# Differential Geometry Seminar Schedule for Spring 2012

## Fridays 3:00 - 3:50pm, SH 6617

### 4/6/2012 Sergey Grigorian, Simons Center at Stony Brook, visiting KITP "Deformations of G2-structures with torsion"

Abstract: We consider non-infinitesimal deformations of G2-structures on 7-dimensional manifolds and derive a closed expression for the torsion of the deformed G2-structure. We then specialize to the case where the deformation lies in the seven-dimensional representation of G2 and is hence defined by a vector v. In this case, we explicitly derive the expressions for the different torsion components of the new G2-structure in terms of the old torsion components and derivatives of v. In particular this gives a set of differential equations for the vector v which have to be satisfied for a transition between G2-structures with particular torsions. For some specific torsion classes we then explore the solutions of these equations.

### 4/13/2012 Yanir Rubinstein, Stanford University "Einstein metrics on Kahler manifolds"

Abstract: The Uniformization Theorem implies that any compact Riemann surface has a constant curvature metric. Kahler-Einstein (KE) metrics are a natural generalization of such metrics, and the search for them has a long and rich history, going back to Schouten, Kahler (30's), Calabi (50's), Aubin, Yau (70's) and Tian (90's), among others. Yet, despite much progress, a complete picture is available only in complex dimension 2. In contrast to such smooth KE metrics, in the mid 90's Tian conjectured the existence of KE metrics with conical singularities along a divisor (i.e., for which the manifold is `bent' at some angle along a complex hypersurface), motivated by applications to algebraic geometry and Calabi-Yau manifolds. More recently, Donaldson suggested a program for constructing smooth KE metrics of positive curvature out of such singular ones, and put forward several influential conjectures. In this talk we will try to give an introduction to Kahler-Einstein geometry and briefly describe some recent work mostly joint with R. Mazzeo that resolves some of these conjectures. One key ingredient is a new C^{2,\alpha} a priori estimate and continuity method for the complex Monge-Ampere equation, borrowing ideas from harmonic maps and the Ricci flow. In particular, it also gives a new proof of the Calabi-Yau theorem. It follows that many algebraic varieties that may not admit smooth KE metrics (e.g., Fano or minimal varieties) nevertheless admit KE metrics bent along a simple normal crossing divisor.

### 4/27/2012 Xianzhe Dai, UCSB "The asymptotic expansion of Bergman kernel for orbifolds"

Abstract: The Bergman kernel in the context of several complex variables has long been an important subject. Its analogue for complex projective manifolds is studied by Tian, Zelditch, Catlin, Lu among others, establishing the diagonal asymptotic expansion for high powers of an ample line bundle. Moreover, the coefficients in the asymptotic expansion encode geometric information of the underlying complex projective manifolds. This asymptotic expansion plays a crucial role in the breakthrough work of Donaldson where the existence of K\"ahler metrics with constant scalar curvature is shown to be closely related to Chow-Mumford stability. There are work extending the asymptotic expansion of Bergman kernel to orbifolds, such as those of Dai-Liu-Ma and Song. However, an analog of Donaldson's theorem remained elusive until the recent work of Ross-Thomas who introduced a weighted Bergman kernel on K\"ahler orbifolds with cyclic quotient singularities. We will explain how the work of Dai-Liu-Ma can be used to directly derive an asymptotic expansion for weighted Bergman kernel which plays an important role in proving an orbifold version of Donaldson's theorem.

### 5/18/2012 Yuanqi Wang, UCSB "On the K\"ahler-Ricci flow near a K\"ahler-Einstein metric"

Abstract: Motivated by recent study of the convergence of Calabi flow near a constant scalar curvature K\"ahler metric, we prove a similar theorem on the stability of the K\"ahler-Ricci flow near a K\"ahler-Einstein metric of positive scalar curvature.

### 6/1/2012 Lee Kennard, UPenn "On the Hopf conjectures with symmetry"

Abstract: We prove results related to the classical Hopf conjectures about positively curved manifolds under the assumption of a logarithmic symmetry rank bound. The main new tool is the action of the Steenrod algebra on cohomology, which we use to formulate a certain notion of periodicity in cohomology and obtain a generalization in this context of Adem's theorem on singly generated cohomology rings.

# Differential Geometry Seminar Schedule for Winter 2012

## Fridays 3:00 - 3:50pm, SH 6617

### 1/13/2012 Will Wylie, Syracuse University "Warped Product Einstein metrics and Ricci solitons"

Abstract: In this talk I'll discuss a characterization of Einstein metrics which are both homogeneous and admit a warped product structure. While metrics of these type are quite restrictive, I'll also show that one can always build such a metric from an algebraic Ricci soliton metric, extending results of Lauret relating algebraic Ricci soliton metrics to Einstein ones. The methods used involve almost no algebra and instead follow from the study of an overdetermined linear system of equations on the manifold which generalize a well known characterization of the sphere due to Obata. This is joint work with Peter Petersen of UCLA and Chenxu He of Lehigh.

### 2/17/2012 Guangxiang Su, Chern Institute of Mathematics, visiting UCSB "Analytic torsion and Reidemeister torsion for flat vector bundles"

Abstract: Reidemeister torsion is a classical topological invariant of manifolds introduced by Reidemeister for 3-manifolds and generalized to higher dimension by Franz and de Rham. Analytic torsion is an invariant of Riemannian manifolds defined by Ray and Singer as an analytic analogue of Reidemeister torsion. In recent years, the complex-valued analytic torsions are defined and studied by several authors. In this talk, I will review the definitions of these torsions, some methods in studying them and the relations between them.

