In this short note, we deal with complete noncompact expanding and steady Ricci solitons of dimension $n\geq 3.$ More precisely, under an integrability assumption, we obtain a characterization for the generalized cigar Ricci soliton and the Gaussian Ricci soliton.

We prove Reilly-type upper bounds for the first nonzero eigenvalue of the Steklov problem associated with the p-Laplace operator on submanifolds of manifolds with sectional curvature bounded from above by a nonnegative constant.

Kanai proved powerful results on the stability under quasi-isometries of numerous global properties (including Liouville property) between Riemannian manifolds of bounded geometry. Since his work focuses more on the generality of the spaces considered than on the two-dimensional geometry, Kanai's hypotheses in many cases are not satisfied in the context of Riemann surfaces endowed with the Poincaré metric. In this work we fill that gap for the Liouville property, by proving its stability by quasi-isometries for every Riemann surface (and even Riemannian surfaces with pinched negative curvature). Also, a key result characterizes Riemannian surfaces which are quasi-isometric to $\mathbb {R}$.

We show that a complete, two-sided, stable immersed anisotropic minimal hypersurface in $\mathbf {R}^4$ has intrinsic cubic volume growth, provided the parametric elliptic integral is $C^2$ -close to the area functional. We also obtain an interior volume upper bound for stable anisotropic minimal hypersurfaces in the unit ball. We can estimate the constants explicitly in all of our results. In particular, this paper gives an alternative proof of our recent stable Bernstein theorem for minimal hypersurfaces in $\mathbf {R}^4$ . The new proof is more closely related to techniques from the study of strictly positive scalar curvature.

We develop the theory of left-invariant generalized pseudo-Riemannian metrics on Lie groups. Such a metric accompanied by a choice of left-invariant divergence operator gives rise to a Ricci curvature tensor, and we study the corresponding Einstein equation. We compute the Ricci tensor in terms of the tensors (on the sum of the Lie algebra and its dual) encoding the Courant algebroid structure, the generalized metric, and the divergence operator. The resulting expression is polynomial and homogeneous of degree 2 in the coefficients of the Dorfman bracket and the divergence operator with respect to a left-invariant orthonormal basis for the generalized metric. We determine all generalized Einstein metrics on three-dimensional Lie groups.

We show that the singularities of the twisted Kähler–Einstein metric arising as the longtime solution of the Kähler–Ricci flow or in the collapsed limit of Ricci-flat Kähler metrics are intimately related to the holomorphic sectional curvature of reference conical geometry. This provides an alternative proof of the second-order estimate obtained by Gross, Tosatti, and Zhang (2020, Preprint, arXiv:1911.07315) with explicit constants appearing in the divisorial pole.

We investigate the pluriclosed flow on Oeljeklaus–Toma manifolds. We parameterize left-invariant pluriclosed metrics on Oeljeklaus–Toma manifolds, and we classify the ones which lift to an algebraic soliton of the pluriclosed flow on the universal covering. We further show that the pluriclosed flow starting from a left-invariant pluriclosed metric has a long-time solution $\omega _t$ which once normalized collapses to a torus in the Gromov–Hausdorff sense. Moreover, the lift of $\tfrac {1}{1+t}\omega _t$ to the universal covering of the manifold converges in the Cheeger–Gromov sense to $(\mathbb H^s\times \mathbb C^s, \tilde {\omega }_{\infty })$ , where $\tilde {\omega }_{\infty }$ is an algebraic soliton.

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.