Abstract
We prove that any two Kähler potentials on a compact Kähler manifold can be connected by a geodesic segment of \(C^{1,1}\) regularity. This follows from an a priori interior real Hessian bound for solutions of the nondegenerate complex Monge-Ampère equation, which is independent of a positive lower bound for the right hand side.
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Acknowledgements
The authors thank J. Song for pointing out a simplification of the proof given in an earlier version of this paper, and M. Păun for interesting discussions. We also thank the referee for some helpful comments. The first-named author would like to thank his advisor G. Tian for encouragement and support.The second-named author was partially supported by NSF Grant DMS-1610278, and the third-named author by NSF Grant DMS-1406164. This work was completed while the second-named author was visiting the Yau Mathematical Sciences Center at Tsinghua University in Beijing, which he would like to thank for the hospitality.
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Chu, J., Tosatti, V. & Weinkove, B. On the 
Regularity of Geodesics in the Space of Kähler Metrics.
Ann. PDE 3, 15 (2017). https://doi.org/10.1007/s40818-017-0034-8
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DOI: https://doi.org/10.1007/s40818-017-0034-8