In this paper we study spaces of Riemannian metrics with lower bounds on intermediate curvatures. We show that the spaces of metrics of positive p-curvature and k-positive Ricci curvature on a given high-dimensional \(\mathrm {Spin}\)-manifold have many non-trivial homotopy groups provided that the manifold admits such a metric.

One way to study the Kronecker coefficients is to focus on the Kronecker cone, which is generated by the triples of partitions corresponding to non-zero Kronecker coefficients. In this article we are interested in producing particular faces of this cone, formed of stable triples (a notion defined by J. Stembridge in 2014), using some geometric notions—principally those of dominant and well-covering

We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid torus, thus answering a question posed by Arnold. We similarly study spaces of unframed links in the 3-sphere, modulo rotations, and spaces of knots in the thickened torus. The subgroup of meridional rotations

We introduce generalized Kazdan-Warner equations on Riemannian manifolds associated with a linear action of a torus on a complex vector space. We show the existence and the uniqueness of the solution of the equation on any compact Riemannian manifold. As an application, we give a new proof of a theorem of Baraglia [5] which asserts that a cyclic Higgs bundle gives a solution of the periodic Toda equation

Let X be a compact connected Riemann surface of genus g, with \(g\, \ge \, 2\), and let \({\text {G}}\) be a connected semisimple affine algebraic group defined over \({\mathbb {C}}\). Given any \(\delta \, \in \, \pi _1({\text {G}})\), we prove that the moduli space of semistable principal \({\text {G}}\)-bundles over X of topological type \(\delta \) is simply connected. More generally, if \({\text

We show that an open subset \({\mathfrak {F}}_4''\) of the \(\mathrm{PU}(2,1)\) configuration space of four points in \(S^3\) is in bijection with an open subset of \({\mathfrak {H}}^{\star }\times {\mathbb {R}}_{>0}\), where \({\mathfrak {H}}^\star \) is the affine-rotational group. Since the latter is a Sasakian manifold, the cone \({\mathfrak {H}}^\star \times {\mathbb {R}}_{>0}\) is Kähler and

A result of I.V. Dolgachev states that the complex homaloidal polynomials in three variables, i.e. the complex homogeneous polynomials whose polar map is birational, are of degree at most three. In this note we describe homaloidal polynomials in three variables of arbitrarily large degree in positive characteristic. Using combinatorial arguments, we also classify line arrangements whose polar map is

This paper studies the geometry and combinatorics of three interrelated varieties: Springer fibers, Steinberg varieties, and parabolic Hessenberg varieties. We prove that each parabolic Hessenberg variety is the pullback of a Steinberg variety under the projection of the flag variety to an appropriate partial flag variety and we give three applications of this result. The first application constructs

Cauchy-compact flat spacetimes with extreme BTZ are Lorentzian analogue of complete hyperbolic surfaces of finite volume. Indeed, the latter are 2-manifolds locally modeled on the hyperbolic plane, with group of isometries \(\mathrm {PSL}_2(\mathbb {R})\), admitting finitely many cuspidal ends while the regular part of the former are 3-manifolds locally models on 3 dimensionnal Minkowski space, with

We prove entropy rigidity for finite volume strictly convex projective manifolds in dimensions \(\ge 3\), generalizing the work of [1] to the finite volume setting. The rigidity theorem uses the techniques of Besson, Courtois, and Gallot’s entropy rigidity theorem. It implies uniform lower bounds on the volume of any finite volume strictly convex projective manifold in dimensions \(\ge 3\).

Let \(\pi :\mathcal {X}\rightarrow M\) be a holomorphic fibration with compact fibers and L a relatively ample line bundle over \(\mathcal {X}\). We obtain the asymptotic of the curvature of \(L^2\)-metric and Qullien metric on the direct image bundle \(\pi _*(L^k\otimes K_{\mathcal {X}/M})\) up to the lower order terms than \(k^{n-1}\), for large k. As an application we prove that the analytic torsion

We build on the constructions in Brady (J Lond Math Soc 60(2):461–480, 1999) and Lodha (A hyperbolic group with a finitely presented subgroup that is not of type FP3, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, pp 67–81, 2017) to give infinite families of hyperbolic groups, each having a finitely presented subgroup that is not of type \(F_3\). By calculating

The orthoscheme complex of a graded poset is a metrization of its order complex such that the simplex of each maximal chain is isometric to the Euclidean simplex of vertices \(0, e_1,e_1+e_2,\ldots , e_1+e_2+ \cdots + e_n\). This notion was introduced by Brady and McCammond in geometric group theory, and has applications in discrete optimization and submodularity theory. We address the question of

We study Thurston’s Lipschitz and curve metrics, as well as the arc metric on the Teichmüller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston’s stretch maps and prove the following: (1) On the Teichmüller space of the torus

We provide an algorithm to determine whether a link L admits a crossing change that turns it into a split link, under some fairly mild hypotheses on L. The algorithm also provides a complete list of all such crossing changes. It can therefore also determine whether the unlinking number of L is 1.

