The paper classifies all 4D warped-product shrinking solitons on a surface times a sphere as the cylinder R^2 x S^2, and constructs non-product multiply warped Ricci flows that develop Type-I singularities modeled on generalized cylinders R^k x S^ell.
Singularity formation of complete Ricci flow solutions
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Local singularities of compact multiply warped Ricci flow solutions
The paper classifies all 4D warped-product shrinking solitons on a surface times a sphere as the cylinder R^2 x S^2, and constructs non-product multiply warped Ricci flows that develop Type-I singularities modeled on generalized cylinders R^k x S^ell.