Conformal invariants of Riemann surfaces and theta functions

在这一年里，首席研究员继续研究de Branges的互补空间，该空间与核函数和的概念密切相关。他分析了de Branges在单位圆盘上的Caratheodory-Fejer扩展问题的证明，证明了一个关于等距和共等距的收缩存在性的引理是证明中必不可少的。利用这个引理，他给出了由Nagy: Foias提出的交换子跳跃定理的简单替代证明。同时，利用初等线性代数的引理，证明了Nara女子大学的Takahashi女士的扩展插值问题存在的充分必要条件。Kubota给出了一个定理，将rosai - rudin关于有限维c复空间上带不动点的全纯自同构的稳定域的结果推广到Banach空间。沟口研究了抛物方程解的零在实线上的渐近性，并证明了零在固定爆破时间内保持有限。Masumoto给出了一个简单的替代证明，证明了他已经得到的结果，即有限正格闭黎曼曲面上的同调类的极值长度满足代数方程。Yanagihara研究了单位圆盘上归一化Bloch函数f(z)在不动点处的值f(a)的变异性区域。当原点处的导数足够小时，他确定了它的形状。

In this year the head investigator continued his study of de Branges' complementation space that is closely related to the notion of the sum of kernel fun cfons. Analyzing the proof by de Branges of the Caratheodory-Fejer extension problem on the unit disk, he showed that a lemma about the existence of the contraction concerning an isometry and coisometry is essential in the proof. Using this lemma he showed simple alternative proof of the commutant lilting theorem due to Nagy : Foias. Also, he showed that the necessary and sufficient condition for the existence of the extended interpolation problem due to Ms. Takahashi in Nara Women 's University is obtained from the lemma by using elementary linear algebra.Kubota showed a theorem that extends to the case of Banach spaces of the result obtained by Rosay-Rudin about the stable domains of the holomorphic automorphisms with fixed points on finite dimensional c omplex spaces.Mizoguchi studied the asymptotic behavior of zeros of solutions for parabolic equations on the real line, and she proved that the zeros remain finite for a fixed blowup time.Masumoto gave a simple alternative proof of the result alrea dy obtained by him that extremal lengths of homology classes on closed Riemann surfaces of finite positive genus satisfy an algebraic equation.Yanagihara studied the variability region of the values f(a) at a fixed point for normalized Bloch functions f(z) on the unit disk. He determined its shape when the derivative at the origin is sufficiently small.