I am now reading 'A Course in Minimal Surfaces' written by Tobias Colding & William Minicozzi with helping the following lecture. The lecturer explains an essential sphere during the lecture but I do not know what is an essential sphere. Thus I want to delve into compressible surface (or incompressible surface) because google shows the keyword compressible surface instead of essential sphere to me. And I think that an essential sphere/surface is same as a compressible sphere/surface.
Anyway, my problem is here : I cannot imagine 'compressing surface' from the Wikipedia (and I cannot find any reference about the concept of compressing surface) To begin with, the definition of compressing disk $D$ is like this,
Definition. Let $S$ be compact surface properly embedded in a smooth $3$-manifold $M$. Then a compressing disk $D$ is a disk embedded in $M$ such that $D\cap S= \partial D$
I can describe the compressing disk like the below figure. (But because $S$ is $3$-manifold, i.e locally looks like Euclidean 3-dimensional space, the below image is a projection image onto the plane) As following definition, the only overlapping region between compact surface $S$ and disk $D$ is only the boundary of $D$
But, the next definition 'a non-trivial disk inside of $S$' is not easy to understand for me.
Definition. If the curve $\partial D $ does not bound a disk inside of $S$, the $D$ is called a non-trivial compressing disk. If $S$ has a non-trivial compressing disk, then we call $S$ a compressible surface in $M$
Here is my attempting to understand the definition of non-trivial disk :
If the above figure exactly describes the compressing disk , the curve $\partial D $ is clearly the lime circle and I pick a disk inside of $S$ like this. For convenience, I denote new disk inside $S$ by $D'$
And what is that the curve $\partial D $ does not bound a disk inside of $S$ ?To begin with, the disk $D'$ in $S$ would never intersect to the disk $D$ except the region of $\partial D $. Because of this, I cannot understand the verb 'bound'. So, the only visualization that I can imagine is that the center of radius of disk $D'$ moves into center of radius $D$.(making concentric circles)
Then since $D'$, the disk inside of $S$, is not bounded by the curve $\partial D$, $D$ is a non-trivial compressing disk. That means, I can construct a disk inside of $S$ whose radius is bigger than those of disk $D$. (and since $S$ has one compressing disk, $S$ is compressible surface in $M$). //
I suggest you try to understand some examples of compressible surfaces instead of trying to visualize the most general ones. Consider the solid torus $M=S^1\times D^2$; its boundary is the torus $T^2=S^1\times S^1$. This boundary surface is compressible in $M$ because of existence of the disk $\{z\}\times D^2$: Its boundary loop $\{z\}\times S^1$ does not bound a disk in $T^2$ but does bound a disk in $M$. As a next example, consider the same solid torus but embedded in $R^3$. Then its boundary $T^2$ is a compressible surface in $R^3$. As a next task, find a compressible surface of genus $g\ge 2$ in $R^3$.