Partial C ⁰-Estimate for Kähler–Einstein Metrics

Abstract

In this short note, we give a proof of our partial C ⁰-estimate for Kähler–Einstein metrics. Our proof uses a compactness theorem of Cheeger–Colding–Tian and L ²-estimate for $\bar{\partial}$-operator.

Keywords

Kähler–Einstein metrics / Gromov–Hausdorff limit / Partial C ⁰-estimate

References

[1] Cheeger J., Colding T., Tian G. Constraints on singularities under Ricci curvature bounds. C. R. Acad. Sci. Paris Sér. I, Math.. 1997, 324 645-649

[2] Colding T., Naber A. Sharp Hoölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications. Ann. Math.. 2012, 176 2 1173-1229

[3] Croke C.B. Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. Ec. Norm. Super., 4 Ser.. 1980, 13 419-435

[4] Donaldson, S., Sun, S.: Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry. arXiv:1206.2609

[5] Li P. On the Sobolev constant and the p-spectrum of a compact Riemannian manifold. Ann. Sci. Ec. Norm. Super., 4 Ser.. 1980, 13 451-469

[6] Tian G. On Kähler–Einstein metrics on certain Kähler manifolds with C ₁(M)>0. Invent. Math.. 1987, 89 225-246

[7] Tian G. On Calabi’s conjecture for complex surfaces with positive first Chern class. Invent. Math.. 1990, 101 101-172

[8] Tian G. Kähler–Einstein on algebraic manifolds. Proceedings of the International Congress of Mathematicians, Vols. I, II. 1991 Tokyo: Math. Soc. Japan. 587-598

[9] Tian G. . Dai . Einstein metrics on Fano manifolds. Metric and Differential Geometry. Proceeding of the 2008 Conference Celebrating J. Cheeger’s 65th Birthday. 2012 Basel: Birkhäuser

[10] Tian, G.: Extremal Kähler metrics and K-stability. Preprint, June 2012

[11] Tian, G.: K-stability and Kähler–Einstein metrics. arXiv:1211.4669