Friday September 29: No talk (department BBQ)

Friday October 6: Xin Zhou (UCSB)

Min-max theory for constant mean curvature hypersurfaces

In this talk, I will present constructions of closed CMC hypersurfaces using min-max method. In particular, given any closed Riemannian manifold, I will show the existence of closed CMC hypersurfaces of any prescribed mean curvature. This is a joint work with Jonathan Zhu.

Friday October 13: Antoine Song (Princeton)

Local min-max surfaces and existence of minimal Heegaard splittings

Let M be a closed oriented 3-manifold not diffeomorphic to the 3-sphere, and suppose that there is a strongly irreducible Heegaard splitting H. Previously, Rubinstein announced that either there is a minimal surface of index at most one isotopic to H or there is a non-orientable minimal surface such that the double cover with a vertical handle attached is isotopic to H. He sketched a natural outline of a proof using min-max, however some steps are non-trivially incomplete and we will explain how to justify them. The key point is a version of min-max theory producing interior minimal surfaces when the ambient manifold has minimal boundary. Some corollaries of the theorem include the existence in any RP^3 of either a minimal torus or a minimal projective plane with stable universal cover. Several consequences for metric with positive scalar curvature are also derived.

Friday October 20: Zhongmin Jin and Yihan Li (UCSB)

There will be two 25min talks.

Zhongmin Jin

Title: Finite homeomorphism type with integral curvature bound.

Abstract: In the 1991 paper of Anderson and Cheeger, they proved the finite diffeomorphism types of the collection of compact Riemannian manifolds with uniform two-sided Ricci curvature bound, diameter upper bound, volume lower bound and integral curvature upper bound. The main tool is the so called $C^{1,\alpha}$ harmonic coordinates and main ingredients are $\epsilon$-regularity theorem and neck theorem. In this talk, we will introduce a related version of finiteness theorem without the Ricci curvature upper bound and with a non-concentration integral curvature assumption by using Reifenberg parametrization, where $C^{1,\alpha}$ harmonic radius type argument fails.

Yihan Li

Title: Asymptotic Spectral flow with Heat kernel.

Title: Finite homeomorphism type with integral curvature bound.

Abstract: In the 1991 paper of Anderson and Cheeger, they proved the finite diffeomorphism types of the collection of compact Riemannian manifolds with uniform two-sided Ricci curvature bound, diameter upper bound, volume lower bound and integral curvature upper bound. The main tool is the so called $C^{1,\alpha}$ harmonic coordinates and main ingredients are $\epsilon$-regularity theorem and neck theorem. In this talk, we will introduce a related version of finiteness theorem without the Ricci curvature upper bound and with a non-concentration integral curvature assumption by using Reifenberg parametrization, where $C^{1,\alpha}$ harmonic radius type argument fails.

Yihan Li

Title: Asymptotic Spectral flow with Heat kernel.

Friday October 27: Qiang Guang (UCSB)

Curvature estimates and compactness for free boundary minimal hypersurfaces

Minimal hypersurfaces with free boundary are critical points of the area functional in compact manifolds with boundary. In this talk, we will present curvature estimates for stable free boundary minimal hypersurfaces. In particular, for embedded stable free boundary minimal surfaces in 3-manifolds, we present a stronger curvature estimate without a prior area bound. We will also discuss the curvature estimates and compactness for free boundary minimal hypersurfaces which have interior touching with the boundary of the ambient manifold.

Friday November 3: Longzhi Lin (UC Santa Cruz)

Energy convexity of intrinsic bi-harmonic map and its heat flow

The theory of harmonic map and its higher dimensional analogues (e.g. bi-harmonic map) has been a classic and intensely researched field in PDE and geometric analysis. In this talk, we will discuss an energy convexity for intrinsic bi-harmonic map and its heat flow with small intrinsic bi-energy from the 4-disc to spheres. This in particular yields the uniqueness of intrinsic bi-harmonic maps on the 4-disc with small bi-energy and the uniform convergence of the intrinsic bi-harmonic map heat flow on the 4-disc with small initial bi-energy. The energy convexity for harmonic maps with small energy was proved earlier by Colding-Minicozzi (c.f. Lamm-Lin) and the uniform convergence of the harmonic map heat flow with small initial energy was proved earlier by myself. This is a recent joint work with Paul Laurain.

