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 \(\mathrm {SL}_2\)-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 \(\mathrm {SL}_n\)-character variety for a natural

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 \(\kappa \)-concave

Inspired by a recent work of Grove and Petersen (Alexandrov spaces with maximal radius, 2018), where the authors studied positively curved Alexandrov spaces with largest possible boundary, namely the round sphere, we study Alexandrov spaces with lower curvature bound 1 and with large boundary other than the sphere. In particular, we classify those spaces with radius equal to \(\pi /2\), and the intrinsic

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

Critical nets in \({\mathbb {R}}^k\) (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, the

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

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 \(\Gamma \le L\) is a torsion-free lattice of a finite connected covering of \(\mathrm{PU}(1

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 ramified coverings. These are among 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 \(\alpha \cup \beta \) on \(M_g\) are simply connected. The length of a filling pair is defined to be the sum of their individual lengths. In Aougab and Huang (Algebr Geom Topol 15:903–932, 2015), Aougab–Huang conjectured

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 \(m \ge 6\) 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.

We consider the evolution of a strictly convex hypersurface by a class of general curvature. We prove that given some Neumann boundary condition, the flow exists for all time and converges to a solution with prescribed general curvature that satisfies the Neumann boundary condition. 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 quasi-isometrically embedded and relatively quasiconvex in vertex groups, we prove that vertex groups are relatively quasiconvex if and only if all the vertex groups have finite relative height in the fundamental group.

We first extend the construction of the pressure metric to the deformation space of globally hyperbolic maximal Cauchy-compact anti-de Sitter structures. We show that, in contrast with the case of the Hitchin components, the pressure metric is degenerate and we characterize its degenerate locus. We then introduce a nowhere degenerate Riemannian metric adapting the work of Qiongling Li on the \(\mathrm

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

Let \({\mathbb {F}}_{\Theta }=U/K_\Theta \) be a partial flag manifold, where \(K_\Theta \) is the centralizer of a torus in U. We study U-invariant almost Hermitian structures on \({\mathbb {F}}_{\Theta }\). The classification of these structures are naturally related with the system \(R_{\mathfrak {t}}\) of \({\mathfrak {t}}\)-roots associated to \({\mathbb {F}}_{\Theta }\). We introduced the notion

We give a complete description of all locally conformally Kähler structures with holomorphic Lee vector field on a compact complex manifold of Vaisman type. This provides in particular examples of such structures whose Lee vector field is not homothetic to the Lee vector field of a Vaisman structure. More generally, dropping the condition of being of Vaisman type, we show that on a compact complex

Weyl (Zur Infinitisimalgeometrie: Einordnung der projektiven und der konformen Auffasung, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Göttinger Akademie der Wissenschaften, Göttingen, 1921) demonstrated that for a connected manifold of dimension greater than 1, if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization

We provide sharp bounds for the squared norm of the second fundamental form of a wide class of Weingarten hypersurfaces in Euclidean space satisfying \(H_r = aH + b\), for constants \(a, b \in \mathbb {R}\), where \(H_r\) stands for the rth mean curvature and H the mean curvature of the hypersurface. Besides we are able to characterize those hypersurfaces for which these bounds are attained by showing

There is an isomorphism between the moduli spaces of \(\sigma \)-stable holomorphic triples and some of the critical submanifolds of the moduli space of k-Higgs bundles of rank three, whose elements \((E,\varphi ^k)\) correspond to variations of Hodge structure, VHS. There are special embeddings on the moduli spaces of k-Higgs bundles of rank three. The main objective here is to study the cohomology

In this paper, we construct the first families of distinct Lagrangian ribbon disks in the standard symplectic 4-ball which have the same boundary Legendrian knots, and are not smoothly isotopic or have non-homeomorphic exteriors.

Sharp inequalities of the parameterized functional \(U_j\) for Borel measures on the unit sphere in \({\mathbb {R}}^n\) are established. As two applications, some inequalities related to cone-volume measures and Schneider’s projection problem are obtained.

As in the case of irreducible holomorphic symplectic manifolds, the period domain Compl of compact complex tori of even dimension 2n contains twistor lines. These are special 2-spheres parametrizing complex tori whose complex structures arise from a given quaternionic structure. In analogy with the case of irreducible holomorphic symplectic manifolds, we show that the periods of any two complex tori

We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the Kronheimer–Mrowka proof of the Thom conjecture.

