Geometric Analysis Seminar

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

April 4, Adam Jacob (UC Davis)
Tangent cones of Yang-Mills connections with isolated singularities

Abstract: Originally developed in the study of minimal surfaces Simon and Almgren, tangent cones have proved useful in applications to many geometric equations, including Yang-Mills connections. In this talk I will discuss how to uniquely identify the tangent cone of a Yang-Mills connection with isolated singularity in the complex setting, with an assumption on the complex structure of the bundle. This is joint work with H. Sa Earp and T. Walpuski.

April 25, Weiyong He (UO)
Constant scalar curvature metric and its weak solution on compact Kahler manifolds

Abstract: We define a notion of weak solution for constant scalar curvature equation on Kahler manifolds for metrics with only bounded coefficients. We develop some linear theory on compact Kahler manifolds for elliptic equations with only bounded coefficients. Using the linear theory, we prove a weak solution is smooth. This in part confirms a conjecture of X.X. Chen, regarding the regularity of minimizers of K-energy. This is joint work with Yu Zeng from University of Rochester

May 16, Micah Warren (UO)
Radial solutions of a fourth order Hamiltonian stationary equation

Abstract: We consider smooth radial solutions to the Hamiltonian stationary equation which are defined away from the origin. We show that in dimension two all radial solutions on unbounded domains must be special Lagrangian. In contrast, for all higher dimensions there exist non-special Lagrangian radial solutions over unbounded domains; moreover, near the origin, the gradient graph of such a solution is continuous if and only if the graph is special Lagrangian. Joint with Jingyi Chen.

May 23, Demetre Kazaras (UO)
Gluing manifolds with boundary and bordisms of positive scalar curvature metrics

Abstract: This thesis presents two main results on geometric and topological aspects of scalar curvature. The first is a gluing theorem for scalar-flat manifolds with vanishing mean curvature on the boundary. Our methods involve tools from conformal geometry and perturbation techniques for nonlinear elliptic PDE. The second part studies bordisms of positive scalar curvature metrics. We present a modification of the Schoen-Yau minimal hypersurface technique to manifolds with boundary which allows us to prove a hereditary property for bordisms of positive scalar curvature metrics. The main technical result is a convergence theorem for stable minimal hypersurfaces with free boundary in bordisms with long collars which may be of independent interest.

June 6, Arunima Bhattacharya (UO)
Regularity Bootstrapping for fourth order non linear equations

Abstract: We consider a C^{2,α} solution (u) of a fourth order non linear elliptic PDE. We show that the solution will be smooth if the PDE is regular. We define a regular PDE as a PDE whose coefficient matrix, a smooth function of the hessian of u, along with it’s derivative w.r.t the hessian satisfy the Legendre Hadamard (L.H) condition of ellipticity. This result can be applied to C^{2,α} minimizers of functionals defined on the hessian space satisfying certain convexity conditions, therefore showing that the minimizers are in fact smooth functions.

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

February 14, John Ma (UBC)
Compactness, finiteness properties of Lagrangian self-shrinkers in $\mathbb R^4$ and Piecewise mean curvature flow.

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

Math Department of Mathematics