy-Littlewood circle method to count "sufficiently free" rational points of bounded height on arbitrary smooth projective hypersurfaces of low degree that are defined over the rational numbers.
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Essential dimension of representations of algebras Comment. Math. Helv. (IF 0.9) Pub Date : 2020-12-07 Federico Scavia
Let $k$ be a field, $A$ a finitely generated associative $k$-algebra and $\operatorname{Rep}_A[n]$ the functor $\operatorname{Fields}_k\to \operatorname{Sets}$, which sends a field $K$ containing $k$ to the set of isomorphism classes of representations of $A_K$ of dimension at most $n$. We study the asymptotic behavior of the essential dimension of this functor, i.e., the function $r_A(n) := \oper
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The Euler characteristic of Out($F_n$) Comment. Math. Helv. (IF 0.9) Pub Date : 2020-12-07 Michael Borinsky, Karen Vogtmann
We prove that the rational Euler characteristic of Out(Fn) is always negative and its asymptotic growth rate is Γ(n− 3 2 )/ √ 2π log n. This settles a 1987 conjecture of J. Smillie and the second author. We establish connections with the LambertW -function and the zeta function.
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On Borel Anosov representations in even dimensions Comment. Math. Helv. (IF 0.9) Pub Date : 2020-12-07 Konstantinos Tsouvalas
We prove that a word hyperbolic group which admits a $P_{2q+1}$-Anosov representation into $\mathsf{PGL}(4q+2, \mathbb{R})$ contains a finite-index subgroup which is either free or a surface group. As a consequence, we give an affirmative answer to Sambarino's question for Borel Anosov representations into $\mathsf{SL}(4q+2,\mathbb{R})$.
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On the decomposability of mod 2 cohomological invariants of Weyl groups Comment. Math. Helv. (IF 0.9) Pub Date : 2020-12-07 Christian Hirsch
We compute the invariants of Weyl groups of type $A_n$, $B_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$ and $G_2$ in mod 2 Milnor $K$-theory and more general cycle modules, which are annihilated by 2. Over a base field of characteristic coprime to the group order and where $-1$ is a square, the invariants decompose as direct sums of the coefficient module. With the exception of invariants coming from $G_2$-components
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Involutions and Chern numbers of varieties Comment. Math. Helv. (IF 0.9) Pub Date : 2020-12-07 Olivier Haution
Consider an involution of a smooth projective variety over a field of characteristic not two. We look at the relations between the variety and the fixed locus of the involution from the point of view of cobordism. We show in particular that the fixed locus has dimension larger than its codimension when certain Chern numbers of the variety are not divisible by two, or four. Some of those results, but
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Algebraic flows on commutative complex Lie groups Comment. Math. Helv. (IF 0.9) Pub Date : 2020-09-15 Tien-Cuong Dinh, Duc-Viet Vu
We recover results by Ullmo-Yafaev and Peterzil-Starchenko on the closure of the image of an algebraic variety in a compact complex torus. Our approach uses directed closed currents and allows us to extend the result for dimension 1 flows to the setting of commutative complex Lie groups which are not necessarily compact. A version of the classical Ax-Lindemann-Weierstrass theorem for commutative complex
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The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed Comment. Math. Helv. (IF 0.9) Pub Date : 2020-09-15 Daniel Cristofaro-Gardiner, Marco Mazzucchelli
A classical theorem due to Wadsley implies that, on a contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed 3-manifold, the following conditions are actually equivalent: (1) every Reeb orbit is closed; (2) all closed Reeb orbits have a common period; (3) the action spectrum has rank 1. We also show
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The Liouville property and random walks on topological groups Comment. Math. Helv. (IF 0.9) Pub Date : 2020-09-15 Friedrich Martin Schneider, Andreas Thom
We study harmonic functions and Poisson boundaries for Borel probability measures on general (i.e., not necessarily locally compact) topological groups, and we prove that a second-countable topological group is amenable if and only if it admits a fully supported, regular Borel probability measure with trivial Poisson boundary. This generalizes work of Kaimanovich--Vershik and Rosenblatt, confirms a
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Hyperbolic surfaces with sublinearly many systoles that fill Comment. Math. Helv. (IF 0.9) Pub Date : 2020-09-15 Maxime Fortier Bourque
For any e>0, we construct a closed hyperbolic surface of genus g=g(e) with a set of at most eg systoles that fill, meaning that each component of the complement of their union is contractible. This surface is also a critical point of index at most eg for the systole function, disproving the lower bound of 2g−1 conjectured by Schmutz Schaller.
