Regularity of fully non-linear elliptic equations on Kähler cones

We derive quantitative boundary estimates, and then solve the Dirichlet problem for a general class of fully non-linear elliptic equations on annuli of Kähler cones over closed Sasakian manifolds. This extends extensively a result concerning the geodesic equations in the space of Sasakian metrics due to Guan–Zhang. Our results show that the solvability is deeply affected by the transverse Kähler structures of Sasakian manifolds. We also discuss possible extensions of the results to equations with right-hand side depending on unknown solutions.

Keywords

Dirichlet problem, degenerate fully non-linear elliptic equations, quantitative boundary estimate, gradient estimate, cone condition, Sasakian manifolds

2010 Mathematics Subject Classification

Primary 35J15. Secondary 35B45, 53C25, 58J05.

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The author is partially supported by the National Natural Science Foundation of China (Grant No. 11801587).

Received 31 July 2019

Accepted 23 January 2020

Published 17 February 2021