A closed surface evolving under mean curvature flow becomes singular in finite time. Near the singularity, the surface resembles a self-shrinker, a surface that shrinks by dilations under mean curvature flow. If the singularity is modeled on a self-shrinker other than a round sphere or cylinder, then the singularity is unstable under perturbations of the flow. One can quantify this instability using

We study multi-moment maps on nearly Kähler six-manifolds with a two-torus symmetry. Critical points of these maps have non-trivial stabilisers. The configuration of fixed-points and one-dimensional orbits is worked out for generic six-manifolds equipped with an \(\mathrm {SU}(3)\)-structure admitting a two-torus symmetry. Projecting the subspaces obtained to the orbit space yields a trivalent graph

We examine distortion of finitely generated normal subgroups. We show a connection between subgroup distortion and group divergence. We suggest a method computing the distortion of normal subgroups by decomposing the whole group into smaller subgroups. We apply our work to compute the distortion of normal subgroups of graph of groups and normal subgroups of right-angled Artin groups that induce infinite

We explicitly construct Brill–Noether general K3 surfaces of genus 4, 6 and 8 having the maximal number of elliptic pencils of degrees 3, 4 and 5, respectively, and study their moduli spaces and moduli maps to the moduli space of curves. As an application we prove the existence of Brill–Noether general K3 surfaces of genus 4 and 6 without stable Lazarsfeld–Mukai bundles of minimal \(c_2\).

Let \(\Sigma _0,\Sigma _1\) be closed oriented surfaces. Two oriented knots \(K_0 \subset \Sigma _0 \times [0,1]\) and \(K_1 \subset \Sigma _1 \times [0,1]\) are said to be (virtually) concordant if there is a compact oriented 3-manifold W and a smoothly and properly embedded annulus A in \(W \times [0,1]\) such that \(\partial W=\Sigma _1 \sqcup -\Sigma _0\) and \(\partial A=K_1 \sqcup -K_0\). This

In this article we show that the only 2-step nilpotent Lie groups which carry a non-degenerate left invariant Killing–Yano 2-form are the complex Lie groups. In the case of 2-step nilpotent complex Lie groups arising from connected graphs, we prove that the space of left invariant Killing–Yano 2-forms is one-dimensional.

We generalize Gruber–Sisto’s construction of the coned-off graph of a small cancellation group to build a partially ordered set \({\mathcal {TC}}\) of cobounded actions of a given small cancellation group whose smallest element is the action on the Gruber–Sisto coned-off graph. In almost all cases \({\mathcal {TC}}\) is incredibly rich: it has a largest element if and only if it has exactly 1 element

Given a holomorphic family of pairs \(\{(X_t,E_t)\}\) where each \(E_t\) is a holomorphic vector bundle over a compact complex manifold \(X_t\), we get a correspondence between the Dolbeault complex of \(E_t\)-valued (p, q)-forms on \(X_t\) and the one of \(E_0\)-valued (p, q)-forms on \(X_0\) for small enough t.

A new method for constructing self-referential tilings of Euclidean space from a graph directed iterated function system (GIFS), based on a combinatorial structure we call a pre-tree, is introduced. For each GIFS, a family of tilings is constructed indexed by a parameter. For what we call a commensurate GIFS, our method is used to define what we refer to as balanced tilings. Under mild conditions on

We show that every plumbing of disc bundles over surfaces whose genera satisfy a simple inequality may be embedded as a convex submanifold in some closed hyperbolic four-manifold. In particular its interior has a geometrically finite hyperbolic structure that covers a closed hyperbolic four-manifold.

A Riemannian manifold is called almost positively curved if the set of points for which all 2-planes have positive sectional curvature is open and dense. We find three new examples of almost positively curved manifolds: \(Sp(3)/Sp(1)^2\), and two circle quotients of \(Sp(3)/Sp(1)^2\). We also show the quasi-positively curved metric of Tapp (J Differ Geom 65:273–287, 2003) on \(Sp(n+1)/Sp(n-1) Sp(1)\)

We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semi-simple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thus extend earlier rigidity results about quasi-isometric embeddings to the setting

The Burau representation is a fundamental bridge between the braid group and diverse other topics in mathematics. A 1974 question of Birman asks for a description of the image; in this paper we give an approximate answer. Since a 1984 paper of Squier it has been known that the Burau representation preserves a certain Hermitian form. We show that the Burau image is dense in this unitary group relative

We provide criteria ensuring that a tunnel number one knot K is not determined by its double branched cover, in the sense that the double branched cover is also the double branched cover of a knot \(K'\) not equivalent to K.

