Pith. sign in

REVIEW 6 cited by

Sobolev inequalities on K\"ahler spaces

Not yet reviewed by Pith; the record is open.

Re-run · record.json Download PDF Read on arXiv ↗

This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.

SPECIMEN: schema-true, not a live event

T0 review · schema-true

One-sentence machine reading of the paper's core claim.

pith:XXXXXXXX · record.json · timestamp

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

Sobolev inequalities on K\"ahler spaces

Bin Guo , Duong H. Phong , Jian Song , Jacob Sturm This is my paper
classification math.DG math.APmath.CV
keywords ahlerestimatessobolevinequalitymetricsspacesbounddiameter
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
0 comments
read the original abstract

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.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Gromov-Hausdorff limits of immortal K\"ahler-Ricci flows

    math.DG 2026-02 unverdicted novelty 8.0

    Normalized Kähler-Ricci flow converges in Gromov-Hausdorff sense to the metric completion of the twisted Kähler-Einstein metric on the canonical model when the canonical bundle is semiample.

  2. Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type

    math.DG 2026-04 unverdicted novelty 7.0

    Chern-Ricci flow on Hermitian minimal models of general type admits uniform estimates yielding subsequential Gromov-Hausdorff convergence under a local Kähler assumption.

  3. Convergence of the Chern-Ricci flow on complex minimal surfaces of general type

    math.DG 2026-05 unverdicted novelty 6.0

    Proves diameter estimates, volume non-collapsing, and Gromov-Hausdorff convergence for normalized Chern-Ricci flow on complex minimal surfaces of general type from arbitrary Hermitian metrics.

  4. Asymptotics of small eigenvalues on degenerations of K\"ahler manifolds

    math.DG 2026-05 unverdicted novelty 6.0

    Exact asymptotic rates for small Laplacian eigenvalues on degenerations of compact Kähler manifolds are derived, generalizing Dai-Yoshikawa to higher dimensions via Skoda inequality and auxiliary Monge-Ampère equations.

  5. Complex Monge-Amp\`ere equation in Orlicz space and Diameter Bound

    math.DG 2026-01 conditional novelty 6.0

    Under general Orlicz-space integrability of the Monge-Ampère measure, the paper proves L∞ and stability estimates for the potential and derives uniform diameter and volume lower bounds for the associated Kähler metrics.

  6. Optimal geometric estimates for compact K\"ahler manifolds of a Nash entropy bound

    math.DG 2026-06 unverdicted novelty 5.0

    Proves optimal Sobolev inequalities and local volume noncollapsing for compact Kähler manifolds with bounded q-Nash entropy.