# Geometric Analysis Seminar

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

### Spring Quarter, 2023

- April 18,
**Xuemiao Chen** (University of Waterloo)

**Zoom Meeting:** 933 0554 5643 (324863)

Tangent cones of admissible Hermitian-Yang-Mills connections
**Abstract**: Admissible Hermitian-Yang-Mills (HYM) connections are singular HYM connections with natural geometric bounds. In higher dimensional gauge theory, they naturally appear on the boundary of the moduli space of Hermitian-Yang-Mills connections over Kaehler manifolds. A fundamental problem was to study the uniqueness of the tangent cones of admissible HYM connections. I will explain joint work with Song Sun which confirms the uniqueness by showing that the tangent cones are algebraic invariants of the underlying reflexive sheaf.

- May 2,
**Paul Allen** (Lewis & Clark)

The singular Yamabe problem for Sobolev-regular metrics.
**Abstract**: This talk concerns complete, conformally-compact manifolds whose local geometry approaches that of hyperbolic space near the conformal boundary.

We work with Sobolev-regular metrics that are locally continuous and develop Fredholm theory for corresponding elliptic geometric operators.

As an application, we show that the Yamabe problem is well posed within the regularity classes of metrics that we define.

This work was done jointly with David Maxwell (UAF) and John M Lee (UW).

- May 9,
**Zhiqin Lu** (Irvine)

**Zoom Meeting:** 933 0554 5643 (324863)

On spectrum of complete Riemannian manifold
**Abstract**: We computed the

spectrum of Laplacians on k-forms on hyperbolic spaces. Moreover, we proved the

boundedness of certain resolvent of Laplacians by assuming the Ricci lower bound and manifold volume growth. This generalized a result of M. Taylor, in which bounded geometry of the manifold is assumed. This is a joint work of N. Charalambous.

- May 16,
**Arunima Bhattacharya** (UNC Chapel Hill)

A priori interior estimates for Lagrangian mean curvature equations
**Abstract**: In this talk, we will introduce the special Lagrangian and Lagrangian mean curvature type equations. We will derive a priori interior estimates for the Lagrangian mean curvature equation under certain natural restrictions on the Lagrangian angle. As an application, we will use these estimates to solve the Dirichlet problem for the Lagrangian mean curvature equation with continuous boundary data on a uniformly convex, bounded domain. We will also briefly introduce the fourth-order Hamiltonian stationary equation and mention some recent results on the regularity of solutions of certain fourth-order PDEs, which are critical points of variational integrals of the Hessian of a scalar function. Examples include volume functionals on Lagrangian submanifolds. This is based on joint works with Connor Mooney and Ravi Shankar.

- May 23,
**Eric Bahuaud** (Seattle University)

A dynamical framework for geometric flows
**Abstract**: In this talk I’ll discuss recent results concerning two geometric flows. The first result concerns the wellposedness of a higher-order geometric flow modeled on the ambient obstruction tensor over complete noncompact manifolds of bounded geometry. The second result concerns “convergence stability” for normalized Ricci flow near rotationally symmetric asymptotically hyperbolic metrics. Our analytic methods exploit semigroup techniques, and I’ll discuss a criterion for a geometric elliptic operator to generate an analytic semigroup on weighted Hölder spaces on manifolds of bounded geometry. This talk is based on joint work with Guenther, Isenberg and Mazzeo.

- May 31,
**Yu Li** (UTSC)

**8-9pm in Zoom Meeting:** 933 0554 5643 (324863)

On Kähler Ricci shrinker surfaces
**Abstract**: We use convergence theories for Ricci shrinkers to show that non-compact Kähler Ricci shrinker surfaces have two distinct canonical neighborhoods outside a compact set. Therefore, we prove that Kähler Ricci shrinker surfaces have uniformly bounded sectional curvature. By combining this curvature estimate with previous research by multiple authors, we achieve a comprehensive classification of all Kähler Ricci shrinker surfaces. This work was done in collaboration with Bing Wang.

- June 6,
**Yu-Shen Lin** (Boston University)

On the moduli spaces of ALH*-gravitational instantons
**Abstract**: Gravitational instantons are defined as non-compact hyperKahler 4-manifolds with L^2 curvature decay. They are all bubbling limits of K3 surfaces and thus serve as stepping stones for understanding the K3 metrics. In this talk, we will focus on a special kind of them called ALH*-gravitational instantons. We will explain the Torelli theorem, describe their moduli spaces and some partial compactifications of the moduli spaces. This talk is based on joint works with T. Collins, A. Jacob, R. Takahashi, X. Zhu and S. Soundararajan.

### 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