Month: November 2021

November 19, 2021

Minimal 2-spheres in ellipsoids of revolution

Motivated by Morse-theoretic considerations, Yau asked in 1987 whether all minimal 2-spheres in a 3-dimensional ellipsoid inside R^4 are planar, i.e., determined by the intersection with a hyperplane. Recently, this was shown not to be the case by Haslhofer and Ketover, who produced an embedded non-planar minimal 2-sphere in sufficiently elongated ellipsoids, combining Mean Curvature Flow and Min-Max methods. Using Bifurcation Theory and the symmetries that arise if at least two semiaxes coincide, we show the existence of arbitrarily many distinct embedded non-planar minimal 2-spheres in sufficiently elongated ellipsoids of revolution. This is based on joint work with P. Piccione.

November 12, 2021

Geodesics on SL(n) with the Hilbert-Schmidt metric, and one application to fluids

One way to analyze the motion of rigid bodies is to study the corresponding trajectory in SO(n); the physical features of the body give rise to a Riemannian metric. If we were to replace rigid bodies with incompressible fluids, the same procedure leads to Arnol’d’s equations on the volume-preserving diffeomorphism group, into which SL(n) sits as a subgroup. For some fluid bodies, this subgroup is totally geodesic; in this case the induced Riemannian metric on SL(n) is the Hilbert-Schmidt metric. Extending previous works by Sideris and Roberts-Shkoller-Sideris, we analyze the geodesic motion in this setting. Results are obtained in collaboration with my REU students Audrey Rosevear (Amherst) and Samuel Sottile (MSU).

November 5, 2021

Koebe circle domain conjecture and the Weyl problem in hyperbolic 3-space

In 1908, Paul Koebe conjectured that every open connected set in the plane is conformally diffeomorphic to an open connected set whose boundary components are either round circles or points. The Weyl type problem, in the hyperbolic setting, asks for isometric embedding of surfaces of curvature at least -1 into the hyperbolic 3-space. We show that there are close relationships among the Koebe conjecture, the Weyl problem and the work of Alexandrov and Thurston on convex surfaces. This is a joint work with Feng Luo.