# Fridays 3:30 - 4:30pm, SH 6617

### 4/2 Xianzhe Dai, UCSB Local Index Theorem and the Asymptotic Expansion of Bergman Kernel"

Abstract : Localization is an important principle in mathematics and local index theorem is an example of such localization. We will discuss the application of the local index Theorem technique to the question of asymptotic expansion of Bergman kernel, which has played an essential role in Donaldson's work on extremal metrics. This is joint work with Kefeng Liu, Xiaonan Ma and Xiaowei Wang.

### 4/23 Krishnan Shankar, University of Oklahama Spherical Rank Rigidity and Blashcke Manifolds"

Abstract: Let $M^n$ be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that $M$ has positive spherical rank if along every geodesic there is a conjugate point at $t = \pi$ (geodesics are assumed to be parametrized by arc-length). In case $M$ has positive spherical rank, the equality discussion of the Rauch Comparison theorem implies that along every geodesic of $M$ we have a spherical Jacobi field i.e., a Jacobi field of the form $J(t) = \sin(t) E(t)$, where is $E(t)$ is a parallel vector field along the geodesic. This notion of rank is analogous to the notion of geometric rank for upper curvature bound 0 or upper curvature bound -1 studied by Ballmann, Burns, Spatzier, Eberlein, Hamenstaedt etc. In the case of spherical rank we show: Let $M^n$ be a complete, simply connected Riemannian manifold with upper curvature bound 1 and positive spherical rank. Then $M^n$ is isometric to a compact rank one symmetric space i.e., isometric to a sphere or projective space. This is joint work with Ralf Spatzier and Burkhard Wilking.

### 5/7 Pengzi Miao, MSRI, Ricci Curvature Rigidity of Asymptotically Hyperbolic Manifolds"

Abstract: Let $n$ be a dimension for which the classical Positive Mass Theorem holds in general relativity. Let $(X^{n+1}, g)$ be a conformally compact manifold whose Ricci curvature is bounded below by $-n$. We show that $(X^{n+1}, g)$ is isometric to the standard hyperbolic space provided the conformal infinity of $(X^{n+1}, g)$ is the standard sphere and $Ric(g) + n$ decays faster than $r^2$ near infinity, where $r$ is any boundary defining function. The proof is based on a quasi-local mass characterization of Euclidean balls, which is essentially a local version of the Positive Mass Theorem. This is a joint work with J. Qing.

# Fridays 3:30 - 4:30pm, SH 6617

### 1/9 Xianzhe Dai, UCSB On the Stability of Riemannian Manifold with Parallel Spinors"

Abstract: Inspired by the recent work of Hertog-Horowitz-Maeda, we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admits nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. In fact, we show that the Lichnerowicz Laplacian, which governs the second variation, is the square of a twisted Dirac operator. Our second result, which is a local version of the first one, shows that any metrics of positive scalar curvature cannot lie too close to a metric with nonzero parallel spinor. As an interesting application, we prove that scalar flat deformations of Calabi-Yau metric must be Calabi-Yau. We also explore the connection with a positive mass theorem of X. Dai. This is a joint work with Xiaodong Wang and Guofang Wei.

### 1/16 Vitali Kapovitch, UCSB Pinching estimates for negatively curved manifolds with nilpotent fundamental groups"

Abstract: We compute optimal pinching constants for negatively curved manifolds with almost nilpotent fundamental groups. Namely, we show that if a manifold $M$ with almost nilpotent fundamental group admits a metric with $-a^2\le sec(M)\le -1$, then the infimum of $a^2$ over all such metrics is equal to square of the nilpotency class of the fundamental group of $M$.

### 1/23 Mark Haskins, IHES Isolated conical singularities of special Lagrangian varieties"

Abstract: we describe recent progress in understanding isolated cone-like singularities of special Lagrangian varieties in dimension 3.

