Calculus of Variation and Geometric Structures on Manifolds

首席研究员Yoe Itokawa，在与Katsuhiro Shiohama的联合研究项目中，证明了正截面曲率的完全非紧流形的完全极小变体必然具有无界像。在环境空间只有非负曲率的情况下，首席研究员已经获得了一些部分结果，但他还没有决定是发表后者的结果还是等待进一步的信息。在另一项工作中，这次是与Ryoichi Kobayashi合作，首席研究员在一些相对较弱的条件下对非负Ricci曲率流形的<普朗克常数>-1维同调群进行了分类，其横截面尺寸的增长率。研究者Masaru Nishihara继续研究了复局部凸空间E上弱连续多项式对其对偶空间E的可拓性。他在《第六届复分析国际学术研讨会论文集》上发表了他的研究结果“关于无限维空间中全纯函数的扩展”。研究者Sinya Nishibata研究了双曲型和椭圆型耦合P.D.E系统。他能够证明熵函数的存在性等价于系统的同时对角性，并且在这种条件下，可以用经典技术来建立解的短期存在性。此外，他还证明了在稳定性假设下，也存在长期解，当t趋于*时，它们收敛于平衡状态。

The head investigator, Yoe Itokawa, in a joint research project with Katsuhiro Shiohama, has shown that complete minimal varieties in complete noncompact manifolds of positive sectional curvature necessarily have unbounded images. The head investigator has obtained some partial results concerning the case where the ambient spaces have only nonnegative curvature, but he is still undecided whether to publish the latter results or wait for further information. In another work, this time with Ryoichi Kobayashi, the head investigator has classified the <planck's constant>-1 -dimensional homology groups of manifolds of nonnegatively Ricci curvature under some relatively weak conditions on the growth rate of their cross-sectional size. The last results were described in the paper"Minimizing currents in open manifolds and <planck's constant>-1 homology of nonnegatively Ricci curved mThe investigator Masaru Nishihara has continued his investigation on the extendability of weakly continuous polynomials on a comples locally convex space E to its bidual space E". He has published his results in the paper "On extensions of holomorphic functions in infinite dimensional spaces" in Proceedings of the Sixth International Colloquium on Complex Analysis.The investigator Sinya Nishibata has studied systems of coupled hyperbokic and elliptic P.D.E.'s. He was able to prove that the existence of entropy function is equivalent to the simultaneous diagonability of the system and that, under this condition, classical techniques can be used to establish the short-term existence of solutions. In addition, he has also shown that under stability assumptions, long term solutions also exist and when t tends to*, they converge to equilibrium states.