International Journal of Mathematics

We prove various classification results for homogeneous locally conformally symplectic manifolds. In particular, we show that a homogeneous locally conformally Kähler manifold of a reductive group is of Vaisman type if the normalizer of the isotropy group is compact. We also show that such a result does not hold in the case of non-compact normalizer and determine all left-invariant lcK structures on reductive Lie groups.

We introduce the de Sitter Schwarz map for the hypergeometric differential equation as a variant of the classical Schwarz map. This map turns out to be the dual of the hyperbolic Schwarz map, and it unifies the various Schwarz maps studied before. An example is also studied.

The Gronwall conjecture, which is still open, asserts that if a 3-web in the plane is linearizable, then the linearization is unique modulo projective transformations. We prove the conjecture for Legendrian d-webs, provided d ≥ 4. Precisely, our theorem states that, if a Legendrian d-web with d ≥ 4 in the (real or complex) 3-dimensional contact manifold is linearizable, then there is a unique linearization of the Legendrian d-web up to a contact projective transformation. For the proof, we use the linearization technique of the third order ordinary differential equations and the Schwarzian derivatives of contact transformations.

We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg–Witten invariant [S. Bauer and M. Furuta, Stable cohomotopy refinement of Seiberg–Witten invariants: I, Invent. Math.155 (2004) 1–19; S. Bauer, Stable cohomotopy refinement of Seiberg–Witten invariants: II, Invent. Math.155 (2004) 21–40.] of connected sums of 4-manifolds with positive first Betti number.

We show a necessary and sufficient condition that the fixed point set of a holomorphic isometry and the intersection of two real forms of a Hermitian symmetric space of compact type are discrete and prove that they are antipodal sets in the cases. We also consider some relations between the intersection of two real forms and the fixed point set of a certain holomorphic isometry.

We prove a second main theorem type estimate in Nevanlinna theory for holomorphic curves f : Y → X from finite covering spaces Y → ℂ of the complex plane ℂ into complex projective manifolds X of maximal albanese dimension. If X is moreover of general type, then this implies that the special set of X is a proper subset of X. For a projective curve C in such X, our estimate also yields an upper bound of the ratio of the degree of C to the geometric genus of C, provided that C is not contained in a proper exceptional subset in X.

Using Busemann function of an Hadamard manifold X we define the barycenter map from the space 𝒫⁺(∂X, dθ) of probability measures having positive density on the ideal boundary ∂X to X. The space 𝒫⁺(∂X, dθ) admits geometrically a fiber space structure over X from Fisher information geometry. Following the arguments in [E. Douady and C. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math.157 (1986) 23–48; G. Besson, G. Courtois and S. Gallot, Entropies et rigidités des espaces localement symétriques de coubure strictement négative, Geom. Funct. Anal.5 (1995) 731–799; Minimal entropy and Mostow's rigidity theorems, Ergodic Theory Dynam. Systems16 (1996) 623–649], we exhibit that under certain geometrical hypotheses a homeomorphism Φ of the ideal boundary ∂X induces, by the aid of push-forward, an isometry of X whose extension is Φ.

We estimate the cardinalities of antipodal sets in oriented real Grassmann manifolds of low ranks. The author reduced the classification of antipodal sets in oriented real Grassmann manifolds to a certain combinatorial problem in a previous paper. So we can reduce estimates of the antipodal sets to those of certain combinatorial objects. The sequences of antipodal sets we obtained in previous papers show that the estimates we obtained in this paper are the best.

We establish a Lehto–Virtanen-type theorem and a rescaling principle for an isolated essential singularity of a holomorphic curve in a complex space, which are useful for establishing a big Picard-type theorem and a big Brody-type one for holomorphic curves.

Given a smooth map from a compact Riemann surface to a complex manifold equipped with a strongly pseudoconvex complex Finsler metric, we define the -energy of the map, whose absolute minimum is attained by a holomorphic map. A harmonic map is then defined to be a stationary map of the -energy functional. We prove that with each harmonic map is associated a holomorphic quadratic differential on the domain, which vanishes if the map is weakly conformal. Also, under the condition that the metric be weakly Kähler, we determine the second variation of the functional, and prove that any -energy minimizing harmonic map from the Riemann sphere to a weakly Kähler Finsler manifold of positive curvature is either holomorphic or anti-holomorphic.