# Fridays 3:30 - 4:30pm, SH 6617

### 10/3 Rugang Ye, UCSB Ricci flow and geometrization of 3-manifolds, introduction"

Abstract: Thurston's geometrization program conjectures that closed 3-manifolds can be cut into peices in a suitable way such that each piece has a homogenous geometric structure. This includes the Poincare conjecture. Hamilton has developed the approach of Ricci flow to carry out this program. Recently, Perelman annouced that he carried out the Ricci flow approach and hence proved Thurston's conjecture and the Poincare conjecture. We will give an introduction to Perelman's papers.

### 10/17 Anda Degeratu, MSRI Geometrical McKay Correspondence"

Abstract: A Calabi-Yau orbifold is locally modeled on C^n/G, where G is a finite subgroup of SL(n, C). One way to handle this type of orbifolds is to resolve them using a crepant resolution of singularities. We use analytical techniques to give a description of the topology of the crepant resolution in terms of the finite group G. This gives a generalization of the McKay Correspondence.

### 10/24 (Special time 3-4pm) Jaeup So, visiting UCSB On G-invariant Minimal Hypersurfaces with Constant Scalar Curvatures in S^5"

Abstract: Let M^n be a closed minimally immersed hypersurface in the unit sphere S^{n+1}, and h its second fundamental form. Denote by R and S its scalar curvature and the square norm of h respectively. It is well known that S=n(n-1)-R from the structure equations of both M^n and S^{n+1}. In particular, S is constant if and only if M has constant scalar curvature. In 1968, J. Simons observed that if S <,= n everywhere and S is constant, then S =0 or n. Clearly, M^n is an equatorial sphere if S=0. And when S=n, M^n is indeed a product of spheres, due to the works of Chern, do Carmo, and Kobayashi. We are concerned about the following conjecture posed by Chern, Chern Conjecture: For any n>2, the set R_n of the real numbers each of which can be realized as the constant scalar curvature of a closed minimally immersed hypersurface in S^{n+1} is discrete. Theorem [Chang, 1993] A closed minimally immersed hypersurface with constant scalar curvature in S^4 is either an equatorial 3-sphere, a product of spheres, or a Cartan's minimal hypersurface. In particular, R_n= {0, 3, 6 }. Theorem [Yang and Cheng, 1998] Let M^n be a closed minimally immersed hypersurface with constant scalar curvature in S^{n+1}. If S>n, then S>,= n+n/3. Let G = O(k)xO(k)xO(q), k=2 or 3 and set 2k+q=n+2. Then W. Y. Hsiang [4] investigated G-invariant, minimal hypersurfaces, M^n in S^{n+1} and showed that there exit infinitely many closed minimal hypersurfaces in S^{n+1} for all n>1. Theorem [Hsiang, 1987] For each dimension n>1, there exist infinitely many, mutually noncongruent closed G-invariant minimal hypersurfaces in S^{n+1}, where G = O(k)xO(k)x(q) and k=2 or 3. We studied G-invariant minimal hypersurfaces with constant scalar curvatures in S^{5}. We prove the following theorem: Theorem: A closed G-invariant minimal hypersurface with constant scalar curvature in S^5 is a product of spheres, where G=O(k)xO(k)xO(q). In particular, S=4.

### 11/14, Guofang Wei, UCSB Laplace spectrum, Length Spectrum and Covering Spectrum"

Abstract: One of the most important subfields of Riemannian Geometry is the study of the Laplace spectrum of a compact Riemannian manifold. Another spectrum defined in an entirely different manner is the length spectrum of a manifold: the set of lengths of smoothly closed geodesics. We define a new spectrum for compact length spaces and Riemannian manifolds called the covering spectrum" which roughly measures the size of the one dimensional holes in the space. We investigate the relationship between these spectrums. We analyze the behavior of the covering spectrum under Gromov-Hausdorff convergence and study its gap phenomenon. This is a joint work with Christina Sormani.

### 12/5 Vitali Kapovitch, UCSB Classification of negatively pinched manifolds with almost nilpotent fundamental groups"

Abstract: We prove that an open manifold with virtually nilpotent fundamental group admits a complete metric of pinched negative curvature if and only if it is diffeomorphic to a product of a line and the total space of a flat vector bundle over a closed infranil manifold.