Minmax for sweepouts by curves
Tobias H. Colding, William P. Minicozzi II,
2007.02.21
http://arxiv.org/abs/math/0702625
Given a Riemannian metric on the 2sphere, sweep the 2sphere out by a
continuous oneparameter family of closed curves starting and ending at point
curves. Pull the sweepout tight by, in a continuous way, pulling each curve as
tight as possible yet preserving the sweepout. We show the following useful
property (see Theorem 1.9 below); cf. [CM1], [CM2], proposition 3.1 of [CD],
proposition 3.1 of [Pi], and 12.5 of [Al]:
Each curve in the tightened sweepout whose length is close to the length of
the longest curve in the sweepout must itself be close to a closed geodesic. In
particular, there are curves in the sweepout that are close to closed
geodesics.
Finding closed geodesics on the 2sphere by using sweepouts goes back to
Birkhoff in the 1920s. The above useful property is virtually always implicit
in any sweepout construction of critical points for variational problems yet it
is not always recorded since most authors are only interested in the existence
of one critical point.
Similar results holds for sweepouts by 2spheres instead of circles; cf.
\cite{CM2}. The ideas are essentially the same in the two cases, though
technically easier for curves where there is no concentration of energy (i.e.,
``bubbling''); cf. \cite{Jo}.
