This paper is about a type of quantitative density of closed geodesics and orthogeodesics on complete finite-area hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic and the shortest doubly truncated orthogeodesic that are $\varepsilon$-dense on a given compact set on the surface.

Let $n\geq 2$ random lines intersect a planar convex domain D. Consider the probabilities $p_{nk}$, $k=0,1, \ldots, {n(n-1)}/{2}$ that the lines produce exactly k intersection points inside D. The objective is finding $p_{nk}$ through geometric invariants of D. Using Ambartzumian’s combinatorial algorithm, the known results are instantly reestablished for $n=2, 3$. When $n=4$, these probabilities are expressed by new invariants of D. When D is a disc of radius r, the simplest forms of all invariants are found. The exact values of $p_{3k}$ and $p_{4k}$ are established.

In this short paper, we show a sufficient condition for the solvability of the Dirichlet problem at infinity in Riemannian cones (as defined below). This condition is related to a celebrated result of Milnor that classifies parabolic surfaces. When applied to smooth Riemannian manifolds with a special type of metrics, which generalize the class of metrics with rotational symmetry, we obtain generalizations of classical criteria for the solvability of the Dirichlet problem at infinity. Our proof is short and elementary: it uses separation of variables and comparison arguments for ODEs.

In the present note, we establish a finiteness theorem for $L^p$ harmonic 1-forms on hypersurfaces with finite index, which is an extension of the result of Choi and Seo (J. Geom. Phys. 129 (2018), 125–132).

Crofton formulas on simply connected Riemannian space forms allow the volumes, or more generally the Lipschitz–Killing curvature integrals of a submanifold with corners, to be computed by integrating the Euler characteristic of its intersection with all geodesic submanifolds. We develop a framework of Crofton formulas with distributions replacing measures, which has in its core Alesker's Radon transform on valuations. We then apply this framework, and our recent Hadwiger-type classification, to compute explicit Crofton formulas for all isometry-invariant valuations on all pseudospheres, pseudo-Euclidean and pseudohyperbolic spaces. We find that, in essence, a single measure which depends analytically on the metric, gives rise to all those Crofton formulas through its distributional boundary values at parts of the boundary corresponding to the different indefinite signatures. In particular, the Crofton formulas we obtain are formally independent of signature.

For subsets in the standard symplectic space $(\mathbb {R}^{2n},\omega _0)$ whose closures are intersecting with coisotropic subspace $\mathbb {R}^{n,k}$ we construct relative versions of the Ekeland–Hofer capacities of the subsets with respect to $\mathbb {R}^{n,k}$, establish representation formulas for such capacities of bounded convex domains intersecting with $\mathbb {R}^{n,k}$. We also prove a product formula and a fact that the value of this capacity on a hypersurface $\mathcal {S}$ of restricted contact type containing the origin is equal to the action of a generalized leafwise chord on $\mathcal {S}$.

We construct one-dimensional foliations which are subfoliations of two-dimensional foliations in $3$-manifolds. The subfoliation is by quasigeodesics in each two-dimensional leaf, but it is not funnel: not all quasigeodesics share a common ideal point in most leaves.

We prove a version of the Fatou theorem for bounded functions with a bounded $\overline \partial _J$ part of the differential on wedge-type domains in an almost complex manifold.

In this paper we introduce two new classes of stationary random simplicial tessellations, the so-called $\beta$- and $\beta^{\prime}$-Delaunay tessellations. Their construction is based on a space–time paraboloid hull process and generalizes that of the classical Poisson–Delaunay tessellation. We explicitly identify the distribution of volume-power-weighted typical cells, establishing thereby a remarkable connection to the classes of $\beta$- and $\beta^{\prime}$-polytopes. These representations are used to determine the principal characteristics of such cells, including volume moments, expected angle sums, and cell intensities.

We construct examples of compact homogeneous Riemannian manifolds admitting an invariant Bismut connection that is Ricci flat and non-flat, proving in this way that the generalized Alekseevsky–Kimelfeld theorem does not hold. The classification of compact homogeneous Bismut Ricci flat spaces in dimension $5$ is also provided. Moreover, we investigate compact homogeneous spaces with non-trivial third Betti number, and we point out other possible ways to construct Bismut Ricci flat manifolds. Finally, since Bismut Ricci flat connections correspond to fixed points of the generalized Ricci flow, we discuss the stability of some of our examples under the flow.

Given $a,\,b\in \mathbb {R}$ and $\Phi \in C^{1}(\mathbb {S}^{2})$, we study immersed oriented surfaces $\Sigma$ in the Euclidean 3-space $\mathbb {R}^{3}$ whose mean curvature $H$ and Gauss curvature $K$ satisfy $2aH+bK=\Phi (N)$, where $N:\Sigma \rightarrow \mathbb {S}^{2}$ is the Gauss map. This theory widely generalizes some of paramount importance such as the ones constant mean and Gauss curvature surfaces, linear Weingarten surfaces and self-translating solitons of the mean curvature flow. Under mild assumptions on the prescribed function $\Phi$, we exhibit a classification result for rotational surfaces in the case that the underlying fully nonlinear PDE that governs these surfaces is elliptic or hyperbolic.

