Mathematical Research Letters

Volume 23 (2016)

Number 1

On convexity of the regular set of conical Kähler–Einstein metrics

Pages: 105 – 126



Ved V. Datar (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.)


In this note we prove convexity, in the sense of Colding–Naber, of the regular set of solutions to some complex Monge–Ampère equations with conical singularities along simple normal crossing divisors. In particular, any two points in the regular set can be joined by a smooth minimal geodesic lying entirely in the regular set. As a consequence, the classical theorems of Myers and Bishop–Gromov extend almost verbatim to this singular setting.

Accepted 4 January 2015

Published 25 May 2016