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Diameter estimates in K\"ahler geometry

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arxiv 2209.09428 v1 pith:RMD2FE3Z submitted 2022-09-20 math.DG math.APmath.MG

Bin Guo , Duong H. Phong , Jian Song , Jacob Sturm This is my paper · ORCID
classification math.DGmath.APmath.MG
keywords diameterestimatesahlerboundsolutionsahler-ricciallowingbounds
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abstract

Diameter estimates for K\"ahler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for $L^\infty$ estimates for the Monge-Amp\`ere equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a consequence, diameter bounds are obtained for long-time solutions of the K\"ahler-Ricci flow and finite-time solutions when the limiting class is big, as well as for special fibrations of Calabi-Yau manifolds.

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  1. Estimates of heat kernels and Sobolev-type inequalities for twisted differential forms on compact K\"ahler manifolds

    math.CV 2025-07 conditional novelty 6.0 of 10

    The paper extends the Sobolev-type inequalities of Guo-Phong-Song-Sturm and Guedj-Tô from functions to twisted differential forms using heat kernel estimates.

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