Liouville’s theorem says that in dimension greater than two, all conformal maps are Möbius transformations. We prove an analogous statement about simplicial complexes, where two simplicial complexes are considered discretely conformally equivalent if they are combinatorially equivalent and the lengths of corresponding edges are related by scale factors associated with the vertices.

We show that an Anosov map has a geodesic axis on the curve graph of the torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any given Anosov map. The application of our result is threefold: (a) to determine

A complex K3 surface or an algebraic K3 surface in characteristics distinct from 2 cannot have more than 16 disjoint nodal curves.

Let X be a minimal projective threefold of general type over \(\mathbb {C}\) with only Gorenstein quotient singularities, and let \(\mathrm {Aut}_{\mathbb {Q}}(X)\) be the subgroup of automorphisms acting trivially on \(H^*(X,\mathbb {Q})\). In this paper, we show that if X is of maximal Albanese dimension, then \(|\mathrm {Aut}_{\mathbb {Q}}(X)|\le 6\). Moreover, if X is nonsingular and \(K_X\) is

While a generic smooth Ribaucour sphere congruence admits exactly two envelopes, a discrete R-congruence gives rise to a 2-parameter family of discrete enveloping surfaces. The main purpose of this paper is to gain geometric insights into this ambiguity. In particular, discrete R-congruences that are enveloped by discrete channel surfaces and discrete Legendre maps with one family of spherical curvature

We show that given a collection \(X=\{f_1\), ..., \(f_m\}\) of pure mapping classes on a surface S, there is an explicit constant N, depending only on X, such that their Nth powers \(\{f_1^N\), ..., \(f_m^N\}\) generate the expected right-angled Artin subgroup of MCG(S). Moreover, we show that these subgroups are undistorted, and that each element is pseudo-Anosov on the largest possible subsurface

Let \({\mathcal {O}}\) be a closed n-dimensional arithmetic (real or complex) hyperbolic orbifold. We show that the diameter of \({\mathcal {O}}\) is bounded above by $$\begin{aligned} \frac{c_1\log \mathrm {vol}({\mathcal {O}}) + c_2}{h({\mathcal {O}})}, \end{aligned}$$ where \(h({\mathcal {O}})\) is the Cheeger constant of \({\mathcal {O}}\), \(\mathrm {vol}({\mathcal {O}})\) is its volume, and constants

Let \(\Gamma \) be a finite simplicial graph such that the flag complex on \(\Gamma \) is a 2-dimensional triangulated disk. We show that with some assumptions, the Dehn function of the associated Bestvina–Brady group is either quadratic, cubic, or quartic. Furthermore, we can identify the Dehn function from the defining graph \(\Gamma \).

The goal of this paper is to construct distinct trisections of the same genus on a fixed 4-manifold. For every \(k \ge 2\), we exhibit infinitely many manifolds with \(2^{k}-1\) non-diffeomorphic (3k, k)-trisections. Here, the manifolds are spun Seifert fiber spaces and the trisections come from Meier’s spun trisections. The technique used to distinguish the trisections parallels a common setup for

We study, for the first time, the maximum modulus set of a quasiregular map. It is easy to see that these sets are necessarily closed, and contain at least one point of each modulus. Blumenthal showed that for entire maps these sets are either the whole plane, or a countable union of analytic curves. We show that in the quasiregular case, by way of contrast, any closed set containing at least one point

Hilbert volume is an invariant of real projective geometry. Polygons inscribed in polygons are considered for the real projective plane. In two-dimensions the correspondence between Fock–Goncharov and Cartesian coordinates is examined. Degeneration and Hilbert area of inscribed quadrilaterals are analyzed. A microlocal condition is developed for bounded Hilbert area under degeneration. The condition

A Correction to this paper has been published: https://doi.org/10.1007/s10711-015-0119-z

It is a well-known result that a stable curve of compact type over \({\mathbb {C}}\) having two components is hyperelliptic if and only if both components are hyperelliptic and the point of intersection is a Weierstrass point for each of them. With the use of admissible covers, we generalize this characterization in two ways: for stable curves of higher gonality having two smooth components and one

For any \(n\ge 3\), the classical Liouville theorem asserts that any local conformal diffeomorphism defined on a connected open subset of \({\mathbb {S}}^n\) can be uniquely extended to a M\(\ddot{\mathrm{o}}\)bius transformation. In this article, we formulate and prove a generalization of this theorem to general rigid geometric structures.

