Participants and talks
Browse the talks and speakers who participated in our spring 2017 workshop at University College Cork.
Minimal isometric immersions into S^2 x R and H^2 x R
For a given simply connected Riemannian surface Sigma, we relate the problem of finding minimal isometric immersions of Sigma into S^2 x R or H^2 x R to a system of two partial differential equations on Sigma. We prove that a constant intrinsic curvature minimal surface in S^2 x R or H^2 x R is either totally geodesic or part of an associate surface of a certain limit of catenoids in H^2 x R. We also prove that if a non constant curvature Riemannian surface admits a continuous one-parameter family of minimal isometric immersions into S^2 x R or H^2 x R, then all these immersions are associate.
Surfaces of critical constant mean curvature and harmonic maps
Minimal surfaces (H=0) in euclidean 3-space and Bryant surfaces (H=1) in hyperbolic 3-space are a special family among all the CMC surfaces in spaces forms. Similarly, surfaces of critical CMC in homogenous 3-spaces present a special behavior among all the CMC surfaces. For example, Fernández and Mira found a hyperbolic Gauss map for surfaces in H2xR that turns out to be harmonic for surfaces of critical CMC. Later on, Daniel discovered a Gauss map for surfaces in Heisenberg space that is also harmonic when the mean curvature is critical. However, both definitions are quite different and it was unclear how to extend them to the general setting. In this talk we will review some properties of critical CMC surfaces in homogeneous 3-spaces and present a unified definition of a Gauss map for surfaces in these ambient spaces that is harmonic when the mean curvature of the surface is critical.
Properly embedded minimal annuli in H2 × R
In this talk we ask for properly embedded minimal annuli in H2 × R which bound a pair of vertical graphs over ∂∞H2 ≡ S1. We present some compactness results for these surfaces. We also give some existence results for proper, Alexandrov-embedded, minimal annuli. Contrary to what might be expected, we show that, in general, one can not prescribe the two components of the boundary at infinity. However, we can prescribe one of the boundary data, the position of the neck and the vertical flux of the annulus. This is a joint work with F. Martín, R. Mazzeo and M. Rodríguez.
Asymptotic methods for finite gap curves in the 3-dimensional space forms
I will discuss how asymptotic methods can be used to study closed curves, and in particular finite gap curves, in the 3-dimensional space forms.
Asymptoticity of Minimal surfaces in Heisenberg space
I will describe the asymptotic behavior of minimal ends using Dirac operator. A joint work with Taimanov.
Complex line bundles over simplicial complexes
Over the last years we could apply bundle theory to several problems in Computer Graphics. This led to a quite good understanding of the underlying discrete theory: We present a discrete analogue of a classification theorem due to Kobayashi and then focus on hermitian line bundles with curvature. For these a discrete analogue of Weil's theorem and a discrete Poincaré-Hopf theorem hold. Furthermore, we generalize the well-known cotan-Laplace operator.
Isometric immersions and integrability
Some examples of surfaces for which isometric immersion to a space form is an integrable system.
Minimal Surfaces in the Heisenberg Space
We discuss the behaviour of some minimal surfaces in the Heisenberg space. In particular, we deal with existence and growth of non compact graphs and stability properties.
Constant mean curvature spheres in homogeneous three-manifolds, I
The aim of these two talks (which are based on joint work with Bill Meeks and Antonio Ros) is to prove the following theorem: any two spheres of the same constant mean curvature immersed in a homogeneous three-manifold only differ by an ambient isometry. Our study will also determine the exact values of the mean curvature for which such CMC spheres exist, together with some of their most important geometric properties. For instance, we will show that CMC spheres in simply connected metric Lie groups have index one, are Alexandrov embedded and maximally symmetric, their left invariant Gauss maps are diffeos, and the corresponding moduli space of CMC spheres is a connected one-dimensional manifold.
Energy quantization for harmonic 2-spheres in non-compact symmetric spaces
It is well known from results by Uhlenbeck, Chern-Wolfson and Burstall-Rawnsley that harmonic 2-spheres in compact symmetric spaces have quantized energies. Using the reformulation of the harmonic map equation as a family of flat connections, we construct an energy preserving duality between harmonic maps into a non-compact symmetric space (typically pseudo-Riemannian) and harmonic maps into compact real forms of its complexification. Applying this construction to the conformal Gauss maps of Willmore 2-spheres in the n-sphere provides a generalization and unifying approach to existing quantization results in special cases: Bryant for n=3; Montiel for n=4; and Ejiri for Willmore 2-spheres admitting a dual Willmore surface.
Constant mean curvature spheres in homogeneous three-manifolds, II
See Pablo Mira.
From Smoke Ring Flow to Real Fluids
The so-called "smoke ring flow" for space curves was introduced in 1910 by Da Rios (who was a PhD student of Levi-Civita) for describing the time evolution of vortex filaments in an ideal fluid. Starting from the 1970's it became clear that the smoke ring flow actually is the most basic integrable system that originates in Differential Geometry. It is closely related to the one-dimensional Landau-Lifshitz equation and to the one-dimensional nonlinear Schrödinger equation. As an asymptotic limit it is also crucial for understanding the geometry of CMC surfaces in space forms.
Vortex filaments are the solitons of fluid dynamics, so in this sense fluid flow can be viewed as a perturbed integrable system. In this talk we will show a method for fluid simulation that reflects this fact. This method is closely related to the three-dimensional Landau-Lifshitz equation and to the three-dimensional nonlinear Schrödinger equation.
Willmore energy of conformal maps f: C/Gamma -> H
We use the quaternionic function theory of Pedit and Pinkall in order to describe conformal maps from an elliptic curve to the quaternions. We describe a natural way to cut the corresponding spectral curves into two halves. Both halves are Riemann surfaces with boundary and have finite genus. We expect that this genus can be bounded in terms of the Willmore functional, like the spectral genus of constant mean curvature tori in 3d space forms. For each elliptic curve we construct 3 conformal maps with Willmore energy not larger than 8pi. These conformal maps are constructed in terms of the solutions of the sinh Gordon equation of spectral genus 2. We conjecture that one of these conformal maps represents the minimum of the Willmore functional on the space of all conformal maps from the given elliptic curve to the quaternions.