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Squeezing Lagrangian tori in dimension 4 Comment. Math. Helv. (IF 0.9) Pub Date : 2020-09-15 Richard Hind, Emmanuel Opshtein
We find the minimal size of 4 dimensional balls and polydisks into which product Lagrangian tori can be mapped by a Hamiltonian diffeomorphism.
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Rigidity of center Lyapunov exponents and $su$-integrability Comment. Math. Helv. (IF 0.9) Pub Date : 2020-09-15 Shaobo Gan, Yi Shi
Let $f$ be a conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism $A$ on $\mathbb{T}^3$. We show that the stable and unstable bundles of $f$ are jointly integrable if and only if every periodic point of $f$ admits the same center Lyapunov exponent with $A$. In particular, $f$ is Anosov. Thus every conservative partially hyperbolic diffeomorphism, which is homotopic
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Fractal geometry of the complement of Lagrange spectrum in Markov spectrum Comment. Math. Helv. (IF 0.9) Pub Date : 2020-09-15 Carlos Matheus, Carlos Gustavo Moreira
The Lagrange and Markov spectra are classical objects in Number Theory related to certain Diophantine approximation problems. Geometrically, they are the spectra of heights of geodesics in the modular surface. These objects were first studied by A. Markov in 1879, but, despite many efforts, the structure of the complement $M\setminus L$ of the Lagrange spectrum $L$ in the Markov spectrum $M$ remained
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Simple groups of birational transformations in dimension two Comment. Math. Helv. (IF 0.9) Pub Date : 2020-06-16 Christian Urech
We classify simple groups that act by birational transformations on compact complex K\"ahler surfaces. Moreover, we show that every finitely generated simple group that acts non-trivially by birational transformations on a projective surface over an arbitrary field is finite.
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Poisson brackets of partitions of unity on surfaces Comment. Math. Helv. (IF 0.9) Pub Date : 2020-06-16 Lev Buhovsky, Alexander Logunov, Shira Tanny
Given an open cover of a closed symplectic manifold, consider all smooth partitions of unity consisting of functions supported in the covering sets. The Poisson bracket invariant of the cover measures how much the functions from such a partition of unity can become close to being Poisson commuting. We introduce a new approach to this invariant, which enables us to prove the lower bound conjectured
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Singular genuine rigidity Comment. Math. Helv. (IF 0.9) Pub Date : 2020-06-16 Luis Florit, Felippe Guimarães
We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain new global rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and Dajczer-Gromoll theorems by showing that any compact $n$-dimensional submanifold of ${\mathbb R}^{n+p}$ is singularly genuinely rigid in ${\mathbb R}^{n+q}$, for
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Rational equivalence and Lagrangian tori on K3 surfaces Comment. Math. Helv. (IF 0.9) Pub Date : 2020-06-16 Nick Sheridan, Ivan Smith
Fix a symplectic K3 surface X homologically mirror to an algebraic K3 surface Y by an equivalence taking a graded Lagrangian torus L in X to the skyscraper sheaf of a point y of Y. We show there are Lagrangian tori with vanishing Maslov class in X whose class in the Grothendieck group of the Fukaya category is not generated by Lagrangian spheres. This is mirror to a statement about the `Beauville--Voisin
