# Geometric Analysis Seminar

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

### Winter Quarter, 2017

- January 10,
**Bing Wang** (University of Wisconsin)

The extension problem of the mean curvature flow
**Abstract**: We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in ℝ3. This is a joint work with H.Z. Li.

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

Regularity of weak solutions of scalar curvature equation.
**Abstract**: We prove some regularity result for weak solutions of scalar curvature equation. This is a joint work with Yu Zeng

- January 31,
**Boris Botvinnik** (UO)
- February 14,
**Jim Isenberg** (UO)

### Fall Quarter, 2016

- September 27, Organizational Meeting
- October 4,
**Micah Warren** (UO)

Increased Regularity for Hamiltonian Stationary submanifolds
**Abstract**: A Hamiltonian Stationary submanifold of complex space is a Lagrangian manifold whose volume is stationary under Hamiltonian variations. We consider gradient graphs (x,Du(x)) for a function u. For a smooth u, the Euler-Lagrange equation can be expressed as a fourth order nonlinear equation in u that can be locally linearized (using a change of tangent plane) to the bi-Laplace. The volume can be defined for lower regularity, however, and computing the Euler-Lagrange equation with less assumed regularity gives a “double divergence” equation of second order quantities. We show several results. First, there is a c_n so that if the Hessian D^2u is c_n close to a continuous matrix-valued function, then the potential must be smooth. Previously, Schoen and Wolfson showed that when the potential was C^(2,α), then the potential u must be smooth. We are also able to show full regularity when the Hessian is bounded within certain ranges. This allows us to rule out conical solutions with mild singularities.

- October 11,
**Peter Gilkey** (UO)

Geodesics on locally homogeneous affine surfaces
**Abstract**: We examine questions of geodesic completeness in the context of locally homogeneous affine surfaces. Any locally homogeneous surface has a local model M where either the Christoffel symbols take the form Γ_{ij}^k are constant and the underlying space is R2 (Type-A) or the Christoffel symbols take the form Γ_{ij}^k=C_{ij}^k/x^1 where the underlying space is R+×R (Type-B). The model space M is said to be ESSENTIALLY GEODESICALLY COMPLETE if there does not exist a complete locally homogeneous surface modeled on M which is geodesically complete. Up to linear equivalence, there are exactly 3 models which are geodesically incomplete but not essentially geodesically incomplete. We classify all the geodesically complete models of Type-A and present some partial results concerning Type-B models. This is joint work in progress with E. Puffini (Krill Institute, Islas Malvinas) and with Daniela Dascanio and Pablo Pisani (Universidad Nacional de La Plata, Argentina)

- October 25,
**Demetre Kazaras** (UO)

Minimal hypersurfaces with free boundary and psc-bordism
**Abstract**: There is a well-known technique due to Schoen-Yau from the late 70s which uses (stable) minimal hypersurfaces to study the topological implications of a (closed) manifold`s ability to support positive scalar curvature metrics. In this talk, we describe a version of this technique for manifolds with boundary and discuss how it can be used to study bordisms of positive scalar curvature metrics.

- November 2,
**Bradley Burdick** (UO)

Perelman`s construction for gluing manifolds with positive Ricci curvature
- November 15,
**Weiyong He** (UO)

The Calabi flow with rough initial data
**Abstract**: We prove the short time existence of the Calabi flow for continuous initial metric. This is the joint work with Yu Zeng.

**Previous Schedule:** 2015 Fall – 2016 Spring, 2015 Spring, 2015 Winter, 2014 Fall, 2014 Spring, 2014 Winter, 2012 Winter, 2011 Fall, 2010 Spring, 2010 Winter