We determine all three-dimensional homogeneous and $1$-curvature homogeneous Lorentzian metrics which are critical for a quadratic curvature functional. As a result, we show that any quadratic curvature functional admits different non-Einstein homogeneous critical metrics and that there exist homogeneous metrics which are critical for all quadratic curvature functionals without being Einstein.

Let $(\mathbb {D}^2,\mathscr {F},\{0\})$ be a singular holomorphic foliation on the unit bidisc $\mathbb {D}^2$ defined by the linear vector field

$$ \begin{align*} z \frac{\partial}{\partial z}+ \unicode{x3bb} w \frac{\partial}{\partial w}, \end{align*} $$

where $\unicode{x3bb} \in \mathbb {C}^*$. Such a foliation has a non-degenerate singularity at the origin ${0:=(0,0) \in \mathbb {C}^2}$. Let T be a harmonic current directed by $\mathscr {F}$ which does not give mass to any of the two separatrices $(z=0)$ and $(w=0)$. Assume $T\neq 0$. The Lelong number of T at $0$ describes the mass distribution on the foliated space. In 2014 Nguyên (see [16]) proved that when $\unicode{x3bb} \notin \mathbb {R}$, that is, when $0$ is a hyperbolic singularity, the Lelong number at $0$ vanishes. Suppose the trivial extension $\tilde {T}$ across $0$ is $dd^c$-closed. For the non-hyperbolic case $\unicode{x3bb} \in \mathbb {R}^*$, we prove that the Lelong number at $0$:

1. (1) is strictly positive if $\unicode{x3bb}>0$;

2. (2) vanishes if $\unicode{x3bb} \in \mathbb {Q}_{<0}$;

3. (3) vanishes if $\unicode{x3bb} <0$ and T is invariant under the action of some cofinite subgroup of the monodromy group.

Campana introduced the class of special varieties as the varieties admitting no Bogomolov sheaves, i.e. rank-one coherent subsheaves of maximal Kodaira dimension in some exterior power of the cotangent bundle. Campana raised the question of whether one can replace the Kodaira dimension by the numerical dimension in this characterization. We answer partially this question showing that a projective manifold admitting a rank-one coherent subsheaf of the cotangent bundle with numerical dimension one is not special. We also establish the analytic characterization with the non-existence of Zariski dense entire curve and the arithmetic version with non-potential density in the (split) function field setting. Finally, we conclude with a few comments for higher codimensional foliations which may provide some evidence towards a generalization of the aforementioned results.

In 1977, Gauduchon proved that on every compact hermitian manifold $(X, \omega )$ there exists a conformally equivalent hermitian metric $\omega _\mathrm {G}$ which satisfies $\mathrm {dd}^{\mathrm {c}} \omega _\mathrm {G}^{n-1} = 0$. In this note, we extend this result to irreducible compact singular hermitian varieties which admit a smoothing.

In this work, we study foliations of arbitrary codimension $\mathfrak{F}$ with integrable normal bundles on complete Riemannian manifolds. We obtain a necessary and sufficient condition for $\mathfrak{F}$ to be totally geodesic. For this, we introduce a special number $\mathfrak{G}_{\mathfrak{F}}^{\alpha}$ that measures when the foliation ceases to be totally geodesic. Furthermore, applying some maximum principle we deduce geometric properties for $\mathfrak{F}$. We conclude with a geometrical version of Novikov’s theorem (Trans. Moscow Math. Soc. (1965), 268–304), for Riemannian compact manifolds of arbitrary dimension.

We provide examples of infinitesimally Hilbertian, rectifiable, Ahlfors regular metric measure spaces having pmGH-tangents that are not infinitesimally Hilbertian.

As is well known, the holomorphic sectional curvature is just half of the sectional curvature in a holomorphic plane section on a Kähler manifold (Zheng, Complex differential geometry (2000)). In this article, we prove that if the holomorphic sectional curvature is half of the sectional curvature in a holomorphic plane section on a Hermitian manifold then the Hermitian metric is Kähler.

In our previous paper (Gaster et al., 2018, arXiv:1810.11932), we showed that the theory of harmonic maps between Riemannian manifolds, especially hyperbolic surfaces, may be discretized by introducing a triangulation of the domain manifold with independent vertex and edge weights. In the present paper, we study convergence of the discrete theory back to the smooth theory when taking finer and finer triangulations, in the general Riemannian setting. We present suitable conditions on the weighted triangulations that ensure convergence of discrete harmonic maps to smooth harmonic maps, introducing the notion of (almost) asymptotically Laplacian weights, and we offer a systematic method to construct such weighted triangulations in the two-dimensional case. Our computer software Harmony successfully implements these methods to compute equivariant harmonic maps in the hyperbolic plane.

We investigate the geometry of Hermitian manifolds endowed with a compact Lie group action by holomorphic isometries with principal orbits of codimension one. In particular, we focus on a special class of these manifolds constructed by following Bérard-Bergery which includes, among the others, the holomorphic line bundles on $\mathbb {C}\mathbb {P}^{m-1}$, the linear Hopf manifolds and the Hirzebruch surfaces. We characterize their invariant special Hermitian metrics, such as balanced, Kähler-like, pluriclosed, locally conformally Kähler, Vaisman and Gauduchon. Furthermore, we construct new examples of cohomogeneity one Hermitian metrics solving the second-Chern–Einstein equation and the constant Chern-scalar curvature equation.