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Algebraic varieties are homeomorphic to varieties defined over number fields Comment. Math. Helv. (IF 0.9) Pub Date : 2020-06-16 Adam Parusiński, Guillaume Rond
We show that every affine or projective algebraic variety defined over the field of real or complex numbers is homeomorphic to a variety defined over the field of algebraic numbers. We construct such a homeomorphism by choosing a small deformation of the coefficients of the original equations. This method is based on the properties of Zariski equisingular families of varieties. Moreover we construct
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Connected components of strata of Abelian differentials over Teichmüller space Comment. Math. Helv. (IF 0.9) Pub Date : 2020-06-16 Aaron Calderon
This paper describes connected components of the strata of holomorphic abelian differentials on marked Riemann surfaces with prescribed degrees of zeros. Unlike the case for unmarked Riemann surfaces, we find there can be many connected components, distinguished by roots of the cotangent bundle of the surface. In the course of our investigation we also characterize the images of the fundamental groups
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The collapsing geometry of almost Ricci-flat 4-manifolds Comment. Math. Helv. (IF 0.9) Pub Date : 2020-04-07 John Lott
We consider Riemannian 4-manifolds that Gromov-Hausdorff converge to a lower dimensional limit space with the Ricci tensor going to zero. Among other things, we show that if the limit space is two dimensional then under some mild assumptions, the limiting four dimensional geometry away from the curvature blowup region is semiflat Kaehler.
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Smooth zero-entropy diffeomorphisms with ergodic derivative extension Comment. Math. Helv. (IF 0.9) Pub Date : 2020-04-07 Philipp Kunde
On any smooth compact and connected manifold of dimension 2 admitting a smooth nontrivial circle action we construct C∞-diffeomorphisms of topological entropy zero whose differential is ergodic with respect to a smooth measure in the projectivization of the tangent bundle. The proof is based on a version of the “approximation by conjugation”-method.
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On the universality of the Epstein zeta function Comment. Math. Helv. (IF 0.9) Pub Date : 2020-04-07 Johan Andersson, Anders Södergren
We study universality properties of the Epstein zeta function $E_n(L,s)$ for lattices $L$ of large dimension $n$ and suitable regions of complex numbers $s$. Our main result is that, as $n\to\infty$, $E_n(L,s)$ is universal in the right half of the critical strip as $L$ varies over all $n$-dimensional lattices $L$. The proof uses an approximation result for Dirichlet polynomials together with a recent
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Boundary rigidity of negatively-curved asymptotically hyperbolic surfaces Comment. Math. Helv. (IF 0.9) Pub Date : 2020-04-07 Thibault Lefeuvre
In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.
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Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero Comment. Math. Helv. (IF 0.9) Pub Date : 2020-04-07 Olivier Benoist, John Christian Ottem
We prove that the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth projective
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Non-existence of geometric minimal foliations in hyperbolic three-manifolds Comment. Math. Helv. (IF 0.9) Pub Date : 2020-04-07 Michael Wolf, Yunhui Wu
In this paper we show that every three-dimensional closed hyperbolic manifold admits no locally geometric $1$-parameter family of closed minimal surfaces.
