Constant-Mean-Curvature Surfaces: Computation and Visualization
College: Liberal Arts and Sciences
Award year: 2000-2001
Many real-world problems can be cast as optimizing a shape; mathematically, these become minimizations of geometric energies. For instance, soap bubbles minimize area while enclosing fixed volumes; this leads to the study of constant-mean-curvature (CMC) surfaces. All interfaces in a froth are CMC surfaces.
Most interesting mathematically are complete surfaces, with no boundary. Complete embedded surfaces are classified by topology: number of handles and ends (at infinity). Two-ended CMC surfaces form a one-parameter family (including the cylinder) of surfaces of revolution called unduloids.
Sullivan has, with Kusner and Grobe-Brauckmann, determined the moduli space of all three-ended genus-zero CMC surfaces (triunduloids). Each end is asymptotic to an unduloid, and the surface has mirror symmetry. Triunduloids are classified by spherical triangles, whose edge lengths are the ends' necksizes. (Symmetry and force balancing are crucial for this classification.) It is rare for the space of solutions to a partial differential equation to be known so explicitly.
As a Faculty Fellow, Sullivan will make use of the NCSA's state-of-the-art computation and visualization facilities to investigate CMC surfaces. Triunduloids, or rather their truncations, form basic building blocks for gluing together all other embedded CMC surfaces. The parameters for truncated triunduloids are known only approximately; thus numerical optimization is needed. This large computation will require the parallelized version of Brakke's Evolver, developed to run on the Origin 2000.
The numerical algorithm computes a minimal surface in spherical space, and conjugates it to give a triunduloid. Working with an RA, Sullivan is now implementing this algorithm in the Evolver. The unstable minimal surfaces which are needed can be best computed by minimizing elastic bending energy for surfaces (as for the sphere eversion in The Optiverse, the video produced at NCSA in 1998).
Traversing a certain loop in the three-dimensional space of triunduloids adds a bubble to one end of the triunduloid. Results like this can be best understood visually, so Sullivan proposes to create a virtual-reality (CAVE) environment. Here a user could navigate through the ordinary geometric space of a single CMC surface, and also navigate through the three-dimensional space of all triunduloids. This will require the computation of thousands of triunduloids in advance on the supercomputer. This visualization environment will also allow the investigation of the more complicated CMC surfaces built from triunduloids. Here, since rigorous understanding is so far lacking, further theoretical progress will require numerical experiments with good visualization.
As an NCSA Fellow, Sullivan will also produce a computer-graphics videotape on triunduloids. His experience producing The Optiverse means that many necessary tools are already at hand; the NCSA post-production facilities and staff will be essential to complete this video. Sullivan will also create an interactive web page, a kind of portal for access to information about CMC surfaces through the Grid.
Sullivan's proposed NCSA fellowship project will explore the frontier of geometry, where problems that may be physically natural are still challenging from theoretical and computational standpoints. The interplay between rigorous proofs and numerical experiments (with interactive visualization) is what allows progress on both fronts.