Global analysis, analysis on manifolds Definition

Birkhoff min-max method

Builds on Riemannian metric Sphere Arc length Closed geodesic Degree of a continuous map

Definition

Let $g$ be a smooth Riemannian metric on the sphere $S^2$ . A sweepout of $S^2$ is a continuous map $c \colon [0, 1] \times S^1 \to S^2$ such that:

each curve $c_t = c(t, \cdot)$ is piecewise smooth, and $t \mapsto c_t$ is continuous for the $C^0$ topology and for the length;
the curves $c_0$ and $c_1$ are constant;
the map $S^2 \to S^2$ induced by $c$ on the quotient of $[0, 1] \times S^1$ that collapses $\set{0} \times S^1$ and $\set{1} \times S^1$ to points has nonzero degree.

The width of $(S^2, g)$ is $W(g) = \inf_{c} \ \max_{t \in [0, 1]} L(c_t),$ the infimum over all sweepouts $c$ of the length of the longest curve. The Birkhoff min-max method shows that $W(g) > 0$ and that $W(g)$ is the length of a closed geodesic. The geodesic is a limit of longest curves of a minimizing sequence of sweepouts. Before taking the limit, a curve-shortening process is applied to all curves of each sweepout at once.

Remarks

The width is positive because curves of small length lie in small discs, so a sweepout by short curves would induce a map of degree $0$ . Birkhoff's curve-shortening process replaces each curve by a broken geodesic through finitely many of its points, and repeats. It acts continuously on all curves of a sweepout and does not increase lengths.

Example: on the round unit sphere, the circles of latitude $c_t(s) = (\sin \pi t \cos s, \sin \pi t \sin s, -\cos \pi t)$ form a sweepout of degree $1$ , whose longest curve is the equator, of length $2\pi$ . Here $W = 2\pi$ , the length of a great circle.

Non-example: a family that starts and ends at constant curves but stays in the southern hemisphere, such as $c_t(s) = (\tfrac12 \sin \pi t \cos s, \tfrac12 \sin \pi t \sin s, -\sqrt{1 - \tfrac14 \sin^2 \pi t})$ , is not a sweepout. The induced map misses the north pole, so its degree is $0$ , and clause 3 fails.

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Tag	`QTHPN6V7XM`
URI	`m2://QTHPN6V7XM`
Kind	definition
Writing	W0 stub
MSC	58E10, 53C22, 49J35
Defines	Birkhoff min-max method
Uses	Riemannian metric, Sphere, Arc length, Closed geodesic, Degree of a continuous map

Birkhoff min-max method

Definition

Remarks

Reading path

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