Speaker
Description
An important class of problems in Riemannian geometry can be stated as follows: given a smooth and orientable Riemanninan manifold M, find an hypersphere U in M having prescribed mean curvature K at each point. We will be mainly focused on the case when the target M is the Euclidean plane and the unknown U is a planar loop. Besides its geometrical interpretation, this (apparently) simple problem naturally arises in the study of the planar motion of an electrified particle that experiences a Lorentz force produced by a magnetostatic field. It can be regarded as a model for a more general question raised by V.I. Arnold in [Uspekhi Mat. Nauk 1986]. We will first discuss some Alexandrov-type uniqueness results in case the prescribed curvature is a positive constant or, more generally, a positive and radially non increasing function. To obtain existence results we will choose a parametric point of view, which will lead us to study certain variational, noncompact systems of second order ODE's for functions on the circle. This will give us the opportunity to briefly introduce some variational (mountain pass lemma) and nonvariational (Lyapunov-Schmidt dimension reduction)basic techniques. In the last part of the course will overview some recent results and open problems in case the target space M is the hyperbolic plane, or the Euclidean/hyperbolic 3-dimensional space.
Program:
The curvature of planar curves. Planar loops and physical interpretation: a related ODE system and Arnol'd problem. Planar loops of positive curvature. Homework: the curvature of circles and ellipses, radially symmetric prescribed curvatures.
The four vertex theorem (Osserman's proof).
Uniqueness results: Alexandrov (1956), Aeppli (1960) and more (2011).
The (pseudo)-length functional and the weighted, signed area functional.
The variational approach. A quick introduction to variational methods.
Palais-Smale condition, the Mountain Pass Lemma, saddle points.
A non-variational approach: the Lyapunov-Schmidt dimension reduction.
Loops of prescribed curvature in the hyperbolic plane. Bubbles in the Euclidean and in hyperbolic spaces: recent results and open problems.
