# Geometric Analysis Seminar 2016

The geometric analysis seminar is held on Tuesdays at 11am in 210 Deady Hall.

### Spring Quarter, 2016

- March 29,
**Heather MacBeth** (MIT/MSRI)

Kähler-Einstein metrics and higher alpha-invariants
**Abstract**: I will describe a condition on the Bergman metrics of a Fano manifold M, which guarantees the existence of a Kähler-Einstein metric on M. I will also discuss a conjectural relationship between this condition and M‘s higher alpha-invariants αm,k(M), analogous to a 1991 theorem of Tian for αm,2(M).

- April 12,
**Greg Drugan** (UO)

Solitons for the inverse mean curvature flow
- April 19,
**Demetre Kazaras** (UO)

Rigidity of area-minimizing surfaces in 3-manifolds
**Abstract:** In this expository talk, we will describe some rigidity results for area-minimizing surfaces in 3-manifolds. For instance, Cai-Galloway (2000) show that a locally area-minimizing (2-sided) torus in a scalar-nonnegative 3-manifold is flat and in fact the ambient manifold is flat near the torus. I will highlight the more recent (2013) result of I. Nunes for hyperbolic surfaces which has a connection to Escobar`s Yamabe problem for manifolds with boundary.

- April 26,
**Christine Escher** (OSU)

Non-negative curvature and torus actions
**Abstract:** The classification of Riemannian manifolds with positive and non-negative sectional curvature is a long-standing problem in Riemannian geometry. In this talk I will summarize recent joint work with Catherine Searle on the classification of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, maximal torus action. This classification has many applications, in particular the Maximal Symmetry Rank conjecture holds for this class of manifolds.

- May 2,
**Peter Gilkey** (UO)

Moduli space of oriented affine manifolds of Type-A
This is joint work with E. Puffini, J.H. Park, E. Garcia-Rio, and M. Brozos-Vazquez.

- May 10,
**Adam Layne** (UO)

Future stable T^2 symmetric metrics of the Einstein Flow
**Abstract:** Study of the dynamic properties of the Einstein Field Equations has recently been dominated by local stability results: the stability of Minkowski space and the linear stability of Kerr. One may on the other hand study global stability questions, although this has only been possible under symmetry assumptions. I will survey such recent results and some current work with Jim Isenberg and Beverly Berger on future stable solutions of the Einstein Flow.

- May 17,
** Yu Zheng**

Deformations of twisted cscK metrics
**Abstract:** On a closed K\”ahler manifold, people have been searching for canonical representative in each K\”ahler class. In 80`s, Calabi has proposed to look for the constant scalar curvature K\”ahler(cscK) metric, or more generally the extremal metric, the critical point of Calabi energy. The cscK metric satisfies a fourth order nonlinear PDE. Recently, X. Chen proposed a new continuity path which connects the cscK metric with the critical point of J-functional. In this talk, I will present various openness results about this path. Mostly, I will describe how to deform from the critical point of J-functional(second order) to a twisted cscK metric(fourth order).

- May 24,
**Weiyong He** (UO)

### Winter Quarter, 2016

- January 6,
**Peter Gilkey** (UO)

The moduli space locally homogeneous affine surfaces
- January 12,
**Lorenzo Foscolo** (Stony Brook University)

Exotic nearly Kähler structures on the 6-sphere and the product of two 3-spheres
**Abstract**: Compact 6-dimensional nearly Kähler manifolds are the cross-sections of Riemannian cones with G2 holonomy. Viewing Euclidean 7-space as the cone over the round 6-sphere endows the latter with a nearly Kähler structure which coincides with the standard G2-invariant almost complex structure induced by octonionic multiplication. A long-standing problem has been the question of existence of complete nearly Kähler 6-manifolds besides the four known homogeneous ones. We resolve this problem by proving the existence of an exotic (inhomogeneous) nearly Kähler structure on the 6-sphere and on the product of two 3-spheres. This is joint work with Mark Haskins, Imperial College London.

- January 26,
**Boris Botvinnik** (UO)

Cheeger-Gromov convergence in the conformal setting
- February 2,
**Yakov Shlapentokh-Rothman** (Princeton)

Time-Periodic Einstein-Klein-Gordon Bifurcations Of Kerr
**Abstract**: For a positive measure set of Klein-Gordon masses mu^2 > 0, we construct one-parameter families of solutions to the Einstein-Klein-Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-periodic. An immediate corollary is that for these Klein-Gordon masses, the Kerr family is not asymptotically stable as a solution to the Einstein-Klein-Gordon equations. This is joint work with Otis Chodosh.

- February 9,
**Qin Hang** (Notre Dame)

Loewner-Nirenberg problem in singular domains
**Abstract**: We study the asymptotic behaviors of solutions of the Loewner-Nirenberg problem in singular domains and prove that the solutions are well approximated by the corresponding solutions in tangent cones at singular points on the boundary. The conformal structure of the underlying equation plays an essential role in the derivation of the optimal estimates.

- February 16,
**Micah Warren** (UO)

Approximate and Coarse Ricci Curvatures
**Abstract**: A few years ago, Tony Ache and I started on a project to recover the Ricci curvature on a submanifold from a sample of points and their extrinsic distances. This led us to study the notion of Coarse Ricci curvature, which is a function on pairs of points. Using the Carre du Champ of Bakry-Emery, we followed this to a somewhat satisfying notion of Coarse Ricci curvature that can be defined on a large class of spaces. Unfortunately, this turns out to be something that does not work when considering an extrinsic distance function. Instead, we consider approximate Ricci curvature, which is really just an approximation of the classical Ricci tensor using Bakry-Emery, applied to the approximate tangent spaces that are obtained using PCA. We will discuss the pros and cons of both approaches.