We prove some injectivity results: that a Coxeter monoid \(\mathbb {Z}\)-algebra (or 0-Hecke algebra) injects in the incidence \(\mathbb {Z}\)-algebra of the corresponding Bruhat poset, for any Coxeter group; that the Hecke algebra of a right-angled Coxeter group injects in the Coxeter monoid \(\mathbb {Z}[q,q^{-1}]\)-algebra (and then in the incidence \(\mathbb {Z}[q,q^{-1}]\)-algebra of the corresponding

The hexagon is the least-perimeter tile in the Euclidean plane. On hyperbolic surfaces, the isoperimetric problem differs for every given area. Cox conjectured that a regular k-gonal tile with 120-degree angles is isoperimetric for its area. We prove his conjecture and more.

In this paper we introduce flat grafting as a deformation of quadratic differentials on a surface of finite type that is analogous to the grafting map on hyperbolic surfaces. Flat grafting maps are generic in the strata structure and preserve parallel measured foliations. The 1-parameter family obtained by flat grafting allows us to explicitly describe a path connecting any pair of quadratic differentials

Let X be a proper geodesic metric space and let G be a group of isometries of X which acts geometrically. Cordes constructed the Morse boundary of X which generalizes the contracting boundary for CAT(0) spaces and the visual boundary for hyperbolic spaces. We characterize Morse elements in G by their fixed points on the Morse boundary \(\partial _MX\). The dynamics on the Morse boundary is very similar

In the theory of cluster algebras, a mutation loop induces discrete dynamical systems via its actions on the cluster \({\mathcal {A}}\)- and \({\mathcal {X}}\)-varieties. In this paper, we introduce a property of mutation loops, called the sign stability, with a focus on the asymptotic behavior of the iteration of the tropical \({\mathcal {X}}\)-transformation. The sign stability can be thought of

In the first page of the published article, the author name is mistakenly spelled out as Klamlein. The correct version of the author name is given below,

We investigate a metric structure on the Thurston boundary of Teichmüller space. To do this, we develop tools in sup metrics and apply Minsky’s theorem.

In this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study 3-syzygy curve arrangements and we present examples that admit unexpected curves.

We show a kind of Obata-type theorem on a compact Einstein n-manifold \((W, \bar{g})\) with smooth boundary \(\partial W\). Assume that the boundary \(\partial W\) is minimal in \((W, \bar{g})\). If \((\partial W, \bar{g}|_{\partial W})\) is not conformally diffeomorphic to \((S^{n-1}, g_S)\), then for any Einstein metric \(\check{g} \in [\bar{g}]\) with the minimal boundary condition, we have that

We study a class of continuous deformations of branched complex projective structures on closed surfaces of genus \(g\ge 2\), which preserve the holonomy representation of the structure and the order of the branch points. In the case of non-elementary holonomy we show that when the underlying complex structure is infinitesimally preserved the branch points are necessarily arranged on a canonical divisor

We investigate the geometry of word metrics on fundamental groups of manifolds associated with the generating sets consisting of elements represented by closed geodesics. We ask whether the diameter of such a metric is finite or infinite. The first answer we interpret as an abundance of closed geodesics, while the second one as their scarcity. We discuss examples for both cases.

Let X be a genus two double covering surface of the standard flat torus \(({\mathbb {C}}/{\mathbb {Z}}[i],dz)\) with two ramified values \(0\ mod\ {\mathbb {Z}}[i]\) and \(\lambda +i\mu \ mod\ {\mathbb {Z}}[i]\). We prove that if the equation \(l+m\lambda +n\mu =0\) has nonzero integer solutions then the Hausdorff dimension of the set of nonergodic directions of X is either 0 or 1/2. And the precise

We classify closed 3-braids which are L-space knots.

Nearly Euclidean Thurston (NET) maps are described by simple diagrams which admit a natural notion of size. Given a size bound C, there are finitely many diagrams of size at most C. Given a NET map F presented by a diagram of size at most C, the problem of determining whether F is equivalent to a rational function is, in theory, a finite computation. We give bounds for the size of this computation

In this paper, we establish two strengthened versions of Klamkin’s inequality for an n-dimensional simplex in Euclidean space \({E}^n\) and give some applications.

