子流形共形高斯映射的几何

The Gauss map is very important for the study of submanifolds in the Euclidean space. This can be generalized in Moebius geometry as below. For a codim-p submanifold M immersed in the n-dimensional sphere, at one point there is a unique codim-p round sphere tangent to M with the same mean curvature vector, called the mean curvature sphere. It is a Moebius invariant object. This defines the conformal Gauss map into the moduli space of codim-p spheres in S^n. This moduli space is well-known to be identical to the Grassmannian manifold Gr(p,L^{n+2}) consisting of p-dimensional spacelike subspaces in the n+2 dimensional Lorentz space. Our propgram aims to investigate Willmore surfaces and Wintgen ideal submanifolds in the sphere by utilizing the conformal Gauss map. These two classes of submanifolds are Moebius invariant objects. The former are critical surfaces with respect to the Willmore functional. The latter are submanifolds attaining equality in the famous DDVV inequality pointwise. In either case, the conformal Gauss map is always a conformal harmonic map from a Riemann surface to the corresponding Grassmannian manifold. (Note that for a Wintgen ideal submanifold of arbitrary dimension or codimension, the image of its conformal Gauss map is always a two-dimensional surface. The harmonicity of this map is proved recently by us.) We will study conformal harmonic maps from Riemann surfaces to non-compact indefinite Grassmannian manifolds, which requires to generalize previous results in the case of compact symmetric spaces. Based on this, we expect to classify Willmore 2-spheres in the n-dimensional sphere and to prove the quantization theorem on the Willmore functional. For Wintgen ideal submanifolds we will provide a general method for their classification and construction.

高斯映射在欧氏空间的子流形几何中非常重要。它在莫比乌斯几何中可如下推广。对球面中的余维p子流形，在任一点取一个圆球与之相切并具有相同中曲率向量，称为中曲率球，为莫比乌斯不变的几何对象；它定义了到全体余维p球面构成的模空间的共形高斯映射。这个模空间可以等同于洛伦兹空间中p维类空子空间构成的格拉斯曼流形。 本项目的研究目的是利用共形高斯映射，研究Willmore曲面和Wintgen ideal子流形。它们是球面中共形不变的理想对象，前者是Willmore泛函的临界曲面，后者逐点取到DDVV不等式中的等号。两者的共形高斯映射已证明都是从黎曼面出发的共形调和映射。 我们需要研究黎曼面到不定度量Grassmann流形的调和映射，推广以往到紧对称空间调和映射的结果。我们期望解决二维Willmore球面的分类问题，证明其Willmore泛函的量子化定理，并对Wintgen ideal子流形获得一般的构造

高斯映射在欧氏空间的子流形几何中非常重要。在莫比乌斯几何中，对球面中的子流形可以在每一点定义一个中曲率球，与其相切并具有相同的平均曲率。它是莫比乌斯不变的几何对象；它定义了到全体余维p球面构成的模空间的共形高斯映射，而这个模空间可以等同于洛伦兹空间中p维类空子空间构成的格拉斯曼流形。..本项目的研究目的是利用共形高斯映射，研究Willmore曲面和Wintgen ideal子流形。我们成功完成了Willmore二维球面的两个分类结果的证明，一个是外围空间为5维球面的情形，另一个是余维任意而加“莫比乌斯齐性”条件的情形。对于 Wintgen ideal 子流形的结构，我们整理发表了两篇论文。..此外，王长平和李同柱、庆杰、谢振肖、王孝振、姬秀等还在Moebius等参超曲面、Lorentz共形平坦超曲面、Moebius齐性超曲面等主题上有诸多进展，均得到了本基金的资助，前后发表和接受的论文有18篇（有基金标注）。除此之外，马翔和指导的博士叶楠、博士后张栋等，还在Lorentz空间的伪凸曲线和伪凸子流形方面做了许多工作，发表两篇论文并多次报告。可以说，本课题的研究成果相当丰富，比较成功。.