Talks

In these two lectures I will explain how loop groups and loop algebras can be used to express the equations for a harmonic map of a Riemann surface into a symmetric space by meromorphic data---a generalized Weierstrass representation. I will discuss how to apply this method to special situations such as the construction of constant mean curvature surfaces in the 3-sphere. The lectures are intended as an introduction into this topic.

In this second talk on Lagrangian mean curvature flow, I will focus on key examples which illustrate the theory from the previous talk. In particular, we will see important objects arising in symplectic and Riemannian geometry, such as the Clifford torus, Whitney sphere and Lawlor necks.

Maximal surfaces in 3-dimensional Lorentz-Minkowski space arise as solutions to the variational problem of local area maximizing among the spacelike surfaces. These surfaces are zero mean curvature surfaces, and maximal surfaces with singularities are called generalized maximal surfaces. Maxfaces are a special class of these generalized maximal surfaces where singularities appear at points where the tangent plane contains a light-like vector. I will present the construction of a new family of maxfaces of high genus that are embedded outside a compact set and have arbitrarily many catenoid or planar ends using the node opening technique. The surfaces look like spacelike planes connected by small necks. Among the examples are maxfaces of the Costa-Hoffman-Meeks type. More specifically, the singular set form curves around the waists of the necks. In generic and some symmetric cases, all but finitely many singularities are cuspidal edges, and the non-cuspidal singularities are swallowtails evenly distributed along the singular curves. This work is conducted in collaboration with Dr. Hao Chen, Dr. Anu Dhochak, and Dr. Pradip Kumar.

Lagrangian mean curvature flow is potentially a powerful tool for tackling problems in symplectic geometry via geometric analysis, by studying the existence problem for minimal Lagrangian submanifolds. In this first talk I will give an overview of Lagrangian mean curvature flow and describe some of the key foundational results.

Harmonic maps to symmetric spaces are used in the non-Abelian Hodge correspondence to bridge surface group representations with Higgs bundles. In special cases, these harmonic maps are conformal and hence give minimal surfaces in a symmetric space. In the first lecture, we look at the case of SL(3,R) and describe some asymptotics via Blaschke metrics.

In the second lecture, we will look at higher genus minimal Lagrangians in CP^2. There will be objects reminiscent of Higgs bundles, but which are not Higgs bundles. This will involve loop group methods and satisfying a closing condition.

In these two lectures I will explain how loop groups and loop algebras can be used to express the equations for a harmonic map of a Riemann surface into a symmetric space by meromorphic data---a generalized Weierstrass representation. I will discuss how to apply this method to special situations such as the construction of constant mean curvature surfaces in the 3-sphere. The lectures are intended as an introduction into this topic.

We describe the asymptotics of high energy harmonic maps from Riemann surfaces to locally symmetric spaces in special classes in two settings: surface group actions on PSL(2,\R) and on SL(3,\R).  The goal is to highlight some aspects of technique, though inevitably we will state some results that follow from the methods. Joint work with Dumas, Loftin, Tamburelli, and Pan, if not others.

We prove the existence of complete minimal surfaces in $\mathbb{R}^3$ of arbitrary genus $p\, >\, 1$ and least absolute curvature with precisely two ends --- one catenoidal and one Enneper-type --- thereby resolving, affirmatively, a conjecture posed by Weber. These surfaces, which are called \emph{Angel surfaces}, generalize the genus-one example constructed earlier by Fujimori and Shoda. We extend the orthodisk method developed by Weber and Wolf, \cite{weber2002teichmuller}, to construct the minimal surfaces. A central idea in our construction is the notion of \emph{partial symmetry}, which enables us to introduce controlled symmetry into the surface. Reference: [Weber and Wolf(2002)] Matthias Weber and Michael Wolf. Teichm¨uller theory and handle addition for minimal surfaces. Annals of mathematics, pages 713–795, 2002.

We study the approximation of eigenvalues for the Laplace-Beltrami operator on closed Riemannian manifolds in the class characterized by bounded Ricci curvature, a lower bound on the injectivity radius, and an upper bound on the diameter. We use an (\epsilon,\rho)-approximation of the manifold by a weighted graph, as introduced by Burago et al. By adapting their methods, we prove that as the parameters \epsilon, \rho and the ratio \frac{\epsilon}{\rho} approach zero, the k-th eigenvalue of the graph Laplacian converges uniformly to the k-th eigenvalue of the manifold's Laplacian for each k.

We describe the asymptotics of high energy harmonic maps from Riemann surfaces to locally symmetric spaces in special classes in two settings: surface group actions on PSL(2,\R) and on SL(3,\R).  The goal is to highlight some aspects of technique, though inevitably we will state some results that follow from the methods. Joint work with Dumas, Loftin, Tamburelli, and Pan, if not others.