Friday November 10: holiday, no talk

Friday November 17:

Friday November 24: Thanksgiving, no talk

Friday December 1: Xuwen Zhu (Stanford)

Deformation of constant curvature conical metrics

In this joint work with Rafe Mazzeo, we would like to understand the deformation theory of constant curvature metrics with prescribed conical singularities on a compact Riemann surface. We construct a resolution of the configuration space, and prove a new regularity result that the family of constant curvature conical metrics has a nice compactification as the cone points coalesce. This is one key ingredient to understand the full moduli space of such metrics with positive curvature and cone angles bigger than $2\pi$.

Friday December 8:

Friday January 13: No talk

Friday January 20: Shoo Seto and Lili Wang (UCSB)

Fundamental gap for convex domains of the sphere

In this talk, we introduce the Laplacian eigenvalue problem and briefly go over its history. Then we will present a recent result which gives a sharp lower bound of the fundamental gap for convex domain of spheres motivated by the modulus of continuity approach introduced by Andrews-Clutterbuck. This is joint work with Lili Wang and Guofang Wei.

Friday January 27: No talk

Friday February 3: Chenxu He (UC Riverside)

Warped product Einstein structure

The construction of Einstein metrics on warped products has been widely used in general relativity, including the Schwarzschild metric. The study of such construction on Riemannian manifolds already appeared in Besse?s book on Einstein manifolds. A new perspective of study was introduced by Case-Shu-Wei in 2008. They studied the Einstein metrics on warped products through the equation for the Ricci curvature of the base manifold. In this talk, I will present a few recent results on warped product Einstein metrics: the classification with interesting geometries, the uniqueness of such metrics, and the connection with non-gradient expanding Ricci solitons. It is based on the joint work with Peter Petersen (UCLA) - William Wylie (Syracuse), and with Qiang Chen.

Friday February 10: Jonathan Zhu (Harvard)

Entropy and self-shrinkers of the mean curvature flow

The Colding-Minicozzi entropy is an important tool for understanding the mean curvature flow (MCF), and is a measure of the complexity of a submanifold. Together with Ilmanen and White, they conjectured that the round sphere minimizes entropy amongst all closed hypersurfaces. We will review the basics of MCF and their theory of generic MCF, then describe the resolution of the above conjecture, due to J. Bernstein and L. Wang for dimensions up to six and recently claimed by the speaker for all remaining dimensions. A key ingredient in the latter is the classification of entropy-stable self-shrinkers that may have a small singular set.

Friday February 17: Special Day in Geometry

10:30-11:30am
Christopher Connell, Indiana University, Bloomington

Title: Simplicial Volume of Affine Manifolds

Title: Simplicial Volume of Affine Manifolds

We show that a broad class of affine manifolds (including all complete affine manifolds) must have vanishing simplicial volume. This provides some further evidence for the veracity of the Auslander Conjecture. Along the way, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups of ?1 to have vanishing simplicial volume. This establishes a special case of a conjecture due to Luck. This is joint work with Jean-Franois Lafont and Michelle Bucher.

1:30-2:30pm
Raquel Perales, Institute of Mathematics, UNAM

Title: Gromov-Hausdorff and Intrinsic Flat Convergence

Title: Gromov-Hausdorff and Intrinsic Flat Convergence

In this talk we study sequences integral current spaces and metric spaces with boundary. We prove theorems demonstrating when the Gromov-Hausdorff and Intrinsic Flat limits of sequences of such metric spaces agree. From these theorems we derive compactness theorems for sequences of oriented Riemannian manifolds with boundary where both the GH and SWIF limits agree. For these sequences we only require nonnegative Ricci curvature, upper bounds on volume, noncollapsing conditions on the interior of the manifold and diameter controls on the level sets near the boundary.

3-4pm
Pablo Surez Serrato, National Autonomous University of Mexico

Title: Yamabe flows and extremal entropy

Title: Yamabe flows and extremal entropy

We introduce curvature-normalized versions of the Yamabe flow on complete manifolds with negative scalar curvature, for which we show long time existence of solutions and convergence of these flows to complete Yamabe metrics.
We apply them to study the extrema of the topological entropy in conformal classes and offer an entropy rigidity theorem for convex-cocompact surfaces: extrema of the entropy are the metrics whose closed geodesics coincide with those of the unique hyperbolic metric conformally equivalent to the initial one. On convex-cocompact manifolds of higher dimension we show that local extrema of the entropy have constant scalar curvature on their nonwandering set.
All of this is joint work with Samuel Tapie, University of Nantes.

