Minimal 2-spheres in 3-spheres

Some Recent Developments in the study of minimal 2-spheres in spheres

We discuss recent progress in the study of the space of harmonic maps from the 2sphere to the unit n-sphere in Euclidean (n+1)-space. We consider the structure of this space as an algebraic variety, the existence of non-manifold points in this space, and the relation between this question and the integrability of Jacobi fields along harmonic maps. One of the main tools used is that of the twist...

EMBEDDED MINIMAL SPHERES IN 3 - MANIFOLDS by Joel Israel Kramer

Consider the following question: Does there exist a three-manifold M for which, given any riemannian metric on M there is an area bound for embedded minimal spheres? We answer this question negatively, and, in fact, find an open set of metrics for which this question is false. This generalizes similar results for surfaces of positive genus shown by W. Minicozzi, T. Colding and B. Dean. In addit...

Many Triangulated 3-Spheres

We construct 2Ω(n 5/4) combinatorial types of triangulated 3-spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2O(n log n) combinatorial types of simplicial 4-polytopes, this proves that asymptotically, there are far more combinatorial types of triangulated 3-spheres than of simplicial 4-polytopes on n vertices. This complements results of Kalai (1988...

Existence of Generalized Totally Umbilic 2-spheres in Perturbed 3-spheres

It was recently shown by R. Souam and E. Toubiana [33] that the (non constantly curved) Berger spheres do not contain totally umbilic surfaces. Nevertheless in this article we show, by perturbative arguments, that all analytic metrics su ciently close to the round metric g0 on S possess generalized totally umbilic 2-spheres, namely critical points of the conformal Willmore functional ∫ Σ |A◦|2 ...

The Existence of Minimal Immersions of 2-Spheres