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Sobolev inequalities on K\"ahler spaces

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arxiv 2311.00221 v1 pith:6K6N5IEO submitted 2023-11-01 math.DG math.APmath.CV

Bin Guo , Duong H. Phong , Jian Song , Jacob Sturm This is my paper · ORCID
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keywords ahlerestimatessobolevinequalitymetricsspacesbounddiameter
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We establish a uniform Sobolev inequality for K\"ahler metrics, which only require an entropy bound and no lower bound on the Ricci curvature. We further extend our Sobolev inequality to singular K\"ahler metrics on K\"ahler spaces with normal singularities. This allows us to build a general theory of global geometric analysis on singular K\"ahler spaces including the spectral theorem, heat kernel estimates, eigenvalue estimates and diameter estimates. Such estimates were only known previously in very special cases such as Bergman metrics. As a consequence, we derive various geometric estimates, such as the diameter estimate and the Sobolev inequality, for K\"ahler-Einstein currents on projective varieties with definite or vanishing first Chern class.

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Cited by 8 Pith papers

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