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We prove that every Jordan curve in $$\mathbb {R}^2$$ inscribes uncountably many rhombi. No regularity condition is assumed on the Jordan curve.

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

Let G be a noncompact connected simple Lie group, and $$G^\sigma $$ the subgroup of the fixed points under the action of $$\sigma $$ on G, where $$\sigma $$ is an automorphism of G of even order. This article shows a necessary condition for the pair $$(G,G^\sigma )$$ satisfying condition (D.D.); that is, there exists at lease one infinite-dimensional simple $$({\mathfrak {g}},K)$$ -module that is discretely decomposable as a $$({\mathfrak {g}}^\sigma ,K^\sigma )$$ -module.

We construct simply connected, minimal, symplectic 4-manifolds with exotic smooth structures and each with one Seiberg-Witten basic class up to sign, on the Noether line and between the Noether and half Noether lines by star surgeries introduced by Karakurt and Starkston, and by using complex singularities. We also construct certain configurations of complex singularities in the rational elliptic surfaces geometrically, without using any monodromy arguments. By using these configurations, we give symplectic embeddings of star shaped plumbings inside (some blow-ups of) elliptic surfaces., Two minor typos are corrected. Accepted version

In this paper, we investigate the gap phenomena for complete Lagrangian submanifolds satisfying $$\nabla ^{*}\varvec{T}\equiv 0$$ in complex space forms $$\mathbf{N}^{n}(4c)$$ , where $$\varvec{T}=\frac{1}{n}\nabla ^{*}\varvec{B}$$ and $$\varvec{B}$$ is the Lagrangian trace-free second fundamental form. We prove that under some pointwise or integral curvature pinching condition, the submanifold is either totally geodesic or isometric to the Whitney sphere.

We give an optimal upper bound of the degree of quasi-smooth hypersurfaces which are invariant by a one-dimensional holomorphic foliation on a compact toric orbifold, i.e. on a complete simplicial toric variety. This bound depends only on the degree of the foliation and of the degrees of the toric homogeneous coordinates.

We show that a smooth embedded paper Moebius band must have aspect ratio at least $$\begin{aligned}\lambda _1= \frac{2 \sqrt{4-2 \sqrt{3}}+4}{\root 4 \of {3} \sqrt{2}+2 \sqrt{2 \sqrt{3}-3}} =1.69497\ldots \end{aligned}$$ This bound comes more than 3/4 of the way from the old known bound of $$\pi /2=1.5708\ldots $$ to the conjectured bound of $$\sqrt{3} = 1.732\ldots $$

Let E be a vector bundle generated by sections and L be an ample line bundle over a smooth projective variety X. We give here a condition for the vanishing of Dolbeault cohomology groups of the form $$H^{p,q}(X,{{\mathcal {S}}}^{\alpha }E\otimes \wedge ^{\beta } E\otimes L)$$ . In most cases the result is stronger than what has been obtained for the condition that E is nef and L is ample. In particular a result of Zhi Jiang is improved. To derive our result we introduce a new spectral sequence, which can be used in a broader context.

We study the branched holomorphic projective structures on a compact Riemann surface X with a fixed branching divisor $$S\, =\, \sum _{i=1}^d x_i$$ , where $$x_i \,\in \, X$$ are distinct points. After defining branched $$\mathrm{SO}(3,{\mathbb C})$$ -opers, we show that the branched holomorphic projective structures on X are in a natural bijection with the branched $$\mathrm{SO}(3,{\mathbb C})$$ -opers singular at S. It is deduced that the branched holomorphic projective structures on X are also identified with a subset of the space of all logarithmic connections on $$J^2((TX)\otimes {\mathcal O}_X(S))$$ singular over S, satisfying certain natural geometric conditions.

We study liftable vector fields of smooth map-germs. We show how to obtain the module of liftable vector fields of any map-germ of finite singularity type from the module of liftable vector fields of a stable unfolding of it. As an application, we obtain the liftable vector fields for the family $$H_k$$ H k in Mond’s list. We then show the relation between the liftable vector fields of a stable germ and its augmentations.

