# Geometric Analysis Seminar

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

### Winter Quarter, 2023

- February 7,
**Gavin Ball** (UW Madison)

Geometry of closed G2-structures
**Abstract**: A G2-structure on a 7-manifold M is given by a 3-form of a specific algebraic type. A G2-structure on M gives rise to a Riemannian metric on M in a non-linear manner and extra conditions imposed on the G2-structure are reflected in the geometry of the induced metric, the most well-known example of this phenomenon being that a closed and coclosed G2-structure induces a Ricci-flat metric. In my talk, I will describe my work on the Riemannian geometry of closed G2-structures, with special attention on two interesting cases: the so-called extremally Ricci-pinched closed G2-structures, and the closed G2-structures with conformally flat metric.

### Fall Quarter, 2022

- September 27,
**Shota Hamanaka** (Chuo University, Tokyo)

limit theorems for total scalar curvatures
**Abstract**: We give some

, or

limit theorems for total scalar curvatures. Specifically, we show that under some assumptions, the lower bound of the total scalar curvatures on a closed manifold is preserved under the

, or

convergence of the Riemannian metrics. We also give some counterexamples to the above theorems on some open manifolds.

- October 4,
**Lei Ni ** (UC San Diego)

Complex Monge-Ampere equation, Grauert tube and applications
**Abstract**: Grauert tube arises in the study of complexification of a Riemannian manifold. It also arises in the study of the complex Monge-Ampere equation. Here I explain an application of the results/theory of Lempert and Szöke and their relevance to Riemannian curvature and volume element estimates.

- October 18,
**Jesse Madnick** (UO)

Associative 3-folds in Squashed 7-Spheres
**Abstract**: The quaternionic Hopf fibration

lets us view the round 7-sphere as a family of round 3-spheres. Dilating these 3-spheres by t > 0 yields a family of Riemannian 7-manifolds

called “squashed 7-spheres.” Each of these spaces admits a natural (highly symmetric) co-closed

-structure, making the squashed 7-spheres fundamental examples in

-geometry.

In this talk, we construct the first non-trivial compact associative 3-folds in for every t > 0. Our examples arise from “twisting” circle bundles over pseudo-holomorphic curves in by a meromorphic function. Time permitting, we explain how our construction generalizes to the larger class of “squashed” 3-Sasakian 7-manifolds. This is joint work with Gavin Ball.

- November 1,
**Micah Warren** (UO)

An isoperimetric flow in the plane
**Abstract**: We consider a fourth order flow of compact curves in the plane. This is the gradient flow of arclength on the space of curves bounding a fixed area, with a particular metric (not the classic

metric) on the space of such curves. While this was first studied as a toy problem meant to build momentum for studying a gradient flow on compact lagrangian submanifolds in higher dimensions, it takes some work to get any result for the one dimensional flow. The approach is reminiscent of the work of Gage and Hamilton on curve shortening flow, without the help of a maximum principle for fourth order equations.

If the curve is near enough to the circle in a sense, the curve converges back to a circle.

- November 7,
**Aaron Kennon** (UC Santa Barbara)

**Special Day/Time:** 9:00am in 105 Fenton Hall

Nearly Parallel G2-Structures from the Perspective of Geometric Flows
**Abstract**: A 3-Sasakian structure on a 7-manifold may be used to define two distinct Einstein metrics: the 3-Sasakian metric and the squashed Einstein metric. Both metrics are induced by nearly parallel G2-structures which may also be expressed in terms of the 3-Sasakian structure. Just as Einstein metrics are critical points for the Ricci flow up to rescaling, nearly parallel G2-structures provide natural critical points of the (rescaled) geometric flows of G2-structures known as the Laplacian flow and Laplacian coflow. We study each of these flows in the 3-Sasakian setting and see that their behaviour is markedly different, particularly regarding the stability of the nearly parallel G2-structures. We also compare the behaviour of the flows of G2-structures with the (rescaled) Ricci flow.

This is joint work with Jason Lotay.

- November 8,
**Annegret Burtscher** (Radboud University, Netherlands)

**Location:**Fenton Lounge

On globally hyperbolic spacetimes
**Abstract**: Riemannian manifolds can be equipped with a natural metric space structure, thanks to which many results involving curvature bounds have been extended to metric (measure) spaces. A crucial ingredient in this process is the Hopf-Rinow Theorem which equates geodesic completeness with metric completeness. Lorentzian manifolds, in contrast, do not admit a canonical metric space structure and geodesic incompleteness is actually a desired feature. Still, the “best” Lorentzian manifolds mimic the good properties of complete Riemannian manifolds in many other ways. They are called globally hyperbolic spacetimes and are also of utmost importance in General Relativity (well-posedness of the initial value formulation of the Einstein equations, singularity theorems of Penrose and Hawking, splitting results etc.). In this talk, we present a surprising new characterization of global hyperbolicity. We show that globally hyperbolic spacetimes are precisely those Lorentzian manifolds for which the null distance is complete. This is joint work with Leonardo García-Heveling.

**Previous Schedule:** 2021, 2020, 2019, 2018, 2017, 2016, 2015 Spring, 2015 Winter, 2014 Fall, 2014 Spring, 2014 Winter, 2012 Winter, 2011 Fall, 2010 Spring, 2010 Winter