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arXiv · 1707.03660

$C^{1,1}$ regularity for degenerate complex Monge-Amp\`ere equations and geodesic rays

Jianchun Chu·Valentino Tosatti·Ben Weinkove
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Abstract

We prove a $C^{1,1}$ estimate for solutions of complex Monge-Amp\`ere equations on compact K\"ahler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong-Sturm. As applications we deduce the local $C^{1,1}$ regularity of geodesic rays in the space of K\"ahler metrics associated to a test configuration, as well as the local $C^{1,1}$ regularity of quasi-psh envelopes in nef and big classes away from the non-K\"ahler locus.

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Jianchun Chu, Valentino Tosatti, Ben Weinkove. 2017-07-12. $C^{1,1}$ regularity for degenerate complex Monge-Amp\`ere equations and geodesic rays. https://doi.org/10.1080/03605302.2018.1446167

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