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Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences Comment. Math. Helv. (IF 0.9) Pub Date : 2020-04-07 Catalin Badea, Sophie Grivaux
Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers $p_{1},\dots,p_{r}$ there exists a continuous probability measure $\mu $ on the unit circle $\mathbb{T}$ such that \[ \inf_{k_{1}\ge 0,\dots,k_{r}\ge 0}|\widehat{\mu }(p_{1}^{k_{1}}\dots p_{r}^{k_{r}})|>0. \] This results applies in particular to the Furstenberg
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Quasi-isometric embeddings of non-uniform lattices Comment. Math. Helv. (IF 0.9) Pub Date : 2020-04-07 David Fisher, Thang Nguyen
Let $G$ and $G'$ be simple Lie groups of equal real rank and real rank at least $2$. Let $\Gamma
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Spectral and Hodge theory of “Witt” incomplete cusp edge spaces Comment. Math. Helv. (IF 0.9) Pub Date : 2019-12-18 Jesse Gell-Redman, Jan Swoboda
Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove that the Hodge-Laplacian on differential forms is essentially self-adjoint, with
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Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups Comment. Math. Helv. (IF 0.9) Pub Date : 2019-12-18 Michael Brandenbursky, Michał Marcinkowski
Let $F_n$ be the free group on $n$ generators and $\Gamma_g$ the surface group of genus $g$. We consider two particular generating sets: the set of all primitive elements in $F_n$ and the set of all simple loops in $\Gamma_g$. We give a complete characterization of distorted and undistorted elements in the corresponding $Aut$-invariant word metrics. In particular, we reprove Stallings theorem and answer
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Rigidity of Busemann convex Finsler metrics Comment. Math. Helv. (IF 0.9) Pub Date : 2019-12-18 Sergei Ivanov, Alexander Lytchak
We prove that a Finsler metric is nonpositively curved in the sense of Busemann if and only if it is affinely equivalent to a Riemannian metric of nonpositive sectional curvature. In other terms, such Finsler metrics are precisely Berwald metrics of nonpositive flag curvature. In particular in dimension 2 every such metric is Riemannian or locally isometric to that of a normed plane.
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Middle dimensional symplectic rigidity and its effect on Hamiltonian PDEs Comment. Math. Helv. (IF 0.9) Pub Date : 2019-12-18 Jaime Bustillo
In the first part of the article we study Hamiltonian diffeomorphisms of $\mathbb{R}^{2n}$ which are generated by sub-quadratic Hamiltonians and prove a middle dimensional rigidity result for the image of coisotropic cylinders. The tools that we use are Viterbo's symplectic capacities and a series of inequalities coming from their relation with symplectic reduction. In the second part we consider the
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Lengths of closed geodesics on random surfaces of large genus Comment. Math. Helv. (IF 0.9) Pub Date : 2019-12-18 Maryam Mirzakhani, Bram Petri
We prove Poisson approximation results for the bottom part of the length spectrum of a random closed hyperbolic surface of large genus. Here, a random hyperbolic surface is a surface picked at random using the Weil-Petersson volume form on the corresponding moduli space. As an application of our result, we compute the large genus limit of the expected systole.
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Characterization of generic projective space bundles and algebraicity of foliations Comment. Math. Helv. (IF 0.9) Pub Date : 2019-12-18 Carolina Araujo, Stéphane Druel
In this paper we consider various notions of positivity for distributions on complex projective manifolds. We start by analyzing distributions having big slope with respect to curve classes, obtaining characterizations of generic projective space bundles in terms of movable curve classes. We then apply this result to investigate algebraicity of leaves of foliations, providing a lower bound for the
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Some analytic aspects of automorphic forms on GL(2) of minimal type Comment. Math. Helv. (IF 0.9) Pub Date : 2019-12-18 Yueke Hu, Paul Nelson, Abhishek Saha
Let $\pi$ be a cuspidal automorphic representation of $PGL_2(\mathbb{A}_\mathbb{Q})$ of arithmetic conductor $C$ and archimedean parameter $T$, and let $\phi$ be an $L^2$-normalized automorphic form in the space of $\pi$. The sup-norm problem asks for bounds on $\| \phi \|_\infty$ in terms of $C$ and $T$. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the $L^2$-mass $|\phi|^2
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On rational cuspidal plane curves and the local cohomology of Jacobian rings Comment. Math. Helv. (IF 0.9) Pub Date : 2019-12-18 Alexandru Dimca
This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and nearly free curves in terms of Tjurina numbers. Finally, we suggest a stronger form of Terao’s conjecture on the freeness of a line arrangement being determined by its