In the present paper we consider Schrödinger equations with variable coefficients and potentials, where the principal part is a long-range perturbation of the flat Laplacian and potentials have at most linear growth at spatial infinity. We then prove local-in-time Strichartz estimates, outside a large compact set centered at origin, without loss of derivatives. Moreover we also prove global-in-space Strichartz estimates under the non-trapping condition on the Hamilton flow generated by the kinetic energy.

We prove that any closed and convex surface in the Minkowski 3-space of class $C^3$ has at least two umbilic points. This shows that the Carathéodory conjecture for surfaces in the Euclidean 3-space is true for surfaces in the Minkowski 3-space.

Let $N \geq 5$ be a prime number. Conrad, Edixhoven and Stein have conjectured that the rational torsion subgroup of the modular Jacobian variety $J_1(N)$ coincides with the 0-cuspidal class group. We prove this conjecture up to 2-torsion. To do this, we study certain ideals of the Hecke algebras, called the Eisenstein ideals, related to modular forms of weight 2 with respect to $\varGamma_1(N)$ that vanish at the 0-cusps.

We correct the proof of Theorem 3.8 in [GP2].

We prove the existence of a new class of constant mean curvature cylinders with an arbitrary number of umbilics by unitarizing the monodromy of Hill's equation.

We consider $M^n$, $n\geq3$, an $n$-dimensional complete submanifold of a Riemannian manifold $(\overline{M}^{n+p},\overline{g})$. We prove that if for all point $x\in M^n$ the following inequality is satisfied

$$S\leq\frac{8}{3} \bigg( \overline{K}_{\min}-\frac{1}{4}\overline{K}_{\max} \bigg)+\frac{n^2H^2}{n-1},$$

with strictly inequality at one point, where $S$ and $H$ denote the squared norm of the second fundamental form and the mean curvature of $M^n$ respectively, then $M^n$ is either diffeomorphic to a spherical space form or the Euclidean space $\mathbb{R}^n$. In particular, if $M^n$ is simply connected, then $M^n$ is either diffeomorphic to the sphere $\mathbb{S}^n$ or the Euclidean space $\mathbb{R}^n$.

Let $M$ be an almost complex manifold and $g$ a periodic automorphism of $M$ of order $p$. Then the rotation angles of $g$ around fixed points of $g$ are naturally defined by the almost complex structure of $M$. In this paper, under the assumption that the fixed points of $g^k$ $(1\leq k\leq p-1)$ are isolated, a calculation formula is provided for the homomorphism $I_D: {\Bbb Z}_p \to {\Bbb R}/{\Bbb Z}$ defined in [8]. The formula gives a new method to study the periodic automorphisms of almost complex manifolds. As examples of the application of the formula, we show the nonexistence of the ${\Bbb Z}_p$-action of specific isotropy orders and examine whether specific rotation angles exist or not.

This paper presents a definition of ${\Bbb C}^*$-equivariant degeneration families of compact complex curves over ${\Bbb C}$. Those families are called ${\Bbb C}^*$-pencils of curves. We give the canonical method to construct them and prove some results on relations between them and normal surface singularities with ${\Bbb C}^*$-action. We also define ${\Bbb C}^*$-equivariant degeneration families of compact complex curves over ${\Bbb P}^1$. From this, it is possible to introduce a notion of dual ${\Bbb C}^*$-pencils of curves naturally. Associating it, we prove a duality for cyclic covers of normal surface singularities with ${\Bbb C}^*$-action.

In this paper, we study the existence and the asymptotic behavior of a positive solution to the one-dimensional stationary shadow system of the Gierer-Meinhardt system with saturation. We equip a reaction term of activator with saturation effect $\kappa_0 \varepsilon^{2\alpha}$ for $\alpha\in (0,1)$ (semi-strong saturation effect). Here, $\varepsilon$ > 0 stands for the diffusion constant of activator. For sufficiently small $\varepsilon$, we show the existence of a new type of solutions which has the following properties:

(a) the solution has an internal transition-layer of $O(\varepsilon)$ in width,

(b) the transition-layer is located in the position of $O(\varepsilon^\alpha)$ from the boundary $x=0$,

(c) the solution concentrates at $x=0$ with the amplitude of the order of $O(\varepsilon^{-\alpha})$ when $\varepsilon \ll 1$.

We give a generalization of the Cartan decomposition for connected compact Lie groups motivated by the work on visible actions of T. Kobayashi [J. Math. Soc. Japan, 2007] for type A group. This paper extends his results to type D group. First, we classify a pair of Levi subgroups $(L,H)$ of a simple compact Lie group $G$ of type D such that $G=LG^{\sigma}H$ where $\sigma$ is a Chevalley–Weyl involution. This gives the visibility of the $L$-action on the generalized 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 a generalized Cartan decomposition $G=LBH$ with $B$ in $G^{\sigma}$ by using the herringbone stitch method which was introduced by Kobayashi in his 2007 paper. Applications to multiplicity-free theorems of representations are also discussed.

We consider the intersection of the conjugacy class of a nilpotent matrix with the space of upper triangular matrices. We give necessary and sufficient conditions for this intersection to be a union of finitely many orbits for the action by conjugation of the group of invertible upper triangular matrices.

We study ‘nondegenerate’ SDE's with jumps. These include SDE satisfying ‘point-wise positive’ condition and that satisfying (nonstationary) Hörmander's condition. We show that solutions of these SDE's have hypoelliptic properties. Our result is based on the Malliavin calculus on the Wiener-Poisson space. In case of continuous SDE, it extends and refines works based on the Malliavin calculus on the Wiener space.