Degeneration of riemannian metrics with curvature bounds

Question

In short, I'm curious to know what modes of degeneration of metric might still keep the curvature bounded. More precisely, assume we are keeping the total volume of the manifold fixed and deform the metric, e.g by a conformal factor that preserves the volume. If we allow the metric to degenerate, namely allow it to become semi-definite, $g_{ij} \geq 0$ on some set, is it possible to keep the Ricci curvature bounded below and yet let the metric degenerate? If not, is it possible to still keep some $L^p$ bound on the sectional, Ricci, or scalar curvature?

Edit: It seems that the post requires some clarification: I am aware that there is whole field of studying degeneration of metrics, convergence etc. I was mostly curious to see examples of degeneration of metrics while curvature is bounded. Examples that could, for example, illustrate when you can keep the scalar curvature bounded but Ricci, or any $L^p$ norm of Ricci, might blow up. More than the conformal deformation I was curious to see an example of the case when metric in a K\"ahler class degenerates as one varies the potential, but one curvature functional, say $\Vert Ric \Vert_p$ remains bounded.

Answer 1

The seminal work on how a Riemannian manifold can degenerate with pointwise bounds on curvature are Cheeger and Gromov's JDG papers, "Collapsing Riemannian manifolds while keeping their curvature bounded" as well as papers by Fukaya, which can be found in the references of the Cheeger-Fukaya-Gromov JAMS paper, "Nilpotent structures and invariant metrics on collapsed manifolds".

You can then find other papers on the subject by looking up on Mathscinet which papers cite these papers. You've mentioned degenerations under conformal deformations and degeneration of Kahler metrics. You should be able to find any existing work on this by doing the above. I haven't looked at this for a long time, so unfortunately I can't help more.

As for degenerations with only an integral bound on curvature, searching on "integral bound and curvature" in the Anywhere box on Mathscinet seems to produce lots of hits but I suspect the vast majority of papers study situations where degeneration does not occur. I have a Duke paper, "Riemannian manifolds with small integral norm of curvature" that proves an $L_p$ analogue of the Cheeger-Gromov collapse theorem and also describes an example of what can happen with only an integral bound but doesn't happen with pointwise bounded curvature. I don't know of any other work in this direction, but I'm hoping that someone else does.

Answer 2

Consider for example the torus $S^1\times S^1$ with the flat product metric $g_\varepsilon$ which gives the first factor length $\varepsilon$ and the second factor length $1/\varepsilon$. Then for any $\varepsilon>0$ the metrics $g_\varepsilon$ have unit volume and zero curvature.

As $\varepsilon\to 0$ there is no smooth limit of $g_\varepsilon$ as tensors, but the manifold converges in the pointed Gromov-Hausdorff limit to $\mathbb{R}$.

Answer 3

John Lott has a number of papers on this sort of thing, many to do with the spectrum of the Laplacian on collapsing manifolds. Nice thing is they are all available from his home page at Berkeley, http://math.berkeley.edu/~lott/papers.html. If you look at the most recent preprint, it should point you in all sorts of interesting directions.