Manifolds with finitely many ends

Question

In the article ' The structure of stable minimal hypersurfaces in $ R^{n+1} $ ( http://arxiv.org/pdf/dg-ga/9709001.pdf) of Cao-Shen-Zhu the remark 2 at page 3 contains a statement that i don't understand (actully it seems me false).

Let $ M $ be a manifold and let $ \{K_n\} $ be an exhaustation by compact sets:

$$ \cup_n K_n = M $$ and

$$ K_n \subset K_{n+1} $$

An end of $ M $ is a collection of subsets (actually open subsets) $ \{E_n\} $ such that $ E_n $ is a connected component of $ M-K_n $ and

$$ E_{n+1} \subset E_n $$

It can be proved that the number of ends is independent from the choice of the exhaustation $\{K_n\} $

Now in the article above is stated that if a manifold has only finitely many ends $ \{ E_{n}^{1} \}, \ldots \{ E_{n}^{k} \} $ there exists $ n_0 $ such that

$$ E_{n}^{j} = E_{n_0}^{j} $$ for every $ j = 1 \ldots k $ and $ n \geq n_0 $

This statement seems me false.

Thanks

Answer 1

You're right, that statement is false. They probably mean that the number of ends stabilizes at some point (as there might be just one component at first, then more, then more, etc.).

Answer 2

I suggest you look up the complete minimal surfaces in $\mathbb R^3$ that illustrate these concepts, in each case count the ends. One collection of pictures HERE. Before Celso Costa, we had the plane, the catenoid, the helicoid, and an example of Riemann that I always referred to as the Riemann Staircase. The Staircase is interesting in that you get a countable collection of ends, one for each evident horizontal near-plane. But there is one more end, corresponding to the slanted core of the thing.

Costa found a three-ended example, immersed by construction, and imbedded at infinity. It took quite a bit of work to make pictures of the intricate central portion of the thing, which gave enough information to prove imbeddedness. Later, Meeks and Hoffman and James Hoffman made an industry of this, similar surfaces with up to about 11 ends. Then they decided to try for infinite number of ends, more symmetric (but also more intricate) versions of the Staircase. And those worked as well.

Anyway, in the minimal case, especially with bounded total curvature, ends happen only in fairly pleasant ways, and it is worth the exercise to examine the case of surfaces, where there are nice pictures.