# Differential Geometry Seminar Schedule for Spring 2014

## Fridays 3:00 - 3:50pm, SH 6635

### 4/4  Bogdan Suceava,  California State University, Fullerton  "From a Question of S.-S. Chern to the Geometry of Strongly Minimal Submanifolds: New Inequalities for Curvature"

Abstract:  In 1968, S.-S. Chern raised to attention the question that there might be further Riemannian obstructions for a manifold to admit an isometric minimal immersion into an Euclidean space. This question triggered a whole new direction of study in the geometry of submanifolds. ‪In the last two decades, there have been important advances in the study of new curvature invariants, many of these efforts due to B.-Y. Chen. We will present a few recent developments in this theory, and we focus our attention on several classes of inequalities in the geometry of submanifolds in Riemannian and Kaehlerian context.

### 4/25  Lihan Wang,  UCI  "HODGE THEORY ON COMPACT SYMPLECTIC MANIFOLDS WITH BOUNDARY"

Abstract: We study Hodge theory for symplectic Laplacians on compact symplectic manifolds with boundary. These Laplacians are novel as they can be associated with symplectic cohomologies and be of fourth-order. We prove various Hodge decompositions and use them to obtain the isomorphisms between the cohomologies and the spaces of harmonic fields with certain prescribed boundary conditions. In order to establish Hodge theory in the non-vanishing boundary case, we are required to introduce some new boundary conditions. As an application, our results can be used to solve boundary value problems of differential forms. This is a joint work with Li-Sheng Tseng.

### 5/2  Jeffrey Streets,  UCI  "On the singularity formation in fourth-order curvature flows"

Abstract: The L2 norm of the Riemannian curvature tensor is a natural energy to associate to a Riemannian manifold, especially in dimension 4.  A natural path for understanding the structure of this functional and its minimizers is via its gradient flow, the "L2 flow."  This is a quasi-linear fourth order parabolic equation for a Riemannian metric, which one might hope shares behavior in common with the Yang-Mills flow.  We verify this idea by exhibiting structural results for finite time singularities of this flow resembling results on Yang-Mills flow.  We also exhibit a new short-time existence statement for the flow exhibiting a lower bound for the existence time purely in terms of a measure of the volume growth of the initial data. As corollaries we establish new compactness and diffeomorphism finiteness theorems for four-manifolds generalizing known results to ones with have effectively minimal hypotheses/dependencies.  These results all rely on a new technique for controlling the growth of distances along a geometric flow, which is especially well-suited to the L2 flow.

### 5/9 Robert Ream,  UCSB  "Minimal Two-Spheres in Manifolds with Pinched Curvature"

Abstract: When the compact manifold $M$ has a suitably pinched, generic Riemannian metric, we show that it has many minimal two-spheres of low energy using Morse inequalities for the $\alpha$-energy of Sacks and
Uhlenbeck.  The difficulty is controlling bad behavior of a sequence of $\alpha$-energy critical points as $\alpha$ approaches one.  The two bad behaviors investigated are convergence toward a bubble tree or a branched cover of a
minimal sphere of lower energy.  We control these difficulties by making estimates on the index of bubble trees and branched covers. This proves  existence of many minimal two-spheres of low index.

### 5/23 Yuanqi Wang,   UCSB  "$C^{2,\alpha}$-estimate for  Monge-Ampere equations with H\"older-continuous right hand side"

Abstract:  Given an  elliptic complex Monge-Ampere equation and H\"older exponents $0<\alpha<\alpha^{\prime}$, suppose  $\phi\in C^{2,\alpha^{\prime}}$ is a solution which  possesses an uniform
$C^{1,1}-$bound, we present a new proof  to the $C^{2,\alpha}$-aprori estimate  for the solution $\phi$. Our estimates   only depends on the $C^{\alpha^{\prime}}-$norm of the right hand side of the equation.  Our new proof also works for   the real Monge-Ampere equations.   Our new proof has direct applications in the theory of complex Monge-Ampere equations with conical singularities, including the conical K\"ahler-Einstein equations,  and the conical K\"ahler-Ricci flows.

