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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 it 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 wished someone would also give 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.

3 added 55 characters in body

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 it requires some clarification: I am aware that there is whole field od of studying degeneration of metrics, convergence etc. I was mostly curious to see examples of degeneration of metric 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. I wished someone would also give an example of the case when metric in a K\"ahler class degenerates as one varies the potential, but the one curvature functional, say $\Vert Ric \Vert_p$ remains bounded.

2 added 531 characters in body

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 it requires some clarification: I am aware that there is whole field od studying degeneration of metrics, convergence etc. I was mostly curious to see examples of degeneration of metric while curvature is bounded. Examples that could, for example, illustrate when you can keep the scalar curvature bounded but Ricci might blow up. I wished someone would give an example of the case when metric in a K\"ahler class degenerates as one varies the potential, but the one curvature functional remains bounded.

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