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 \(Sp(n+1)/Sp(n-1) Sp(1)\)

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

Let \(\text {Mod}(S_g)\) be the mapping class group of the closed orientable surface \(S_g\) of genus \(g\ge 2\). In this paper, we derive necessary and sufficient conditions for two finite-order mapping classes to have commuting conjugates in \(\text {Mod}(S_g)\). As an application of this result, we show that any finite-order mapping class, whose corresponding orbifold is not a sphere, has a conjugate

Let G be a group. The orbits of the natural action of \({{\,\mathrm{Aut}\,}}(G)\) on G are called “automorphism orbits” of G, and the number of automorphism orbits of G is denoted by \(\omega (G)\). We prove that if G is a soluble group of finite rank such that \(\omega (G)< \infty \), then G contains a torsion-free radicable nilpotent characteristic subgroup K such that \(G = K \rtimes H\), where

In this article we study the holonomy groups of flat solvmanifolds. It is known that the holonomy group of a flat solvmanifold is abelian; we give an elementary proof of this fact and moreover we prove that any finite abelian group is the holonomy group of a flat solvmanifold. Furthermore, we show that the minimal dimension of a flat solvmanifold with holonomy group \({\mathbb {Z}}_n\) coincides with

We prove that conformal immersion of a Riemannian product \(M_0^{n_0}\times M_1^{n_1}\) as a hypersurface in a Euclidean space must be an extrinsic product of immersions, under the assumption that \(n_0, n_1 \ge 2\) and that \(M^{n_0}_0\times M^{n_1}_1\) is not conformally flat. We also state a similar theorem for an arbitrary number of factors, more precisely, a conformal immersion \(f:M^{n_0}_0 \times

We characterize Riemannian orbifolds and their coverings in terms of metric geometry. In particular, we show that the metric double of a Riemannian orbifold along the closure of its codimension one stratum is a Riemannian orbifold and that the natural projection is an orbifold covering.

The Wiman–Edge pencil is the universal family \(C_t, t\in {\mathcal {B}}\) of projective, genus 6, complex-algebraic curves admitting a faithful action of the icosahedral group \(\mathfrak {A}_5\). The curve \(C_0\), discovered by Wiman in 1895 (Ueber die algebraische Curven von den Geschlecht \(p=4,5\) and 6 welche eindeutige Transformationen in sich besitzen) and called the Wiman curve, is the unique

An immersed surface in \({{\mathbb {R}}}^4\) is said to has constant Jordan angles (CJA) if the angles between its tangent planes and a fixed plane do not depend on the choice of the point. The constant Jordan angles surfaces in \({{\mathbb {R}}}^4\) has been proved to exist, Bayard et al. (Geom Dedicata 162:153–176, 2013), but there are only explicit examples of non planar surfaces for the extremal

Let \(\Omega \subset {\mathbb {R}}^{n-1}\) be a bounded open set, \(X=\Omega \times {\mathbb {R}}\subseteq {\mathbb {R}}^{n}\) be the infinite strip. Let L be a second order uniformly elliptic operator of divergence form acting on a function \(f\in W_{\text {loc}}^{1,2}(X)\) given by \(Lf=\sum _{i,j=1}^{n}\frac{\partial }{\partial x_{i}}\bigl (a^{ij}(x)\frac{\partial f}{\partial x_{j}}\bigr )\). It

We give a condition for a function to produce a Möbius invariant weighted inner product on the tangent space of the space of knots, and show that some kind of Möbius invariant knot energies can produce Möbius invariant and parametrization invariant weighted inner products. They would give a natural way to study the evolution of knots in the framework of Möbius geometry.

We prove a general structural theorem about rectangles inscribed in Jordan loops. One corollary is that all but at most 4 points of any Jordan loop are vertices of inscribed rectangles. Another corollary is that a Jordan loop has an inscribed rectangle of every aspect ratio provided it has 3 points which are not vertices of inscribed rectangles.

We derive a lower bound on the size of finite non-cyclic quotients of the braid group that is superexponential in the number of strands. We also derive a similar lower bound for nontrivial finite quotients of the commutator subgroup of the braid group.

We prove existence and regularity results for energy minimizing maps between ideal hyperbolic 2-dimensional simplicial complexes. The spaces in question were introduced by Charitos–Papadopoulos, who describe their Teichmüller spaces and some compactifications. This work is a first step in introducing harmonic map theory into the Teichmüller theory of these spaces.