- March 1,
**Greg Drugan** (UO)

Mean curvature flow of an entire graph evolving away from the heat flow
**Abstract**: We present an initial graph over the entire plane for which the mean curvature flow behaves different from the heat flow. This is a joint work with Xuan Hien Nguyen.

- March 8,
**Xiaohua Zhu**

Steady Ricci solitons with positively curved curvature
**Abstract**: In this talk, I will discuss the rigidity problem of κ-noncollapsing steady solitons. First, we show that any κ-noncollapsing steady Kaehler-Ricc soliton with nonnegative sectional curvature must be flat. Secondly, we prove that any κ-noncollapsing steady Ricc soliton with nonnegative curvature operator and horizontally ϵ-pinched Ricci curvature should be rotationally symmetric.

### Fall Quarter, 2015

- September 17,
**Hojoo Lee** (Korea Institute for Advanced Study)

Sweeping Out Minimal Cones in Euclidean Space
**Abstract**: In 1867, Riemann discovered a family of complete, embedded, singly periodic minimal surfaces (in the three dimensional Euclidean space) foliated by circles and lines. He also proved that his staircases, planes, catenoids, and helicoids are the only minimal surfaces fibered by circles or lines in parallel planes. We explicitly construct generalized helicoids in odd dimensional Euclidean space, and minimal cones in even dimensional Euclidean space. Our minimal varieties unify various interesting examples: helicoids foliated by straight lines, Choe-Hoppe’s minimal hypersurfaces foliated by Clifford cones, Barbosa-Dajczer-Jorge’s ruled minimal submanifolds, and Harvey-Lawson’s twisted normal cone over Clifford torus. This is joint work with Eunjoo Lee.

- September 29,
**Valentino Tosatti** (Northwestern)

Complex Monge-Ampere type equations on Hermitian manifold
**Abstract**: The Calabi conjecture, proved by Yau in 1976, says that the complex Monge-Ampere equation on a compact Kahler manifold is always solvable, and this can be interpreted geometrically as constructing Kahler metrics with prescribed volume form (or equivalently prescribed Ricci curvature). In recent years there has been much interest in extending this theory to general compact Hermitian manifolds, which may not admit any Kahler metric. I will describe several results in this direction (joint with B. Weinkove) as well as a recent solution of a conjecture of Gauduchon (joint with G. Szekelyhidi and B. Weinkove).

- October 13,
**Peter Gilkey** (UO)

Affine gradient Ricci solitions and affine Killing vector fields on homogeneous affine surfaces
This is joint work with M. Bozos-Vazquez, E. Garcia-Rio, and E. Puffini.

- October 20,
**Micah Warren** (UO)

Regularity for Hamiltonian stationary submanifolds
**Abstract**: Suppose that L is a Hamiltonian stationary Lagrangian submanifold submanifold of complex space, that is, a manifold which is a critical point for volume under Hamiltonian variations. Suppose that L can be locally represented by the graph of a C1,α function over its tangent plane on a ball of radius r. Then then manifold enjoys a priori interior derivative estimates of all orders based on r and the C1,α bound.

- October 27,
**Greg Drugan** (UO)
- November 3,
**Demetre Kazaras** (UO)

Positive scalar curvature and free boundary minimal surfaces
**Abstract**: In 1979, Schoen and Yau defined a large class of closed manifolds which do not admit a metric of positive scalar curvature (PSC) by showing the following in dimensions 3 through 7: a codimension 1 integral homology class of a PSC manifold can be represented by a submanifold which admits a PSC metric. We will discuss how this technique can be applied to manifolds with non-empty boundary.

- November 10,
**HaoTian Wu** (UO)

Metrics with non-negative Ricci curvature on convex three-manifolds
**Abstract**: We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path-connected; and, moreover, that the associated moduli space (i.e., modulo orientation-preserving diffeomorphisms of the three-ball) is contractible. As an application, using results of Maximo, Nunes, and Smith (2013), we show the existence of properly embedded free boundary minimal annulus on any three-ball with non-negative Ricci curvature and strictly convex boundary. This is joint work with Antonio Ache (Princeton) and Davi Maximo (Stanford).

- November 17,
**Chong SONG** (Xiamen U. and UW)

Skew Mean Curvature Flow
**Abstract**: The skew mean curvature flow(SMCF) or binormal flow, which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. In this talk, I will show the basic properties of the SMCF and prove the existence of a short-time solution to the SMCF of surfaces in Euclidean space R^4 . If time permits, I will also talk about a moving frame method which transforms the SMCF to a non-linear Schrödinger system.

- November 24,
**Weiyong He** (UO)

Geometric flows on almost Hermitian manifolds
- December 1,
**Robert Lipshitz** (UO)

Holomorphic curves, Floer theory, and degenerations
**Abstract**: Lagrangian intersection Floer homology is an invariant of Lagrangian submanifolds defined by counting pseudo-holomorphic curves (solutions to a particular elliptic PDE). It can be used to define invariants of 3-manifolds in several ways; one such invariant is called Heegaard Floer homology. In this talk, we will recall the basic results about moduli spaces of pseudo-holomorphic curves needed to define Lagrangian intersection Floer homology. We will then talk about a second framework for these results, and the underlying analysis needed to define an extension of Heegaard Floer homology, called bordered Heegaard Floer homology. (This is joint work with Peter Ozsváth and Dylan Thurston.) Finally, we will mention some (slightly vague) open problems which seem to require further analytic developments.

**Previous Schedule:** 2010 Winter, 2010 Spring, 2011 Fall, 2012 Winter, 2014 Winter, 2014 Spring, 2014 Fall, 2015 Winter, 2015 Spring