### 5/30  Lee Kennard,  UCSB  "What rational homotopy theory tells us about the Wilhelm conjecture"

Abstract:  A conjecture of Fred Wilhelm is the following: Given a Riemannian manifold M with positive sectional curvature, if M admits a Riemannian submersion to another manifold B, then the dimension of M is less than twice the dimension of B. We discuss some examples supporting this conjecture and some evidence for it. I will then discuss new evidence for the conjecture obtained using topological methods. This is joint work with Manuel Amann.

# Differential Geometry Seminar Schedule for Winter 2014

## Fridays 3:00 - 3:50pm, SH 6635

### 1/31 Guofang Wei,  UCSB "Monotonicity Formulas for Ricci Curvature"

Abstract: Bishop-Gromov's volume comparison has been a very useful tool for studying manifolds with Ricci curvature lower bound. Its essential information is contained in the monotonicity of a volume ratio. Perelman obtained several monotonicity formulas along the Ricci flow which are key tools in his work on the geometrization conjecture. Recently Colding and Minicozzi-Colding introduced some monotonicity formulas associated with harmonic functions. Joint with B. Song and G. Wu we extend and generalize these to Bakry-Emery Ricci curvature.

### 2/7  Changliang Wang,  UCSB "Linear and dynamical stability of Einstein metrics"

Abstract: There are two kinds of stability problems for Einstein metrics,  i.e. linear stability and dynamical stability. Natasa Sesum proved that in Ricci flat case dynamical stability implies linear stability and linear stability together with an integrability assumption imply dynamical stability. Recently, Robert Haslhofer and Reto Muller improved Sesum's result by getting rid of the integrability assumption. Then more recently, Klaus Kroncker generalized Haslhofer and Muller's result to general Einstein metrics.

### 2/14 Yuanqi Wang,  UCSB "Liouville theorem for complex Monge-Ampere equations with conic singularities"

Abstract:  Following Calabi, Pogorelov, Evans-Krylov-Safanov, and Trudinger's pioneer work on interior regularities and liouville theorems for Monge-Ampere equations, we prove the  Liouville theorem for conic K\"ahler-Ricci flat metrics. We
also discuss various applications of this Liouville theorem to conic K\"ahler geometry.

### 2/21 Peng Wu, Cornell University "Einstein four-manifolds of three-positive curvature operator"

Abstract: We will show that Einstein four-manifolds of three-positive
curvature operator are isometric to $(S^4, g_0)$, $(\mathbb{R}P^4, g_0)$, or
$(\mathbb{C}P^2, g_{FS})$. This is joint work with Xiaodong Cao.

### 2/28 Yuanqi Wang,  UCSB "Liouville theorem for complex Monge-Ampere equations with conic singularities II"

Abstract:  Following Calabi, Pogorelov, Evans-Krylov-Safanov, and Trudinger's pioneer work on interior regularities and liouville theorems for Monge-Ampere equations, we prove the  Liouville theorem for conic K\"ahler-Ricci flat metrics. We
also discuss various applications of this Liouville theorem to conic K\"ahler geometry.

### 3/7 Doug Moore,  UCSB "Morse Theory for Minimal Tori"

Abstract: Morse theory might have developed in three stages.  The first stage would have described the relationship between critical points on a finite dimensional manifold and the topology of the manifold.  The second might have been "calculus of variations in the large" for ODE's including the Morse theory of geodesics.  Palais and Smale formulated this in terms of infinite-dimensional manifolds, and Smale expressed the hope that this approach might lead to a similar theory for minimal surfaces, and related PDE's. Sacks and Uhlenbeck described how to perturb the theory of minimal surfaces and develop a Morse theory for the perturbed problem, initiating the third stage.

In this talk I will describe what happens when the perturbation is turned off.  One gets a partial Morse theory in the limit.  Applications to the existence of minimal tori in Riemannian manifolds will be presented.

