We generalize the Alexandrov-Toponogov comparison theorem to the case of complete Riemannian manifolds referred to warped product models. We prove the maximal diameter theorem and the rigidity theorem. In particular, we discuss collapsing phenomena where the curvature explosion may occur.

We study Hankel operators on the harmonic Bergman spaces on bounded smooth domains, and obtain a necessary and sufficient condition for Hankel operators to be bounded or compact on both harmonic Bergman space and its dual space.

We examine the structure of the coefficient matrix in the functional equation of the zeta distribution of a self-adjoint prehomogeneous vector space over a non-Archimedean local field. Under a restrictive assumption on the generic stabilizers, we show that this matrix is block upper-triangular with almost symmetric blocks; this generalizes a result of Datskovsky and Wright for the space of binary cubic forms.

There are many papers on the uniqueness theory of meromorphic functions in the whole plane $\boldsymbol{C}$. However, the uniqueness theory concerned with shared sets in an angular domain does not yet seem widely investigated. In this paper, we deal with the problem of uniqueness for meromorphic functions in $\boldsymbol{C}$ under some conditions in an angular domain instead of the whole plane. Moreover, examples show that those conditions are necessary.

We find and study a two-parameter family of coupled Painlevé II systems in dimension four, which can be obtained by a degeneration from the systems of type $A_4^{(1)}$. These systems are compared with other types of coupled Painlevé II systems from the viewpoint of the local index. We also give the phase spaces for these systems.

We generalize and complete some of Maxim's recent results on Alexander invariants of a polynomial transversal to the hyperplane at infinity. Roughly speaking, and surprisingly, such a polynomial behaves, both topologically and algebraically (e.g., in terms of the variation of MHS on the cohomology of its smooth fibers), like a homogeneous polynomial.

We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane $\boldsymbol{C}^2$ by using Legendre curves in the 3-sphere and in the anti de Sitter 3-space or, equivalently, by using spherical and hyperbolic curves, respectively. Among this family, we characterize minimal, constant mean curvature, Hamiltonian-minimal and Willmore surfaces in terms of simple properties of the curvature of the generating curves. As applications, we provide explicitly conformal parametrizations of known and new examples of minimal, constant mean curvature, Hamiltonian-minimal and Willmore surfaces in $\boldsymbol{C}^2$.

In this paper, we give a complete classification of all immersed hypersurfaces in the unit sphere with parallel Blaschke tensors. For this classification, two kinds of new examples are constructed.

Let $G$ be a simply connected semisimple algebraic group and let $K$ be the subgroup of points fixed by an involution of $G$. For certain representations containing a line $r$ preserved by $K$, we study the normality of the closure of the set of vectors which are $G$ conjugate to a vector in $r$. Some applications of our result to the normality of certain classical varieties are given.