We give a necessary and sufficient condition on a d-dimensional affine subspace of \({\mathbb {R}}^n\) to be characterized by a finite set of patterns which are forbidden to appear in its digitization. This can also be stated in terms of local rules for canonical projection tilings, or subshift of finite type. This provides a link between algebraic properties of affine subspaces and combinatorics of

We prove that a general determinantal hypersurface of dimension 3 is nodal. Moreover, in terms of Chern classes associated with bundle morphisms, we derive a formula for the intersection homology Euler characteristic of a general determinantal hypersurface.

We consider a finitely generated group endowed with a word metric. The group acts on itself by isometries, which induces an action on its horofunction boundary. The conjecture is that nilpotent groups act trivially on their reduced boundary. We will show this for the Heisenberg group. The main tool will be a discrete version of the isoperimetric inequality.

In this short note, we use the Bochner technique and the Hodge theory in complex differential geometry to prove several injectivity results for the cohomology of holomorphic vector bundles on compact Kähler manifolds, which generalize Enoki’s original injectivity theorem.

In this paper we translate the necessary and sufficient conditions of Tanaka’s theorem on the finiteness of effective prolongations of a fundamental graded Lie algebras into computationally effective criteria, involving the rank of some matrices that can be explicitly constructed. Our results would apply to geometries, which are defined by assigning a structure algebra on the contact distribution.

Let \(\varSigma \) be a compact, orientable surface of negative Euler characteristic, and let h be a complete hyperbolic metric on \(\varSigma \). A geodesic curve \(\gamma \) in \(\varSigma \) is filling if it cuts the surface into topological disks and annuli. We propose an efficient algorithm for deciding whether a geodesic curve, represented as a word in some generators of \(\pi _1(\varSigma )\)

The trigonal curves of genus 5 can be represented by projective plane quintics that have one singularity of delta invariant one. Combining this with a partial sieve method for plane curves we count the number of such curves over any finite field. The main application is that this gives the motivic Euler characteristic of the moduli space of trigonal curves of genus 5.

The purpose of this paper is to investigate relationship between the automorphism group of a rational surface and that of its Hilbert scheme of n points.

We explore hybrid subgroups of certain non-arithmetic lattices in \({\text {PU}}(2,1)\). In particular, we show that all of Mostow’s lattices are virtually hybrids; moreover, we show that some of these non-arithmetic lattices are virtually hybrids of two non-commensurable arithmetic lattices in \({\text {PU}}(1,1)\).

Let M be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by \(\sigma \). Let \(f: M \rightarrow {\mathbb {R}}\) be a non-constant eigenfunction of the Laplacian. The random wave conjecture suggests that in certain situations, the value distribution of f under \(\sigma \) is approximately Gaussian. Write \(\mu \) for the measure whose density with respect to \(\sigma

The action dimension of a discrete group G is the minimum dimension of a contractible manifold, which admits a proper G-action. In this paper, we study the action dimension of general Artin groups. The main result is that if an Artin group with the nerve L of dimension n for \(n \ne 2\) satisfies the \(K(\pi , 1)\)-Conjecture and the top cohomology group of L with \({\mathbb {Z}}\)-coefficients is

This paper is devoted to extend some Hu–Keel results on Mori dream spaces (MDS) beyond the projective setup. Namely, \(\mathbb {Q}\)-factorial algebraic varieties with finitely generated class group and Cox ring, here called weak Mori dream spaces (wMDS), are considered. Conditions guaranteeing the existence of a neat embedding of a (completion of a) wMDS into a complete toric variety are studied,

We use tools from combinatorial group theory in order to study actions of three types on groups acting on a curve, namely the automorphism group of a compact Riemann surface, the mapping class group acting on a surface (which now is allowed to have some points removed) and the absolute Galois group \(\mathrm {Gal}({\bar{{\mathbb {Q}}}}/{\mathbb {Q}})\) in the case of cyclic covers of the projective

We construct a natural prequantization space over a monotone product of a toric manifold and an arbitrary number of complex Grassmannians of 2-planes in even-dimensional complex spaces, and prove that the universal cover of the identity component of the contactomorphism group of its total space carries a nonzero homogeneous quasi-morphism. The construction uses Givental’s nonlinear Maslov index and

In this paper we give a complete description of the horofunction boundary of the infinite dimensional real hyperbolic space, and characterise its Busemann points.