### 2/24/2012 Michael Williams, UCLA "Stability of solutions of Ricci flow"

Abstract: The Ricci flow is an important tool in geometry, and a main problem is to understand the stability and convergence of solutions of the flow. We consider a class of left-invariant metrics on nilpotent Lie groups for which it is possible to explicit describe the behavior of Ricci flow. We also describe a more general technique for determining stability and convergence, which has applications to solutions on manifolds with certain types of bundle or warped-product structures, as well as to Ricci flow coupled with other geometric flows.

### 3/2/2012 Nicolaos Kapouleas, Brown University, visiting Stanford University "Gluing constructions for minimal surfaces and self-shrinkers"

Abstract: In the first part of the talk I will discuss doubling constructions. In particular I will discuss in some detail a recent doubling construction for an equatorial two-sphere in the round three-sphere, and also potential generalizations for self-shrinkers of the Mean Curvature flow and doubling constructions in higher dimensions. In the second part of the talk I will briefly discuss the current understanding of desingularization constructions for minimal surfaces and self-shrinkers. In the third and final part I will discuss open uniqueness questions for closed embedded minimal surfaces in the round three-sphere inspired by the above constructions.

### 3/23/2012 Catherine Searle, OSU "Non-negatively curved 5-manifolds with almost maximal symmetry rank"

Abstract: We consider closed, non-negatively curved, simply-connected 5-manifolds with an isometric $T^2$ action. We show that such a manifold is diffeomorphic to one of $S^5$, $S^3\times S^2$, $S^3\tilde{\times} S^2$ (the non-trivial $S^3$ bundle over $S^2$) or $SU(3)/SO(3)$, commonly known as the Wu manifold. This is joint work with Fernando Galaz-Garcia.

# Differential Geometry Seminar Schedule for Fall 2011

## Fridays 3:00 - 3:50pm, SH 6617

### 10/14 Xianzhe Dai, UCSB "Eta Invariant and Holonomy"

Abstract: The eta invariant is introduced by Atiyah-Patodi-Singer in their seminal series of papers as the correction term from the boundary for the index formula on a manifold with boundary. It is a spectral invariant associated to the natural geometric operator on the (boundary) manifold and it vanishes for even dimensional manifolds. Witten discovered that the holonomy of determinant line bundle is related to the adiabatic limit of eta invariant. The so called Witten's holonomy theorem has been proved by Bismut-Freed and Cheeger independently. We will discuss our joint work with Weiping Zhang for the even dimensional manifolds. We introduce eta invariants for even dimensional manifolds, which plays the same role as the eta invariant of Atiyah-Patodi-Singer. It is associated to $K^1$ representatives on even dimensional manifolds. The intrinsic spectral interpretation of this new invariant is more complicated. The general formulation relates the (mod $\mathbb Z$ reduction of) eta invariant for even dimensional manifolds with the holonomy of the determinant line bundle of a natural family of Dirac type operators. In this sense our result might be thought of as an even dimensional analogue of Witten's holonomy theorem.

### 10/28 Martin Li, University of British Columbia "Stability of minimal surfaces in complete 4-manifolds with positive isotropic curvature"

Abstract: In the past few decades, there has been a lot of success in using minimal surfaces to study the geometry and topology of manifolds with positive curvatures. Positive isotropic curvature (PIC) is a natural condition arising from the second variation of minimal surfaces. Just like positive sectional curvature tends to make geodesics unstable, PIC tends to make minimal surfaces unstable. This idea has been applied to give some information about the fundamental group of PIC manifolds. In this talk, I will give a new result about complete minimal surfaces in PIC manifolds, which also applies to the noncompact setting. The proof involves constructing holomorphic sections with slow growth using Hormander's weighted L^2 method and a weighted second variation argument.

### 11/4 Yuanqi Wang, UCSB "On four-dimensional anti-self-dual gradient Ricci solitons"

Abstract: Classification of 4-dim gradient Ricci solitons is important to the study of 4-dim Ricci flow with surgeries. My talk will be based on our classification of anti-self-dual gradient shrinking Ricci solitons and our results on anti-self-dual steady Ricci solitons. This is highly related to the analyticity of Ricci solitons. I will also discuss something on anti-self-dual Ricci flows.

### 11/18 Yi Wang, MSRI "The Aleksandrov-Fenchel inequalities of k+1 convex domains"

Abstract: In this talk, I will report a joint work with Sun-Yung Alice Chang in which we partially generalize the Aleksandrov-Fenchel inequalities for quermassintegrals from convex domains in the Euclidean space to a class of non-convex domains.

### 12/1 Please attend RTG seminar by Tobias Colding "Monotonicity formulas revisited"

Abstract: In this talk I will discuss several new monotonicity formulas for manifolds with a lower Ricci curvature bound. The monotonicity formulas are related to the classical Bishop-Gromov volume comparison theorem and Perelman's celebrated monotonicity formula for the Ricci flow. I will explain the connection between all of these. Moreover, I will explain how these new monotonicity formulas are linked to a new sharp gradient estimate for the Green's function. This is parallel to that Perelman's monotonicity is closely related to the sharp gradient estimate for the heat kernel of Li-Yau. In addition, there are obvious parallels between our monotonicity and the positive mass theorem of Schoen-Yau and Witten. Return to Conference and Seminars Page