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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.

In this talk, first we will show that for every spacelike CMC surface of revolution (except spacelike cylinders and standard hyperboloids) about a timelike axis or spacelike axis, which is either an unduloid or a nodoid, in the Lorentz-Minkowski space L^3 there is an associated Weierstrass-P function. Next, using this association, we will show that unduloid and nodoid cannot be algebraic. A similar result is obtained for CMC surfaces of revolution in the Euclidean space E^3.

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.

I shall consider minimal hypersurfaces inside the unit ball whose boundary on the sphere is a small perturbation of the link of a minimizing quadratic cone. In this talk, I will discuss a recent result where I show that such minimal surfaces are uniquely determined by their boundary condition. In particular the solutions of the Plateau problem are unique for boundary conditions given by small perturbations of the link of a quadratic cone. Building on previous results we can characterize these unique surfaces. This is a joint work with Gábor Székelyhidi.

We will discuss $\rho$-equivariant conformal minimal immersions of finite energy into $\mathbb{CH}^2$. We prove that the induced metric is complete at a puncture $p$ if and only if the holonomy of $\rho$ around $p$ is parabolic. In this case $f$ is proper and $\partial f$ has pole at each puncture, so on the compactification $\Sigma$ we obtain a parabolic $\mathrm{PU}(2,1)$–Higgs bundle $(E,\Phi)$ with nilpotent residues and zero parabolic weights. This yields an Osserman-type extension principle for minimal surfaces in $\mathbb{CH}^2$ with complete ends. We further provide an algebro–geometric characterization of these immersions in terms of their associated parabolic Higgs data. This is a joint work with Indranil Biswas and John Loftin.

The unit sphere minimizes the first positive Laplacian eigenvalue among all compact n-dimensional manifolds with Ricci curvature bounded below by n−1, as a consequence of Lichnerowicz's Theorem. This naturally raises a question about the higher Laplacian eigenvalues: Does the unit sphere minimize all the Laplacian eigenvalues? In this talk, we will explore a class of manifolds whose Laplacian eigenvalues are strictly greater than the corresponding eigenvalues of the unit sphere. These examples provide evidence for such a general minimization in dimensions two and three.