Schedule of the Workshop: Navigating the Space of Surfaces

Christoph Bohle: Conformal deformations and elliptic boundary value problems

Francis Everett Burstall: Variational problems in parabolic geometries

Analogues of the Willmore functional in projective and Lie sphere geometry have been known since the work of Blaschke and Thomsen in the 1920's and share many features (integrability, harmonic Gauss map) with the Willmore functional. More recently, Sung Ho Wang introduced a functional on Legendre surfaces in the 5-sphere which turns out to be closely related both to the Blaschke-Thomsen functionals and the Willmore functionals of Montiel-Urbano. I shall describe a uniform approach to these matters.

Keenan Crane: Robust Fairing using Conformal Surface Flows

Smoothing or "fairing" discrete surfaces is often achieved by applying steepest descent methods to an energy functional like the Willmore energy. However, existing discretizations often have prohibitive time step restrictions and do not respect important features of the discrete surface such as element quality. Based on recent developments in conformal geometry processing, we present a robust, explicit procedure for reducing Willmore energy that exhibits excellent stability and closely preserves the quality of the original mesh.

Laura Desideri: Describing minimal surfaces using isomonodromic deformations

We explain the correspondence due to R. Garnier between minimal disks with a polygonal boundary curve and a certain class of ordinary differential equations on the Riemann sphere, and how this correspondence can be used to describe the minimal disks, and to solve the Plateau problem. We will then discuss the possibility to extend this point of view to minimal annuli, whose associated equations are defined on an elliptic curve.

Laurent Hauswirth: Geometry of embedded minimal annuli in S² x R via integrable systems

Sebastian Heller: Integrable system methods for higher genus minimal surfaces

In this talk I will discuss the notion of a spectral curve for higher genus minimal and CMC surfaces. In the case of tori the spectral curve parametrizes the eigenlines of the associated family of flat connections. This does not work for higher genus surfaces anymore because of their non-abelian first fundamental groups. Nevertheless, using the eigenlines of a Higgsfield corresponding to the Hopf differential, one can abelinize the theory which leads naturally to the notion of a spectral curve.

Martin Kilian: Constrained Willmore tori of spectral genus one

A constrained Willmore torus of spectral genus one is a constant mean curvature torus in the 3-sphere. I will discuss their moduli space and show that the minimal tori in this moduli space are unstable critical points for the Willmore energy.

Felix Knöppel: Globally Optimal Smooth k-Direction Fields and Curvature Lines

We introduce the notion of a discrete hermitean line bundle on a triangulated surface. Under a suitable smoothness condition on the triangulation the Poincare-Hopf theorem holds.
We construct quadratic energies on these discrete bundles with help of which we find globally optimal smooth sections. Including an additional term measuring alignment with a given section leads to optimal smooth curvature direction aligned line or cross elds.
This is joint work with K. Crane, U. Pinkall, and P. Schröder.

Robert Kusner: The space of soap bubbles

We'll discuss what's known about the topology and smoothness for the moduli spaces of complete embedded surface of constant mean curvature.

Katrin Leschke: Simple factor dressing and the associated family of minimal surfaces

There are various harmonic maps which are canonically associated to a minimal surface in Euclidean 4-space, e.g., the left and right normal of the immersion and the conformal Gauss map. We will discuss how the well-known dressing operation on harmonic maps applied to these harmonic maps is related to transformations of the minimal surface. In particular, the simple factor dressing is connected to both the associated family of minimal surfaces and a family of Willmore surfaces associated to the minimal surface.

Helmut Pottmann: Shape Space Exploration of Constrained Meshes

We present a general computational framework to locally characterize any shape space of meshes implicitly prescribed by a collection of non-linear constraints. 

We computationally access such manifolds, typically of high dimension and co-dimension, through first and second order approximants, namely tangent spaces and quadratically parameterized osculant surfaces. Exploration and navigation of desirable subspaces of the shape space with regard to application specic quality measures are enabled using approximants that are intrinsic to the underlying manifold and directly computable in the parameter space of the osculant surface. 

We demonstrate our framework on shape spaces of various types of meshes which are important in freeform architecture and indicate directions for future research beyond this specic application. 

This is joint research with Yongliang Yang, Yijun Yang and Niloy Mit.

Martin Schmidt: The closure of spectral data of CMC tori in S^{^3}

The spectral curve correspondence for finite-type solutions of the sinh-Gordon equation describes how they arise from and give rise to hyperelliptic curves with a real structure. Constant mean curvature (CMC) 2-tori in S^{^3} result when these spectral curves satisfy periodicity conditions. We prove that the spectral curves of CMC tori are dense in the space of smooth spectral curves of finite-type solutions of the sinh-Gordon equation.

Nicholas Schmitt: Finite type spacecurves

Finite type spacecurves are stationary with respect to all but finitely many flows of the nonlinear Schroedinger hierarchy. These spacecurves are closely related to constant mean curvature and (constrained) Willmore tori; for example elastica generate such equivariant tori as profile curves. Flows through finite type spacecurves and their generated tori will be described, computed and visualized using Whitham deformations of the spectral curve.

Peter Schröder: Conformal Deformation of Surfaces

Limiting deformations of surfaces in the context of geometric modeling to conformal deformations has many advantages. Among them maintenance of mesh quality and no unsightly shearing artifacts in attached texture maps. As it turns out such deformations can be characterized completely by being in the kernel of a computationally very attractive first order differential operator, the Dirac operator for embedded surfaces. In this talk I will go through the underlying theory and demonstrate computational results from geometric modeling and geometric flow and will discuss how the underlying representation provides a parameterization of the shape space of surfaces of a given conformal class.

Boris Springborn: Combinatorial Ricci flow on triangulated surfaces

I will review Feng Luo's "Combinatorial Yamabe flow on surfaces" (in two dimensions, Ricci flow is the same as Yamabe flow) and show how one can avoid the problem of degenerating triangles to obtain a flow that is defined for all positive times and converges. This is based on the connection with three-dimensional hyperbolic polyhedra that was developed in a paper with A. Bobenko and U. Pinkall.

Chuu-Lian Terng: Integrable curve flows

I will explain a systematic method for constructing integrable curve flows from soliton theory. Examples include Schrödinger flow on S^{^2}, smoke ring equation, central affine curve flow in the plane, and the Hodge star mean curvature flow in flat 3-space.