We show that complete uniform visibility manifolds of finite volume with sectional curvature $- 1\leq K\leq 0$ have positive simplicial volume. This implies that their minimal volume is nonzero.

We prove the classification of the real vector subspaces of a quaternionic vector space by using a covariant functor which associates, to any pair formed of a quaternionic vector space and a real subspace, a coherent sheaf over the sphere.

We study a class of Hermitian metrics on complex manifolds, recently introduced by Fu, Wang and Wu, which are a generalization of Gauduchon metrics. This class includes the class of Hermitian metrics for which the associated fundamental 2-form is ∂∂-closed. Examples are given on nilmanifolds, on products of Sasakian manifolds, on S1-bundles and via the twist construction introduced by Swann.

We prove a vanishing theorem for the twisted de Rham cohomology of a compact manifold.

In this paper, we give a classification of spacelike submanifolds with parallel normalised mean curvature vector field and linear relation $R= aH+ b$ of the normalised scalar curvature $R$ and the mean curvature $H$ in the de Sitter space ${ S}_{p}^{n+ p} (c)$.

We prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky’s theory of hyperholomorphic sheaves and a study of the cohomology algebra of Hilbert schemes of K3 surfaces.

Slant curves are introduced in three-dimensional warped products with Euclidean factors. These curves are characterised by the scalar product between the normal at the curve and the vertical vector field, and an important feature is that the case of constant Frenet curvatures implies a proper mean curvature vector field. A Lancret invariant is obtained and the Legendre curves are analysed as a particular case. An example of a slant curve is given for the exponential warping function; our example illustrates a proper (that is, not reducible to the two-dimensional) case of the Lancret theorem of three-dimensional hyperbolic geometry. We point out an eventuality relationship with the geometry of relativistic models.

We study Fredholm properties and index formulas for Dirac operators over complete Riemannian manifolds with straight ends. An important class of examples of such manifolds are complete Riemannian manifolds with pinched negative sectional curvature and finite volume.

This work is a continuation of the author’s previous paper [Greatest lower bounds on the Ricci curvature of toric Fano manifolds, Adv. Math. 226 (2011), 4921–4932]. On any toric Fano manifold, we discuss the behavior of the limit metric of a sequence of metrics which are solutions to a continuity family of complex Monge–Ampère equations in the Kähler–Einstein problem. We show that the limit metric satisfies a singular complex Monge–Ampère equation. This gives a conic-type singularity for the limit metric. Information on conic-type singularities can be read off from the geometry of the moment polytope.

We give a generalisation of the Cartan decomposition for connected compact Lie groups of type B motivated by the work on visible actions of Kobayashi [‘A generalized Cartan decomposition for the double coset space $(U(n_{1})\times U(n_{2})\times U(n_{3})) \backslash U(n)/ (U(p)\times U(q))$’, J. Math. Soc. Japan59 (2007), 669–691] for type A groups. Suppose that $G$ is a connected compact Lie group of type B, $\sigma $ is a Chevalley–Weyl involution and $L$, $H$ are Levi subgroups. First, we prove that $G=LG^{\sigma }H$ holds if and only if either (I) both $H$ and $L$ are maximal and of type A, or (II) $(G,H)$ is symmetric and $L$ is the Levi subgroup of an arbitrary maximal parabolic subgroup up to switching $H$ and $L$. This classification gives a visible action of $L$ on the generalised flag variety $G/H$, as well as that of the $H$-action on $G/L$ and of the $G$-action on $(G\times G)/(L\times H)$. Second, we find an explicit ‘slice’ $B$ with $\dim B=\mathrm {rank}\, G$ in case I, and $\dim B=2$ or $3$ in case II, such that a generalised Cartan decomposition $G=LBH$holds. An application to multiplicity-free theorems of representations is also discussed.

We prove the existence and uniqueness of harmonic maps between rotationally symmetric manifolds that are asymptotically hyperbolic.

Let ℱ be a Kähler foliation on a compact Riemannian manifold M. If the transversal scalar curvature of ℱ is nonzero constant, then any transversal conformal field is a transversal Killing field; and if the transversal Ricci curvature is nonnegative and positive at some point, then there are no transversally holomorphic fields.

The spaces of Sp(n)-, Sp(n) · U(1)- and Sp(n) · Sp(1)-invariant, translation-invariant, continuous convex valuations on the quaternionic vector space ℍn are studied. Combinatorial dimension formulae involving Young diagrams and Schur polynomials are proved.

The purpose of this paper is to classify nonharmonic biharmonic curves and surfaces in de Sitter 3-space and anti-de Sitter 3-space.

Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a Kähler group on the real hyperbolic space of dimension at least three factors through a homomorphism onto a cocompact discrete subgroup of PSL2(ℝ). We also study actions of Kähler groups on infinite-dimensional real hyperbolic spaces, describe some exotic actions of PSL2(ℝ) on these spaces, and give an application to the study of the Cremona group.

We study the geometric properties of a base manifold whose unit tangent sphere bundle, equipped with the standard contact metric structure, is H-contact. We prove that a necessary and sufficient condition for the unit tangent sphere bundle of a four-dimensional Riemannian manifold to be H-contact is that the base manifold is 2-stein.

We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least five. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using an existence result of E. Legendre [Toric geometry of convex quadrilaterals, J. Symplectic Geom. 9 (2011), 343–385], we show that a co-oriented compact toric contact 5-manifold whose moment cone has four facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on S2×S3 admitting two non-isometric and non-transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.

Complete minimal immersions satisfying the Omori–Yau maximum principle are investigated. It is shown that the limit set of a proper immersion into a convex set must be the whole boundary of the convex set. In case of a nonproper and nonplanar immersion we prove that the convex hull of the immersion is a half-space or ℝ3.

We find approximate solutions (chord–arc curves) for the system of equations of geodesics in Ω∩ℍ for every Denjoy domain Ω, with respect to both the Poincaré and the quasi-hyperbolic metrics. We also prove that these chord–arc curves are uniformly close to the geodesics. As an application of these results, we obtain good estimates for the lengths of simple closed geodesics in any Denjoy domain, and we improve the characterization in a 1999 work by Alvarez et al. on Denjoy domains satisfying the linear isoperimetric inequality.

Let M=G/K be a generalized flag manifold, that is, an adjoint orbit of a compact, connected and semisimple Lie group G. We use a variational approach to find non-Kähler homogeneous Einstein metrics for flag manifolds with two isotropy summands. We also determine the nature of these Einstein metrics as critical points of the scalar curvature functional under fixed volume.