I have what I believe is a general question relating compact and non-compact Riemannian Manifolds that comes up in a specific context in a paper by Alexei Kovalev titled Twisted Connected Sums and Special Riemannian Holonomy. I have seen similar reasoning used on a number of cases and meerly seek to use this paper to see if the reasoning used in a number of instances is more general.
The General Theory I have been using is as follows: 1) Hopf-Rinow Theorem: Closed and Bounded implies compact 2) Compact subspaces of Hausdorff Spaces are closed 3) Closed subsets of compact spaces are compact
The point of Kovalev's paper is to build a compact manifold out of non-compact building blocks. I have questions regarding two steps of the approach, the reasoning from which the building blocks are non-compact and the reasoning justifiying how to make a compact space out of the non-compact building blocks.
1) Concerning the building blocks being non-compact. The building blocks W are of the form W'\D where W' is compact Kahler and D is a K3 Surface. Does the fact that W is non-compact follow simply from the fact that W' and D are compact, or is it a more subtle consequence of the specific geometry?
2) Building the Compact Space out of building blocks. Kovalev defines a map i: R(>0) x S1 x D -> W and uses it to define a new manifold basically of the following form
M=W\Image(i) x S1
The claim is that M is compact (and then two such M will be sewn together). Does this follow from W being non-compact and the compactness of the image of i?