Construction of submanifold with constant mean curvature, and its applications

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Construction of submanifold with constant mean curvature, and its applications

Format:
Grant
Kyushu Univ. Production Kyushu Univ. Production
Title(Other Language):
定平均曲率部分多様体の構成と応用
Responsibility:
山田 光太郎(九州大学・大学院・数理学研究院・教授)
YAMADA Kotaro(九州大学・大学院・数理学研究院・教授)
Language:
Japanese
Project Year:
1998-2000
Latest Report:
We investigated properties of minimal surfaces in the three dimensional euclidean space using the Weierstrass representation formula, and generalizations of them. First, we gave an affirmative result for an inverse problem of flux for minimal surfaces in the three dimensional euclidean space. Moreover, as a generalization of (a complex analytic) flux, we defined a new homology invariant, which is also called as "flux", for surfaces of constant mean curvature one in the hyperbolic three space. Using the balancing formula of the flux, we proved some non-existence results for constant mean curvature one surface in hyperbolic space. As a continuation of this non-existence results, we tried to classify the complete constant mean curvature one surface in hyperbolic space with low total absolute curvature, and we obtained the complete classification for surfaces with total absolute curvature less than or equal to 4π. On the other hand, as a generalization of the Weierstrass-type representation formula for minimal surface with higher dimensional euclidean space, we defined a notion of surfaces with holomorphic right gauss map in some non-compact type symmetric space, and obtained the Weierstrass-Bryant type representation formula. As an application of this formula, we obtained an Osserman-type inequality for total absolute curvature. Read more
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