In this paper, we define non-linear versions of Banach–Mazur distance in the contact geometry set-up, called contact Banach–Mazur distances and denoted by $$d_\mathrm{CBM}$$ . Explicitly, we consider the following two set-ups, either on a contact manifold $$W \times S^1$$ where W is a Liouville manifold, or a closed Liouville-fillable contact manifold M. The inputs of $$d_\mathrm{CBM}$$ are different in these two cases. In the former case the inputs are (contact) star-shaped domains of $$W \times S^1$$ which correspond to the homotopy classes of positive contact isotopies, and in the latter case the inputs are contact 1-forms of M inducing the same contact structure. In particular, the contact Banach–Mazur distance $$d_\mathrm{CBM}$$ defined in the former case is motivated by the concept, relative growth rate, which was originally defined and studied by Eliashberg–Polterovich. The main results are the large-scale geometric properties in terms of $$d_\mathrm{CBM}$$ . In addition, we propose a quantitative comparison between elements in a certain subcategory of the derived categories of sheaves of modules (over certain topological spaces). This is based on several important properties of the singular support of sheaves and Guillermou–Kashiwara–Schapira’s sheaf quantization.

We construct a family of complete n-dimensional ( $$n\ge 4$$ ) manifolds with nonnegative Ricci curvature and infinite topological type. Moreover, the curvature decay, diameter growth and volume growth can be precisely calculated. This is actually the generalization of Menguy’s examples.

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces $$(M=G/H,g)$$ whose geodesics are orbits of one-parameter subgroups of G. The corresponding metric g is called a geodesic orbit metric. We study the geodesic orbit spaces of the form (G/H, g), such that G is one of the compact classical Lie groups $${{\,\mathrm{SO}\,}}(n)$$ , $$\mathrm{U}(n)$$ , and H is a diagonally embedded product $$H_1\times \cdots \times H_s$$ , where $$H_j$$ is of the same type as G. This class includes spheres, Stiefel manifolds, Grassmann manifolds and real flag manifolds. The present work is a contribution to the study of g.o. spaces (G/H, g) with H semisimple.

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 pairs—and results of N. Ressayre. This extends a result obtained independently by L. Manivel and E. Vallejo in 2014 or 2015, expressed in terms of additive matrices. To illustrate the fact that it allows to produce quite a few new faces of the Kronecker cone, we give at the end of the article details about what our results yield for “small dimensions”.

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 which asserts that a cyclic Higgs bundle gives a solution of the periodic Toda equation., The numbering of theorems, propositions, examples, corollaries, lemmas, definitions, and remarks has been made the same as in the published version. 23 pages

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 Kahler and thus $${\mathfrak {F}}_4''$$ inherits this Kahler structure.

The star transform is a generalized Radon transform mapping a function of two variables to its integrals along “star-shaped” trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in mathematical models of various imaging modalities based on scattering of elementary particles. The paper presents a comprehensive study of the inversion of the star transform. We describe the necessary and sufficient conditions for invertibility of the star transform, introduce a new inversion formula and discuss its stability properties. As an unexpected bonus of our approach, we prove a conjecture from algebraic geometry about the zero sets of elementary symmetric polynomials.

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 Euler characteristic of the hyperbolic groups constructed, we prove that infinitely many of them are pairwise non isomorphic. We further show that the first of these constructions cannot be generalised to dimensions higher than 3.

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 lines are discussed.

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 distinguishing Heegaard splittings. In particular, we show that the Nielsen classes of the generators of the fundamental group, obtained from spines of the 4-dimensional 1-handlebodies of the trisection, are isotopy invariants of the trisection. If we additionally consider the action of the automorphism group on the Nielsen classes we obtain diffeomorphism invariants. Once the invariance is established, we analyze the Nielsen classes of the spun trisections in terms of the Nielsen classes of the original Heegaard splitting, and leverage work of Lustig, Moriah, and Rosenberger on Nielsen classes of Fuchsian groups in order to distinguish the trisections.

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.