Friday February 24: Xiangwen Zhang (UC Irvine)

The Anomaly Flow and Strominger systems

The anomaly flow is a geometric flow which implements the Green-Schwarz anomaly cancellation mechanism originating from superstring theory, while preserving the conformally balanced condition of Hermitian metrics. I will discuss criteria for long time existence and convergence of the flow on toric fibrations with the Fu-Yau ansatz. This is joint work with D.H. Phong and S. Picard.

Friday March 3: Chao Li (Stanford University)

Index and topology of minimal hypersurfaces in R^n

In this talk, we consider immersed two-sided minimal hypersurfaces in R^n with finite total curvature. We prove that the sum of the Morse index and the nullity of the Jacobi operator is bounded from below by a linear function of the number of ends and the first Betti number of the hypersurface. When n=4, we are able to drop the nullity term by a careful study for the rigidity case. I would also like to discuss two applications of this index estimates: one in the study of index one minimal hypersurface, the other in a compactness and finiteness results of minimal hypersurfaces in R^4 with bounded index.

Friday March 10:

The 33rd Pacific Coast Gravity Meeting

Friday March 17: David Morrison (UCSB)

Degenerations of K3 surfaces, gravitational instantons, and M-theory

The detailed study of degenerations of K3 surfaces as complex manifolds goes back more than forty years and is fairly complete. Much less is known about the analogous problem in differential geometry of finding Gromov--Hausdorff limits for sequences of Ricci-flat metrics on the K3 manifold. I will review recent work of H.-J. Hein and G. Chen--X. Chen on gravitational instantons with curvature decay, and describe applications to the K3 degeneration problem. M-theory suggests an additional geometric structure to add, and I will give a conjectural sketch of how that structure should clarify the limiting behavior.

Thursday September 8: Pablo Su\'arez-Serrato (Instituto de Matem\'aticas UNAM, Mexico City)

On the topology and geometry of higher graph manifolds

Our understanding of $3$--manifolds has illuminated two distinct classes of importance; hyperbolic manifolds and graph manifolds. These are by now considered the basic blocks featured in the geometrisation program of Thurston, famously consolidated by Perelman.
From one perspective graph manifolds are exactly the manifolds that {\it collapse}, in the sense that they admit a family of smooth metrics whose volumes tend to zero while their sectional curvatures remain bounded. Historically the term {\it graph manifold} was introduced by Waldhausen in the 1960's. It highlighted the fact that the fundamental group can be described as a graph of groups and that the manifolds were built up from fundamental pieces that are (heuristically) described in terms of circle bundles over 2-orbifolds.
In a recent, and fundamental, paper Frigerio, Lafont and Sisto proposed a family of {\it generalised graph manifolds}; products of $k$--tori with hyperbolic $(n-k)$--manifolds with truncated cusps are glued along their common $n$--toral boundaries. They explored multiple topological aspects of this family and raised some questions. For example, in their definition $k$ is allowed to equal zero, so that one subfamily is made up of hyperbolic manifolds glued along truncated cusps. They asked if the minimal volume of such manifolds is achieved by the sum of the hyperbolic volumes of the pieces.
Together with Chris Connell we answered this question positively. In so doing we realised that a natural family we termed {\it higher graph manifolds} could be defined; bundles of infranilpotent manifolds over negatively curved bases are glued along boundaries (when possible). This family further extends the one proposed by Frigerio, Lafont and Sisto. We first characterise the higher graph manifolds that admit volume collapse, by explicitly constructing sequences of metrics with volume collapse (this builds on earlier work by Fukaya). Various results about the simplicial volume and volume entropy of this family are calculated. Then we exploit the graph structure of the fundamental group to show that these manifolds obey the coarse Baum-Connes conjecture, have finite asymptotic dimension and do not admit metrics of positive scalar curvature. Finally we use several of the produced results to prove that when the infranilmanifold fibre has positive dimension the Yamabe invariant vanishes.

