Research for manifolds with conformal structure

1.协调平坦的超曲面。我们研究了4维空间形式中的共形平坦超曲面，发现每个超曲面在4维标准球面上都有一个很好的结构。根据超曲面的结构，将共形平坦超曲面分为抛物线类、椭圆类和双曲类。我们证明了这些类在球面的共形变换下是不变的，并且相应的类由三维空间欧几里德空间、双曲空间或空间中的常曲率曲面构成的共形平坦超曲面组成。统计流形的共形射影变换。本文得到了如下结果：统计流形的共形-射影变换留下所有脐点和任何超曲面的Ricci曲率的反对称分量；此外，这一性质刻画了co…当统计流形的维度大于2时，更多的非形式射影变换。我们还发现了一个张量场，它在任何共形射影变换下都是不变的，并且如果基本的统计流形是通常的黎曼流形，它就退化为共形曲率张量。三维空间形式中常平均曲率曲面(CMC曲面)的表示公式及其高斯映射。研究了目标为非正曲率无界的非完备黎曼空间的调和映射的存在性问题。在这种情况下，我们证明了无穷远处Dirichlet问题调和映射的存在唯一性定理。作为应用，我们构造了三维双曲空间形式的CMC曲面。从可积系统理论的观点推广了一类CMC曲面。我们在三维空间形式中定义了具有调和逆平均曲率的曲面(HIMC曲面)，并证明了HIMC曲面之间存在类似于Lawson对应的对应关系，而Lawson对应是这类CMC曲面的特征之一。我们还从变分的角度研究了HIMC曲面族与H-曲面族之间的关系，H-曲面族是CMC曲面族的推广。因此，我们证明了HIMC曲面是由H曲面的规范理论方程在一定的约化条件下得到的。较少

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