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., Comment: 13 pages, 2 figures; added arXiv link to November paper

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

We show a kind of Obata-type theorem on a compact Einstein n-manifold $$(W, \bar{g})$$ ( W , g ¯ ) with smooth boundary $$\partial W$$ ∂ W . Assume that the boundary $$\partial W$$ ∂ W is minimal in $$(W, \bar{g})$$ ( W , g ¯ ) . If $$(\partial W, \bar{g}|_{\partial W})$$ ( ∂ W , g ¯ | ∂ W ) is not conformally diffeomorphic to $$(S^{n-1}, g_S)$$ ( S n - 1 , g S ) , then for any Einstein metric $$\check{g} \in [\bar{g}]$$ g ˇ ∈ [ g ¯ ] with the minimal boundary condition, we have that, up to rescaling, $$\check{g} = \bar{g}$$ g ˇ = g ¯ . Here, $$g_S$$ g S and $$[\bar{g}]$$ [ g ¯ ] denote respectively the standard round metric on the $$(n-1)$$ ( n - 1 ) -sphere $$S^{n-1}$$ S n - 1 and the conformal class of $$\bar{g}$$ g ¯ . Moreover, if we assume that $$\partial W \subset (W, \bar{g})$$ ∂ W ⊂ ( W , g ¯ ) is totally geodesic, we also show a Gursky-Han type inequality for the relative Yamabe constant of $$(W, \partial W, [\bar{g}])$$ ( W , ∂ W , [ g ¯ ] ) .

We study a class of continuous deformations of branched complex projective structures on closed surfaces of genus $g\geq 2$, 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, and we establish a partial converse for hyperelliptic structures., Comment: 29 pages, 2 figures; v2: funding information updated

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 criterion for the two possibilities is provided.

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.

The Hitchin component Hit_n(S) of a closed surface S is a preferred component of the character variety X_PSL_n(R)(S) consisting of homomorphisms from the fundamental group pi_1(S) to the Lie group PSL_n(R)(S), whose elements enjoy remarkable geometric and dynamical properties. We consider a certain type of deformations of the elements of Hit_n(S), called shearing deformations, and compute their pairing for the Atiyah-Bott-Goldman symplectic form of the character variety., Comment: 25 pages

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., 6 pages, A typo is corrected. To appear in Geometriae Dedicata

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 of connectedness by triples with zero sum in a general subset of a vector space and proved that the set of $${\mathfrak {t}}$$ -roots satisfies this property. Using this result, the invariant $${\mathcal {G}}_1$$ structures on $${\mathbb {F}}_{\Theta }$$ are completely classified.

We give a complete description of all locally conformally K\"ahler 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 manifold, any lcK metric with potential and with holomorphic Lee vector field admits a potential which is positive and invariant along the anti-Lee vector field., Comment: 16 pages

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 can be joined by a {\em generic} chain of twistor lines. We also prove a criterion of twistor path connectivity of loci in $Compl$ where a fixed second cohomology class stays of Hodge type (1,1). Furthermore, we show that twistor lines are holomorphic submanifolds of $Compl$, of degree $2n$ in the Pl\"ucker embedding of $Compl$., Comment: 29 pages, 2 figures. Theorem 2 and Corollary 3 have been added in the new version

For parameters $n,\delta,B,$ and $C$, let $X=(X_{k\ell})$ be the random uniform contingency table whose first $\lfloor n^{\delta} \rfloor $ rows and columns have margin $\lfloor BCn \rfloor$ and the last $n$ rows and columns have margin $\lfloor Cn \rfloor$. For every $0B_{c}$. Our main result shows that $\mathbb{E}[X_{11}]$ is uniformly bounded for $BB_{c}$. We also establish a strong law of large numbers for the row sums in top right and top left blocks., Comment: 24 pages, 4 figures. Earlier version is accepted for publication in Transactions of the AMS. This version contains an appendix on asymptotic independence of the entries

The Following is a Research Thesis by the author of the 8T. The new Infinitytheory aims to present the subject of Infinities as analyzedfrom a sync of set theory and the author forte, Calculus of Variations/Lagrangians. Insert V1.1 - Decays of Infinities - Page 73. The file include three parts, first the Variational approach on Infinities, the second part -classification of Infinitiesand the third andlast part is the Nature of Infinites. Not all theorems are .proven, this will be presented at later versions. The Theory contains four main equations, whichare (1),(1.2) and equation(2) The Thesis is written in a form of self refection, allowing the reader to experience how the ideas varied over time.&nbsp

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 $$\mathrm {SU}(3)$$ SU ( 3 ) -structure admitting a two-torus symmetry. Projecting the subspaces obtained to the orbit space yields a trivalent graph. We illustrate this result concretely on the homogeneous nearly Kähler examples.