We apply the existence and special properties of Gauduchon metrics to give several applications. The first one is concerned with the implications of algebro-geometric nature under the existence of a Hermitian metric with nonnegative holomorphic sectional curvature. The second one is to show the non-existence of holomorphic sections on Hermitian vector bundles under certain conditions. The third one

In 1983 Culler and Shalen established a way to construct essential surfaces in a 3-manifold from ideal points of the $$\mathrm {SL}_2$$ SL 2 -character variety associated to the 3-manifold group. We present in this article an analogous construction of certain kinds of branched surfaces (which we call essential tribranched surfaces ) from ideal points of the $$\mathrm {SL}_n$$ SL n -character variety

We study the expected volume of random polytopes generated by taking the convex hull of independent identically distributed points from a given distribution. We show that for log-concave distributions supported on convex bodies, we need at least exponentially many (in dimension) samples for the expected volume to be significant and that super-exponentially many samples suffice for concave measures

Inspired by a recent work of Grove-Petersen in [GP18], where the authors studied Alexandrov spaces with largest possible boundary. We study Alexandrov spaces with lower curvature bound 1 and with small boundary. When the radius of X is \pi/2, and the boundary has diameter \pi/2, we classify the total space X.

The Hitchin component $$\mathrm {Hit}_n(S)$$ Hit n ( S ) of a closed surface S is a preferred component of the character variety $$\mathcal {X}_{\mathrm {PSL}_n(\mathbb {R})}(S)$$ X PSL n ( R ) ( S ) consisting of homomorphisms from the fundamental group $$\pi _1(S)$$ π 1 ( S ) to the Lie group $$\mathrm {PSL}_n(\mathbb {R})$$ PSL n ( R ) , whose elements enjoy remarkable geometric and dynamical properties

Critical nets in $${\mathbb {R}}^k$$ R k (sometimes called geodesic nets) are embedded graph with the property that their embedding is a critical point of the total (edge) length functional and under the constraint that certain 1-valent vertices have a fixed position. In contrast to what happens on generic manifolds, we show that, if the embedding is bounded and n is the number of 1-valent vertices

We study the homology of Riemannian manifolds of finite volume that are covered by a product $(\mathbb{H}^2)^r = \mathbb{H}^2 \times \ldots \times \mathbb{H}^2$ of the real hyperbolic plane. Using a variation of a method developed by Avramidi and Nyguen-Phan, we show that any such manifold $M$ possesses, up to finite coverings, an arbitrarily large number of compact oriented flat totally geodesic $r$-dimensional

Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let $\mathbf{G}$ be a semisimple algebraic $\mathbb{R}$-group such that $G=\mathbf{G}(\mathbb{R})^\circ$ is of Hermitian type. If $\Gamma \leq L$ is a torsion-free lattice of a finite connected covering of $\text{PU}(1,1)$, given

We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, namely the ones that arise through a ramified covering. These are the main known examples of bundles with non-zero signature.

A pair $(\alpha, \beta)$ of simple closed geodesics on a closed and oriented hyperbolic surface $M_g$ of genus $g$ is called a filling pair if the complementary components of $\alpha\cup\beta$ in $M_g$ are simply connected. The length of a filling pair is defined to be the sum of their individual lengths. We show that the length of any filling pair on $M$ is at least $\frac{m_{g}}{2}$, where $m_{g}$

In response to Sanki-Vadnere, we present a short proof of the following theorem: a pair of simple curves on a hyperbolic surface whose complementary regions are disks has length at least half the perimeter of the regular right-angled $(8g-4)$-gon.

Let $m \geqslant 6$ be an even integer. In this short note we prove that the Jacobian variety of a quasiplatonic Riemann surface with associated group of automorphisms isomorphic to $C_2^2 \rtimes_2 C_m$ admits complex multiplication. We then extend this result to provide a criterion under which the Jacobian variety of a quasiplatonic Riemann surface admits complex multiplication.

In this paper, we prove long time existence and convergence results for a class of general curvature flows with Neumann boundary condition. This is the first result for the Neumann boundary problem of non Monge-Ampere type curvature equations. Our method also works for the corresponding elliptic setting.

Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups embed quasi-isometrically in vertex groups. We prove that a vertex group is relatively quasiconvex if and only if all the vertex groups have finite height in the fundamental group.

In this short note we explain how to adapt the construction of two Riemannian metrics on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component to the deformation space of globally hyperbolic anti-de Sitter structures: the pressure metric and the Loftin metric (studied by Qiongling Li). We show that the former is degenerate and we characterize its degenerate locus, whereas the latter is nowhere degenerate

A long-standing conjecture asserts that any Anosov diffeomorphism of a closed manifold is finitely covered by a diffeomorphism which is topologically conjugate to a hyperbolic automorphism of a nilpotent manifold. In this paper, we show that any closed 4-manifold that carries a Thurston geometry and is not finitely covered by a product of two aspherical surfaces does not support (transitive) Anosov