Symmetry of hypersurfaces with ordered mean curvature in one direction
授课人：李岩岩教授 (美国Rutgers 大学)
时 间：7月6日至29，每周二、四，8:30—10:00
Abstract: For a connected n-dimensional compact smooth hypersurface M without boundary embedded in Rn+1, a classical result of A.D. Aleksandrov shows that it must be a sphere if it has constant mean curvature. Nirenberg and I studied a one-directional analog of this result: if every pair of points (x′,a), (x′,b) ∈ M with a < b has ordered mean curvature H(x′, b) ≤ H(x′, a), then M is symmetric about some hyperplane x_{n+1}= c under some additional conditions. Our proofs were made by the method of moving planes and some variations of the Hopf Lemma.
In my recent joint work with Xukai Yan and Yao Yao, we studied the problem with a variational approach and solved an open problem raised by Nirenberg and I in 2006.
In this mini-course, I will present these studies, starting from the result of Aleksandrov, and will give expositions to some basic methods like the method of moving planes, Hopf Lemma etc. I will also present a number of open problems on variations of the Hopf Lemma raised by Nirenberg and I, as well as some problems on the above mentioned variational approach.
The mini-course is accessible to first year graduate students and advanced undergraduate students.
李岩岩教授简介：1988年博士毕业于纽约大学柯朗研究所，美国Rutgers大学数学系杰出教授（Distinguished Professor），并从2010年起，担任Rutgers大学非线性分析中心主任。李教授是非线性分析和偏微分方程领域的国际顶尖专家，在国际顶尖数学期刊《Acta Math》，《Invent Math》，《Comm Pure Appl Math》，《Duke Math J.》及其它国际数学主流期刊上发表论文150余篇，并从2004年起，成为ISI高引用研究者（ISI Highly Cited Researcher）。至今，李教授在美国培养17位博士，他们第一份工作的单位有：普林斯顿大学、芝加哥大学、牛津大学、美国普林斯顿大学高等研究院、德克萨斯州大学奥斯汀分校、威斯康星大学麦迪逊分校、明尼苏达大学、英属哥伦比亚大学、约翰•霍普金斯大学、佐治亚理工，布朗大学等，并大部分选择继续从事数学研究，如在牛津大学、澳大利亚国立大学、佛罗里达大学、俄克拉荷马大学、香港科技大学等拿到终身职位。
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