极小曲面
The investigation and construction of surfaces with special geometric properties has always been an important subject in differential geometry. Of particular interest are minimal surfaces and constant mean curvature (CMC) surfaces in space forms. Global properties surfaces were first considered by Hopf, showing that all CMC spheres are round. This result was generalized by Alexandrov [2] in the 1950s, who showed that the round spheres are the only embedded compact CMC surfaces in $\mathbb R^3$, while there do not exist any compact minimal surfaces in euclidean space and hyperbolic 3-space due to the maximum principle. In contrast, there are many compact and embedded minimal surfaces in the 3-sphere, the best known examples being the Clifford torus and the Lawson surfaces, in addition to the totally geodesic 2-sphere, and a full classification is beyond the current knowledge.
One reason for the beauty and depth of minimal surface theory is that there are many different methods and tools with which to construct, study, and classify these surfaces.
In this course we will first derive basic properties of minimal surfaces in $\mathbb R^3$
and introduce some general techniques, and then move on to CMC surfaces in $\mathbb R^3$ and minimal surfaces in $\mathbb S^3.$
We mainly use differential geometric and complex analytical methods in this course.
One reason for the beauty and depth of minimal surface theory is that there are many different methods and tools with which to construct, study, and classify these surfaces.
In this course we will first derive basic properties of minimal surfaces in $\mathbb R^3$
and introduce some general techniques, and then move on to CMC surfaces in $\mathbb R^3$ and minimal surfaces in $\mathbb S^3.$
We mainly use differential geometric and complex analytical methods in this course.
讲师
日期
2022年09月21日 至 12月14日
网站
修课要求
It is necessary to be familiar with the basic concepts of linear algebra and calculus. In order to be able to follow the course throughout, it is beneficial to have some basic knowledge about differential geometry or manifolds, and to be familiar with some complex analysis. However, it is also possible to make up for this within the course.
课程大纲
1. minimal surfaces in $\mathbb R^3:$ Euler-Lagrange equations, curvature of surfaces in space, monotonicity formulas, examples
and partial classifications, Weierstrass formulas
2. CMC surfaces in $\mathbb R^3:$ Gauss-Codazzi equations, associated families of CMC surfaces, examples, Theorems of Alexandrov and Hopf
3. minimal surfaces in $\mathbb S^3:$ Gauss-Codazzi equations, Lawson correspondence, examples, generalized Weierstrass representations and/or classification of embedded minimal tori
and partial classifications, Weierstrass formulas
2. CMC surfaces in $\mathbb R^3:$ Gauss-Codazzi equations, associated families of CMC surfaces, examples, Theorems of Alexandrov and Hopf
3. minimal surfaces in $\mathbb S^3:$ Gauss-Codazzi equations, Lawson correspondence, examples, generalized Weierstrass representations and/or classification of embedded minimal tori
参考资料
[1] S. Brendle Minimal surfaces in S3, a survey of recent results, Bulletin of Mathematical Sciences 3, 133{171 (2013)
[2] T. Colding, W. Minicozzi, A course in minimal surfaces, Graduate Texts in Mathematics
[3] M. do Carmo, Differential Geometry of Curves and Surfaces
[4] N. Hitchin, Harmonic maps from a 2-torus to the 3-sphere, J. Differential Geom. 31 (1990), no. 3, 627-710.
[5] H.B. Lawson, Complete minimal surfaces in S3, Ann. Math. 92(1970), 335{ 374.
[2] T. Colding, W. Minicozzi, A course in minimal surfaces, Graduate Texts in Mathematics
[3] M. do Carmo, Differential Geometry of Curves and Surfaces
[4] N. Hitchin, Harmonic maps from a 2-torus to the 3-sphere, J. Differential Geom. 31 (1990), no. 3, 627-710.
[5] H.B. Lawson, Complete minimal surfaces in S3, Ann. Math. 92(1970), 335{ 374.
听众
Undergraduate
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
Sebastian Heller于2008年获得德国洪堡大学的博士学位，后于2014年获得德国图宾根大学的特许任教资格，在2014-2022年期间在德国海德堡大学任研究员，自2022年9月起任北京雁栖湖应用数学研究院研究员。S. Heller教授致力于微分几何、代数和复杂几何领域研究，特别是紧黎曼曲面上的可积偏微分方程研究，该研究被广泛应用到最小曲面理论、调和映射领域和理论物理学中。在谐波映射、最小和CMC表面领域的科研能力处于国际前沿水平，证明了完整的紧凑型CMC曲面族的存在；展示了沿着变形所有表面都是嵌入的，并获得了关于依赖于保形类型的平均曲率和威尔莫尔能量的更多信息，这是关于3球体中紧凑嵌入CMC表面空间的第一个全局结果；发现了一种用于根据某些积分计算Lawson曲面的复分析数据的递归算法。这些研究成果使得在理解3球体中致密恒定平均曲率(CMC)表面的几何和（复）分析特性方面迈出了重要一步，得到领域专家学者普遍认可。已在《Comm. Math. Phys.》、 《Journal of Integrable Systems》、《Math. Ann.》、《Journal of Differential Geometry》、《Proceedings of the Royal Society A》, 《Journal of Differential Geometry》等国际重要期刊发表论文43篇，引用202余次，H因子8。在国际会议、论坛等做了21场次的主题和邀请口头报告，并组织6次会议。