Research for manifolds with conformal structure

1. 合形平面超曲面。我们研究了四维空间形式的共形平面超曲面，并在四维标准球上找到了每个超曲面的良好结构。根据其结构，将共形平面超曲面集划分为抛物型、椭圆型和双曲型三类。我们证明了这类在球面的共形变换下是不变的，并且这类由三维空间形式之一的常曲率曲面构成的共形平面超曲面组成：欧几里德空间、双曲空间或球面。统计流形的共形投影变换。本文得到了以下结果：统计流形的保形投影变换留下了所有的脐点和任意超曲面的Ricci曲率的偏对称分量；此外，当统计流形的维数大于2时，这一性质描述了更多的非形式射影变换。我们还发现了一个张量场，它在任何共形投影变换下都是不变的，并且如果底层的统计流形是通常的黎曼流形，它就可以化为共形曲率张量。三维空间形式中等平均曲率曲面的表示公式及其高斯映射。研究了目标为从下无界非正曲率的非完全黎曼空间时调和映射的存在性问题。在这种情况下，我们给出了无穷远处Dirichlet问题调和映射的存在性和唯一性定理。作为应用，我们构造了三维双曲空间形式的CMC曲面。从可积系统理论的角度对CMC曲面类的扩展。我们在三维空间形式中定义了调和平均逆曲率曲面（HIMC曲面），并证明了HIMC曲面之间存在类似于CMC曲面类特征之一劳森对应关系的对应关系。我们还从变分的角度研究了HIMC曲面类与h曲面类之间的关系，h曲面是CMC曲面类的扩展。结果证明了在一定的约化条件下，由h曲面的量规理论方程可以得到HIMC曲面。少

1. Conformably flat hypersurfaces. We studied conformally flat, hypersurfaces in the space forms of dimension 4, and found a good structure on the 4-dimensional standard sphere for each hypersurface. According to the structure, the set of conformally flat hypersurfaces is divided into three classes : the parabolic class, the elliptic class, and the hyperbolic class. We showed that the classes are invariant under conformal transformations of the sphere and the respective class consists of conformally flat hypersurfaces constructed by surfaces of constant curvature in one of the 3-dimensional space forms : the Euclidean space, the hyperbolic space, or the sphere.2. Conformal-projective transformations of statistical manifolds. In this study, we obtained the following result : A conformal-projective transformation of a statistical manifold leaves all umbilical points and the skew-symmetric component of the Ricci curvature of any hypersurfaces ; moreover, this property characterizes the co … More nformal-projective transformations when the dimension of the statistical manifold is greater than 2. We also found a tensor field that is invariant under any conformal-projective transformations and that reduces to the conformal curvature tensor if the underlying statistical manifold is a usual Riemannian manifold.3. A representation formula of surfaces with constant mean curvature (CMC surfaces) in a 3-dimensional space form and their Gauss map. The existence problem of harmonic maps was studied in the case where the destination is a non-complete Riemannian space with non-positive curvature unbounded from below. In this situation, we showed tile existence and the uniqueness theorems of harmonic maps for a Dirichlet problem at infinity. As an application, we constructed CMC surfaces in the 3-dimensional hyperbolic space form.4. An extension of the class of CMC surfaces from the viewpoint of the theory of integrable systems. We defined surfaces with harmonic inverse mean curvature (HIMC surfaces) in the 3-dimentional space forms, and showed that there exists a correspondence among the HIMC surfaces similar to the Lawson correspondence, one of the features of the class of CMC surfaces. We also studied the relation between the class of HIMC surfaces and the class of H-surfaces, which is an extension of the class of CMC surfaces from the variational viewpoint. As a result, we proved that HIMC surfaces are obtained from the gauge-theoretic equation for H-surfaces with a certain condition of reduction. Less