We study linear series on a general curve of genus g, whose images are exceptional with respect to their secant planes. Each such exceptional secant plane is algebraically encoded by an included linear series, whose number of base points computes the incidence degree of the corresponding secant plane. With enumerative applications in mind, we construct a moduli scheme of inclusions of limit linear

Dilation surfaces are generalizations of translation surfaces where the geometric structure is modelled on the complex plane up to affine maps whose linear part is real. They are the geometric framework to study suspensions of affine interval exchange maps. However, though the \(SL(2,\mathbb {R})\)-action is ergodic in connected components of strata of translation surfaces, there may be mutually disjoint

We study three different topologies on the moduli space \(\mathcal {H}^\mathrm{loc}_m\) of equivariant local isometry classes of m-dimensional locally homogeneous Riemannian spaces. As an application, we provide the first examples of locally homogeneous spaces converging to a limit space in the pointed \(\mathcal {C}^{k,\alpha }\)-topology, for some \(k>1\), which do not admit any convergent subsequence

In this article we obtain the rigid isotopy classification of generic rational curves of degre 5 in \({\mathbb {R}}{\mathbb {P}}^{2}\). In order to study the rigid isotopy classes of nodal rational curves of degree 5 in \({\mathbb {R}}{\mathbb {P}}^{2}\), we associate to every real rational nodal quintic curve with a marked real nodal point a nodal trigonal curve in the Hirzebruch surface \(\Sigma

We study smooth rational closed embeddings of the real affine line into the real affine plane, that is algebraic rational maps from the real affine line to the real affine plane which induce smooth closed embeddings of the real euclidean line into the real euclidean plane. We consider these up to equivalence under the group of birational automorphisms of the real affine plane which are diffeomorphisms

Let X be a smooth complex projective curve, and let \(x\,\in \, X\) be a point. We compute the automorphism group of the moduli space of framed vector bundles on X of rank \(r\, \ge \, 2\) with a framing over x. It is shown that this automorphism group is generated by the following three: (1) pullbacks using automorphisms of the curve X that fix the marked point x, (2) tensorization with certain line

We investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately related to the class of geodesic metric disc retracts satisfying comparable bounds. This kind of discs naturally come up in the context of the solution

We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths are not fixed. This extends the first author’s earlier result to the with-boundary setting. In the second part of the paper we comment on another direction for extending

We prove that any Fano manifold of coindex three admitting nef tangent bundle is homogeneous.

Given a simple closed curve \(\gamma \) on a connected, oriented, closed surface S of negative Euler characteristic, Mirzakhani showed that the set of points in the moduli space of hyperbolic structures on S having a simple closed geodesic of length L of the same topological type as \(\gamma \) equidistributes with respect to a natural probability measure as \(L \rightarrow \infty \). We prove several

This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the complete classification of such groups up to quasi-isometries and we compare the quasi-isometric classification with the bi-Lipschitz classification. On the other

Tête-à-tête graphs were introduced by N. A’Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed tête-à-tête graphs provide a generalization which define mixed tête-à-tête twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed tête-à-tête twists as those pseudo-periodic automorphisms that have a power which is a product of right-handed

Mark and Paupert devised a general method for obtaining presentations for arithmetic non-cocompact lattices, \(\Gamma \), in isometry groups of negatively curved symmetric spaces. The method involves a classical theorem of Macbeath applied to a \(\Gamma \)-invariant covering by horoballs of the negatively curved symmetric space upon which \(\Gamma \) acts. In this paper, we will discuss the application

Let G be a compact semisimple linear Lie group. We study the action of \(\text {Aut}(F_r)\) on the space \(H_*(G^r; {\mathbb {Q}})\). We compute the image of this representation and prove that it only depends on the rank of \({\mathfrak {g}}\). We show that the kernel of this representation is always the Torrelli subgroup \(\text {IA}_r\) of \(\text {Aut}(F_r)\).

We define flag structures on a real three manifold M as the choice of two complex lines on the complexified tangent space at each point of M. We suppose that the plane field defined by the complex lines is a contact plane and construct an adapted connection on an appropriate principal bundle. This includes path geometries and CR structures as special cases. We prove that the null curvature models are

Let G be a connected Lie group and \(\varGamma \subset G\) a lattice. Connection curves of the homogeneous space \(M=G/\varGamma \) are the orbits of one parameter subgroups of G. To block a pair of points \(m_1,m_2 \in M\) is to find a finite set \(B \subset M{\setminus } \{m_1, m_2 \}\) such that every connecting curve joining \(m_1\) and \(m_2\) intersects B. The homogeneous space M is blockable

We construct in projective differential geometry of the real dimension 2 higher symmetry algebra of the symplectic Dirac operator acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overdetermined operators of arbitrarily high order acting on symmetric tensors. The higher symmetry algebra structure corresponds

A collection \( \Delta \) of simple closed curves on an orientable surface is an algebraic k-system if the algebraic intersection number \( \langle \alpha , \beta \rangle \) is equal to k in absolute value for every \( \alpha , \beta \in \Delta \). Generalizing a theorem of Malestein et al. (Geom Dedicata 168(1):221–233, 2014. doi:10.1007/s10711-012-9827-9) we compute that the maximum size of an algebraic