It is shown that certain Kummer surface admits a Levi flat hypersurface whose complement is strongly pseudoconvex.

In this paper, we consider the uniqueness in the Cauchy problem for systems of differential operators with partial analytic coefficients. By proving Carleman's estimate, we obtain the uniqueness theorem which contains the classical Holmgren's theorem for systems of differential operators.

Let $T_{\varepsilon}$ be the lifespan of solutions to the initial value problem for the one dimensional, derivative nonlinear Schrödinger equations with small initial data of size $O(\varepsilon)$. If the nonlinear term is cubic and gauge invariant, it is known that $\liminf_{\varepsilon \to +0} \varepsilon^{2} \log T_\varepsilon$ is positive. In this paper we obtain a sharp estimate of this lower limit, which is explicitly computed from the initial data and the nonlinear term.

In this article we study a priori upper bounds of subsolutions satisfying a certain differential inequality (*) below on a non-compact complete Riemannian manifold $(M,g)$ without any Ricci curvature condition. Our method depends on a volume estimate of open subsets where those solutions satisfy a certain strong subharmonicity. Several applications in conformal deformation of metrics and value distribution of harmonic maps are given.

Let $X$ be a norm curve in the $\mathit{SL}(2,\mathbb{C})$-character variety of a knot exterior $M$. Let $t = \|\beta\| / \|\alpha\|$ be the ratio of the Culler-Shalen norms of two distinct non-zero classes $\alpha, \beta \in H_1(\partial M,\mathbb{Z})$. We demonstrate that either $X$ has exactly two associated strict boundary slopes $\pm t$, or else there are strict boundary slopes $r_1$ and $r_2$ with $|r_1| > t$ and $|r_2| < t$. As a consequence, we show that there are strict boundary slopes near cyclic, finite, and Seifert slopes. We also prove that the diameter of the set of strict boundary slopes can be bounded below using the Culler-Shalen norm of those slopes.

Let $\Sigma(K; p/q)$ be the result of $p/q$-surgery along a knot $K$ in a homology 3-sphere $\Sigma$. We investigate the Reidemeister torsion of $\Sigma(K; p/q)$. Firstly, when the Alexander polynomial of $K$ is the same as that of a torus knot, we give a necessary and sufficient condition for the Reidemeister torsion of $\Sigma(K; p/q)$ to be that of a lens space. Secondly, when the Alexander polynomial of $K$ is of degree $2$, we show that if the Reidemeister torsion of $\Sigma(K; p/q)$ is the same as that of a lens space, then $\varDelta_K(t)=t^2-t+1$.

We construct automorphic functions on the real $3$-dimensional hyperbolic space $\mathbb{H}^3$ for the Whitehead-link-complement group $W\subset \mathit{GL}_2(\mathbb{Z}[i])$ and for a few groups commensurable with $W$. These automorphic functions give embeddings of the orbit spaces of $\mathbb{H}^3$ under these groups, and arithmetical characterizations of them.

The isospectral property of the algebro-geometric double Darboux transformation is studied by an algebraic method. This is established by verifying that the algebro-geometric double Darboux transformation of the $n$-th algebro-geometric potential solves the evolution equation of the higher order KdV type. Applying these results, the second Darboux-Lamé potential, which can be regarded as the 3-elliptic solitons, is explicitly constructed by an elementary method.

Some quotient algebras arising from the quantum toroidal algebra $U_{q}(\mathit{sl}_{n+1}(\mathcal{C}_{\gamma}))$ ($n\ge 2$) are considered. They are related to integrable highest weight representations of the algebra and are shown to be isomorphic to tensor products of two algebras of symmetric Laurent polynomials and Macdonald's difference operators.

In this paper, we prove that the Mostow fibration of a compact complex parallelizable pseudo-Kähler solvmanifold is a complex torus bundle over a complex torus.

We study the well-posedness issue of the generalized Benjamin-Ono-Burgers (gBO-B) equations. We solve the initial-value problem (IVP) of the gBO-B equations with data below $L^2 (\mathbf{R})$. Our proof is based on the method of L. Molinet and F. Ribaud, which is analogous to that of J. Bourgain, and C.E. Kenig, G. Ponce, and L. Vega. It is known that such a method cannot be applied to the Benjamin-Ono equation.

Assuming the minimal model program, we prove that there exists a positive integer $\nu_{n}$ depending only on $n$ such that for every smooth projective $n$-fold of general type $X$ defined over complex numbers, $|mK_{X}|$ gives a birational rational map from $X$ into a projective space for every $m\geq \nu_{n}$.