A continuous map $\sigma:S^1 \times [0,1] \to M$ is called a sweepout on $M$, if
- For each $t$ the map $\sigma(\cdot,t)$ is $W^{1,2}$;
- The map $t \to \sigma(\cdot,t)$ is continuous from $[0,1]$ to $W^{1,2}$;
- $\sigma$ maps $S^1 \times \{0\}$ and $S^1 \times \{1\}$ to points.
This is the definition given in the book "A Course in Minimal Surfaces". I'm having trouble understanding this concept geometrically, I couldn't find other references besides articles. I was able to reason when $M$ is the closed disk, but not for other manifolds, I would like more examples to see if I can understand this concept well.
Say $M$ is also oriented, compact, has no boundary so we can assign a degree to any map $u: \mathbb{S}^1 \to M$. The degree is homotopy invariant. A better way to rephrase the second point is that $\sigma(x,0)$ and $\sigma(x,0)$ are constant maps.
Now we turn our eye to the problem of finding closed geodesics on $M$, which is the task to find non-trivial critical values of the energy functional $$ E(\sigma(\cdot,t))=\int_{\mathbb{S}^1}|\partial_x \sigma (x,t)|^2dx $$ This energy is well defined for the Sobolev space you just mentioned. Now I have to skip a lot of details, but lets consider all degree $1$ curves in a a sweepout $\Omega$, which I define as the union of all curves in your sweepout. So we take all curves such that
$$ deg(\sigma(\cdot,t))=1 $$ Let us denote all those degree $1$ curves with $\Omega_1.$ On the endpoints, the degree is $0$, since constant maps have degree $0$, so they aren't part of your sweepout. However, now we can study the quantitiy $$ \inf_{\sigma \in \Omega_1} \sup_{t \in [0,1]}E(\sigma (x,t)) $$ Say we could now do Mountain Pass theory for it (i.e. verifying the Palais-Smale condition) as well as proving some kind of topological lower bound for all degree $1$ curves (here we need homotopy invariance of the degree) $$ \sup_{t \in [0,1]}E(\sigma (\cdot,t)) > \epsilon > 0, $$ then we would already have a closed geodesic as a non-trivial critical curve of the energy functional.
If you have any questions, feel free to ask. I can also revise this answer a bit, since it is already late over here.
As a simple example, think of it as as the $2$ endpoints being north and southpol and the sweepout curves being equatorial lines around the globe.