The ECNU Differential Geometry Seminar takes place usually on Thursdays at 1:00 pm in Room 205 Building No. 3 on the Minhang Campus.

This seminar is organized by Meng Zhu.

- Date: December 14, 2017
Title: Diameter rigidity for Kähler manifolds with positive bisectional curvature

Speaker: Yuan Yuan (Syracuse University)

Abstract: I will discuss the recent work with Gang Liu on the diameter rigidity for Kähler manifolds with positive bisectional curvature.

- Date: December 7, 2017
Title: The existence of Kahler-Einstein metrics on K-polystable Q-Fano varieties with non-positive discrepancies

Speaker: Feng Wang (Zhejiang University)

Abstract: We will prove the YTD's conjecture for $Q-$Fano varieties X which has a log smooth resolution $M$ with non-positive discrepancies. At first, we extend Tian's work to the log smooth case. After proving the log K-stability, we get the existence of conic KE metrics on $M$. Then we show that these metrics converges to the singular KE metric on X. This is a joint work with Professsor Tian and Chi Li.

- Date: November 30, 2017
Title: Harmonic maps into CAT(1) spaces

Speaker: Yingying Zhang (Yau Mathematical Science Center, Tsinghua University)

Abstract: In this talk, we will discuss existence and regularity results of harmonic maps into CAT(1) space. In their famous work, Sacks-Uhlenbeck discovered a "bubbling phenomena" for harmonic maps from a Riemann surface to a compact Riemannian manifold. We generalized this result when the target space is a compact locally CAT(1) space. Our proof adapted a local harmonic replacement technique.

We also discuss regularity results of harmonic maps from a Riemannian polyhedra into a CAT(1) space. (The talk is based on the joint work with C. Breiner, A. Fraser, L. Huang, C. Mese, and P. Sargent.)

- Date: November 23, 2017
Title: Introduction to Elliptic genus

Speaker: Bo Liu (East China Normal University)

Abstract: In this talk, I'll introduce the elliptic genus from the point of view of cobordism, elliptic function and the quantum field theory and explain the relations of it with modular forms and the Monster moonshine. At last, we prove the rigidity of elliptic genus for Z/k manifolds, which is a conjecture in 1996. This is a joint work with Yu Jianqing.

- Date: November 16, 2017
Title: Some universal inequalities for Dirichlet eigenvalues on subgraphs of Lattices

Speaker: Bobo Hua (Fudan University)

Abstract: In this talk, we prove some analogues of Payne-Polya-Weinberger, Hile-Protter and Yang's inequalities for Dirichlet (discrete) Laplace eigenvalues on any subset in the integer lattice Zn. This is a joint work with Yong Lin and Yanhui Su.

- Date: November 9, 2017
Title: Li-Yau gradient estimates without Ricci curvature lower bound

Speaker: Meng Zhu (East China Normal University)

Abstract: In their celebrated work, P. Li and S.-T. Yau proved the famous Li-Yau gradient estimate for positive solutions of the heat equation on manifolds with Ricci curvature bounded from below. Since then, Li-Yau type gradient bounds has been widely used in geometric analysis, and become a powerful tool in deriving geometric and topological properties of manifolds.

In this talk, we will present our recent works on Li-Yau type gradient bounds for positive solutions of the heat equation on complete manifolds with certain integral curvature bounds, namely, |Ric_| in L^p for some p>n/2 or certain Kato type of norm of |Ric_| being bounded together with a Gaussian upper bound of the heat kernel. We also study the Li-Yau estimate for the heat equation under the Ricci flow with bounded scalar curvature. These assumptions allow the lower bound of the Ricci curvature to tend to negative infinity, which is weaker than the assumptions in the known results on Li-Yau bounds. These are joint works with Qi S. Zhang.

- Date: November 2, 2017
Title: Existence of Ricci flow on noncompact manifolds

Speaker: Fei He (Xiamen University)

Abstract: Though the Ricci flow has been extensively studied, its short-time existence on noncompact manifolds with potentially unbounded curvature is still not fully understood. There has been some important progress on this problem by many authors. We will discuss recent results and their applications.

- Date: October 26, 2017
Title: The Determinant of Laplace Operators and the Analytic Torsion

Speaker: Bo Liu (East China Normal University)

Abstract: In this talk, we introduce the Ray-Singer analytic torsion as the determinant of Laplace operators and the extended Cheeger-Mueller Theorem by Bismut-Zhang which gives the explicit relation between the Ray-Singer analytic torsion and the Reidemeister torsion. Note that the Reidemeister torsion is the first topological invariant in the history distinguishing the homotopy equivalent but not homeomorphic manifolds. At last, we explain the complex valued torsion, Burghelea-Haller torsion and Cappell-Miller torsion, and the resent results by Liu-Yu and Su-Zhang.

- Date: September 27, 2017 (A Special Time)
Title: Quantitative maximal local rewinding volume rigidity with Ricci curvature bounded below II

Speaker: Lina Chen (East China Normal University)

Abstract: For a metric ball $B_r(x)$ in a Riemannian manifold, its local rewinding volume is the volume of $B_r(x^*)$, where $B_r(x^*)\subset (U^*, x^*)$, the (uncomplete) Riemannian universal cover of $(B_r(x), x)$. A compact manifold with Ricci curvature bounded below by $(n-1)H$ is isometric to a space form with constant curvature $H$ if and only if every $\rho$-ball ($\rho$ fixed) achieves the maximal local rewinding volume. In this talk, we will prove that if a compact manifold $M$ with Ricci curvature lower bound $(n-1)H$ satisfies that the universal cover space $\tilde M$ is non-collapsing (there exist a positive lower bound of the volume of a unit ball in $\tilde M$) and each $\rho$-ball almost achieves the maximal local rewinding volume, then this manifold is diffeomorphic and close to a space form with $H$-constant curvature. This is joint work with Xiaochun Rong and Shicheng Xu.

- Date: September 20, 2017 (A Special Time)
Title: Quantitative maximal local rewinding volume rigidity with Ricci curvature bounded below

Speaker: Lina Chen (East China Normal University)

Abstract: For a metric ball $B_r(x)$ in a Riemannian manifold, its local rewinding volume is the volume of $B_r(x^*)$, where $B_r(x^*)\subset (U^*, x^*)$, the (uncomplete) Riemannian universal cover of $(B_r(x), x)$. A compact manifold with Ricci curvature bounded below by $(n-1)H$ is isometric to a space form with constant curvature $H$ if and only if every $\rho$-ball ($\rho$ fixed) achieves the maximal local rewinding volume. In this talk, we will prove that if a compact manifold $M$ with Ricci curvature lower bound $(n-1)H$ satisfies that the universal cover space $\tilde M$ is non-collapsing (there exist a positive lower bound of the volume of a unit ball in $\tilde M$) and each $\rho$-ball almost achieves the maximal local rewinding volume, then this manifold is diffeomorphic and close to a space form with $H$-constant curvature. This is joint work with Xiaochun Rong and Shicheng Xu.