### 3/14 Tiancong Chen,  UCSB "Prescribing Curvature to Graphs"

Abstract: We review various theorems on the existence of graphs whose curvature is prescribed. Then we present a new theorem. The is a joint work with Q. Han.

# Differential Geometry Seminar Schedule for Fall 2013

## Fridays 3:00 - 3:50pm, SH 6635

### 10/4 Yuanqi Wang, UCSB "On the long time behavior of the Conical K\"ahler-Ricci flow"

Abstract: We show that the conical K\"abler-Ricci flow actually exists for all time. This long time existence results gives
convergence Theorems when the twisted C_{1,beta}(M) is nonpositive, thus   completely settled down existence problem
of K\"ahler-Einstein metrics log Calabi-Yau pairs. We also discuss some related topics.

### 10/11 Xin Zhou, MSRI and MIT  "On the min-max hypersurface in manifold of positive Ricci curvature"

Abstract: In this talk, we will discuss the shape of the min-max minimal hypersurface produced by Almgren-Pitts corresponding to the fundamental class of a Riemannian manifold (M^{n+1},g) of positive Ricci curvature with 2 ≤ n ≤ 6. We characterize the Morse index, volume and multiplicity of this min-max hypersurface.

### 10/25 Maree Jaramillo, UCSB "Fundamental Groups of Spaces with Bakry-Emery Ricci Tensor Bounded Below"

Abstract: The Bakry-Emery Ricci tensor is a natural extension of Ricci curvature on smooth metric measure spaces. Since topological and geometric information can be obtained for manifolds with Ricci curvature bounded from below, it is natural to ask if the same information holds true for smooth metric measure spaces with Bakry-Emery Ricci tensor bounded from below. Using Guofang Wei and Will Wylie’s comparison theorems and an extension of Kevin Brighton’s gradient estimate on smooth metric measure spaces, we extend the Almost Splitting Theorem of Cheeger-Colding to the smooth metric space setting. Using this Almost Splitting theorem, we
show that the fundamental group of the smooth metric measure space with a lower bound on volume has almost abelian fundamental group. We also show that the number of generators of the fundamental group of a smooth metric measure space with Bakry-Emery Ricci tensor bounded from below is uniformly bounded. The results on the fundamental group are extensions of theorems which hold for Riemannian manifolds with Ricci curvature bounded from below.

### 11/8  Lee Kennard, UCSB "On Chern's question with symmetry"

Abstract: In 1965, Chern asked the following: If M is a compact manifold with positive sectional curvature, is every abelian subgroup of the fundamental group cyclic? Shankar proved in 1998 that the answer is no. I wil discuss conditions under which the answer is yes. The proof involves three obstructions to free actions on positively curved rational homology spheres.

### 11/15 Paul Lee, The Chinese University of Hong Kong, visiting MSRI "Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds"

Abstract: In this talk, we introduce a type of Ricci curvature lower bound for a natural sub-Riemannian structure on Sasakian manifolds. We will show that analogues of measure contraction properties, Bishop volume comparison theorem, and Laplacian comparison theorem hold under this lower bound.

### 11/22  Xianzhe Dai, UCSB "Analytic torsion and degeneration via Eguchi-Hanson instanton"

Abstract: Analytic torsion is a combination of determinants of the Laplacians on differential forms introduced by Ray-Singer as an analytic counterpart of the topological invariant, the Reidemeister torsion. It is trivial in even dimensions, but there is a complex version defined for complex manifolds. Motivated by questions from modular forms and mirror symmetry, we study the degeneration of the analytic torsion when the underlying manifolds develop orbifold singularities (via Eguchi-Hanson instanton). This is joint work with Ken-Ichi Yoshikawa.

### 12/6 Hao Fang, University of Iowa   "A spectral approach to singularity theory"

Abstract: In a joint work in progress with Fuijun Fan, motivated by LG mirror symmetry theory in physics, we study a modified d-bar operator over complete Kahler manifolds. We study its spectral properties  and establish a corresponding index theorem. We construct new torsion type invariants and point out possible applications to singularity theory.