2021-2022第一学期几何讨论班
时间：2021.9.15 主讲人：卢小格 地点：腾讯会议 (486692022)
题目：The locally conformal flat hypersurfaces in S^2×S^2.
摘要：In this paper, using the moving frame method, we get the integral conditions of hypersurfaces in S^2×S^2, obtain classification theorem of locally conformal flat hypersurfaces with constant angle in S^2×S^2.
时间：2021.9.22 主讲人：陈慧昭 地点：腾讯会议 (104 366 247)
题目：On Spacelike complete H = 0 genus 0 Surfaces with Regular Flat Embedded Ends in R^4_1.
摘要：This paper concerns spacelike complete H = 0 genus 0 surfaces with regular flat embedded ends in R^4_1. Let k be the number of regular flat embedded ends. We prove that: when k = 1, the surface is the plane; there exist no spacelike H = 0 genus 0 surface with k regular flat embedded ends when k = 2, 3; we gives 2−family of examples of spacelike complete H = 0 genus 0 surfaces with k regular flat embedded ends in R^4_1, when k 6= 1, 2, 3, 5, 7.
时间：2021.9.29 主讲人：王丽莉 地点：腾讯会议 (113547166)
题目：Spectral gap estimates on compact manifolds.
摘要: For a compact Riemannian manifold with boundary, its mass gap is the difference between the first and second smallest Dirichlet eigenvalues. In this paper, taking a variational approach, we obtain an explicit lower bound estimate of the mass gap for any compact manifold in terms of geometric quantities.
时间：2021.10.6 主讲人：王鹏 地点：知明楼401
题目：A Brief Introduction to Special Relativity and Minkowski Space.
摘要：In this talk, we will introduce the Lorentz transformations and geometry of Minkowski space time. Some interesting problems in special relativity will be discussed as well. This talk is based on the note/ lectures of J. Corvino at MSRI.
时间：2021.10.13 主讲人：林和子 地点：知明楼401
题目：Sobolev inequality in manifolds with nonnegative curvature.
摘要：We prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a Michael-Simon inequality for submanifolds in manifolds with nonnegative sectional curvature. Both inequalities depend on the asymptotic volume ratio of the ambient manifold.
时间：2021.10.20 主讲人：钟景洋 地点：知明楼401
题目：Willmore hypersurfaces in a sphere.
摘要： In this paper, through study of the Euler-Lagrange equation of the Willmore functional, we obtain an integral inequality of Simon's type for Willmore hypersurfaces in S^n+1 and give a characterization of Willmore tori by use of our integral formula. We also classify all isoparametric Willmore hypersurfaces in S^n+1.
时间：2021.10.27 主讲人：杨鸿立 地点：知明楼401
题目：Vanishing Theorems of p-harmonic Forms on Locally Conformally Flat Riemannian Manifolds.
摘要：It proves the vanishing theorems of p-harmonic forms on a complete, noncompact, simply connected locally conformally flat Riemannian manifold. Firstly, we assume that the scalar curvature of the manifold M^n is nonnegative, and the L^n\2 norm of the traceless Ricci tensor is less than some positive number, then there are no nontrivial L p p-harmonic forms on the manifold. Secondly, if the manifold is even dimensional and the scalar curvature of the manifold is nonnegative, then there are no nontrivial L^q p-harmonic m-forms on M ^2m . Finally, we assume that the scalar curvature of the manifold M ^n is not positive and the L^n\2 norm of the Ricci curvature tensor is less than some positive number, then there are no nontrivial harmonic forms on the manifold.
时间：2021.11.10 主讲人：石玉林 地点：知明楼401
题目：Example of A Complete Minimal Immersion in R^3 of Genus One And Three Embedded Ends.
摘要：In this work we will construct an example of a complete minimal immersion of the torus punctured at three points in R^3 with embedded ends. The total curvature of such an immersion is -12pei. The result is a consequence of the application to minimal surfaces of the theory of elliptic functions of the complex plane C through the Weierstrass representation.
时间：2021.11.17 主讲人：姚中伟 地点：知明楼401
题目：A Willmore functional for compact surfaces in the complex projective plane.
摘要：We propose the study of a conformally invariant functional for surfaces of complex projective plane which is closely related to the classical Willmore functional. We show that minimal surfaces of complex projective plane are critical for this functional and construct some minima for it via the twistor spaces of complex projective plane. Also, we find lower bounds for this functional and for its restriction to the class of Lagrangian surfaces and characterize the complex lines and the Lagrangian totally geodesic surfaces and the Whitney spheres as the only attaining those bounds.
时间：2021.12.1 主讲人：林和子 地点：知明楼401
题目：Gradient estimate and Liouville theorems for p-harmonic maps.
摘要：In this paper, we first obtain an L^q gradient estimate for p-harmonic maps, by assuming the target manifold supporting a certain function, whose gradient and Hessian satisfy some analysis conditions. From this L^q gradient estimate, we get a corresponding Liouville type result for p-harmonic maps. Secondly, using these general results, we give various geometric applications to p-harmonic maps from complete manifolds with nonnegative Ricci curvature to manifolds with various upper bound on sectional curvature, under appropriate controlled images.
时间：2021.12.15 主讲人：钟景洋 地点：知明楼401
题目：A note on static spaces and related problems.
摘要：In this paper we study static spaces, we have made progress in solving the problem raised in Fischer and Marsden (1975) of classifying vacuum static spaces and in proving the conjecture proposed in Besse (1987) concerning manifolds admitting solutions to the critical point equation in general dimensions. We obtain even stronger results in dimension 3.