Friday September 30: Qiang Guang (UCSB)

Rigidity and curvature estimates for almost stable self-shrinkers

Self-shrinkers model the singularities of the mean curvature flow. A Bernstein-type result for self-shrinkers says that any entire self-shrinking graph must be a hyperplane. In this talk, I will discuss a stronger rigidity result that any self-shrinker which is graphical in a large, but compact, ball must be a hyperplane. In fact, this rigidity holds within a larger class of almost stable self-shrinkers. A key ingredient of this result is a curvature estimate for almost stable shrinkers. This is joint work with Jonathan Zhu.

Friday October 7: Xin Zhou (UCSB)

Min-max minimal hypersurfaces with free boundary

Minimal surfaces with free boundary are natural critical points of the area functional in compact smooth manifolds with boundary. In this talk, I will describe a general existence theory for minimal surfaces with free boundary. In particular, I will show the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. The minimal surfaces with free boundary were constructed using the celebrated min-max method. I will explain the basic ideas behind the min-max theory as well as our new contributions. At the end, I will introduce a few conjectures that can be approached based on my work.

Friday October 14: Jesus Angel Nunez Zimbron (UCSB)

Equivariant geometry of Alexandrov 3-spaces

Alexandrov spaces constitute a synthetic generalization of Riemannian manifolds with a lower bound on sectional curvature to the class of metric spaces. They provide a natural setting to study many notions of global Riemannian geometry, and therefore, a fundamental problem is that of extending to Alexandrov spaces what is known for Riemannian manifolds. As in the Riemannian case, one may investigate Alexandrov spaces via their symmetries. Since the isometry group of a compact Alexandrov space is a compact Lie group, this point of view naturally leads to the study of isometric Lie group actions on Alexandrov spaces. Berestovski i showed that finite-dimensional homogeneous Alexandrov spaces actually are Riemannian manifolds. Later, Galaz-Garcia and Searle studied Alexandrov spaces of cohomogeneity one (i.e. those with an isometric action of a compact Lie group whose orbit space is one-dimensional) and classified them in dimensions at most 4.

In this talk I will present an equivariant and topological classification of closed Alexandrov spaces of dimension 3 admitting isometric actions of cohomogeneity 2. As an application of this result I will talk about a version of the Borel conjecture for Alexandrov 3-spaces with circle symmetry. I will also talk about an application of the classification to address some aspects of the geometry (in the sense of Thurston) of Alexandrov 3-spaces. The results presented here are joint work with Fernando Galaz-GarcĂa and Luis Guijarro.

In this talk I will present an equivariant and topological classification of closed Alexandrov spaces of dimension 3 admitting isometric actions of cohomogeneity 2. As an application of this result I will talk about a version of the Borel conjecture for Alexandrov 3-spaces with circle symmetry. I will also talk about an application of the classification to address some aspects of the geometry (in the sense of Thurston) of Alexandrov 3-spaces. The results presented here are joint work with Fernando Galaz-GarcĂa and Luis Guijarro.

Friday October 21: No seminar this weak

Thursday October 27: Distinguished Lecture Series, Richard Schoen (UC Irvine)

Lecture I: Geometries that optimize eigenvalues: closed surfaces

Friday October 28: Distinguished Lecture Series, Richard Schoen (UC Irvine)

Lecture II: Geometries that optimize eigenvalues: surfaces with boundary

When we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases it makes sense to attempt to find
critical points in the space of metrics. In these talks we will survey two cases in which progress has been made, focusing in the first lecture on closed surfaces and in the second on surfaces with boundary. We will describe the geometric structure of the critical metrics which turn out to be the induced metrics on certain special classes of minimal (mean curvature zero) surfaces in spheres and euclidean balls. The eigenvalue extremal problem is thus related to other questions arising in the theory of minimal surfaces. We will also discuss some of these.

Friday November 4: John Lott (Berkeley)

Some almost theorems about four-dimensional curvature

I will describe some topological conditions to ensure that a noncollapsed Riemannian 4-manifold, which is almost Ricci-flat, in fact admits a Ricci-flat metric. This is joint work with Vitali Kapovitch.

Friday November 18: Douglas Moore (UCSB)

Minimal surfaces in four-manifolds

The theory of minimal surfaces in four-manifolds simplifies when one assumes that the Riemannian metric on the four-manifold is generic. I will discuss the relationship between minimal surfaces in such manifolds and the topological invariants described by Seiberg and Witten.