Conformal invariants of Riemann surfaces and theta functions

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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.