Mathematical Research Letters

Volume 19 (2012)

Number 5

Weak geodesics in the space of Kähler metrics

Pages: 1127 – 1135

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n5.a13

Authors

Tamás Darvas (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

László Lempert (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

Abstract

Given a compact Kähler manifold $(X,\omega_0)$, according to Mabuchi, the set $\mathcal{H}_0$ of Kähler forms cohomologous to $\omega_0$ has the natural structure of an infinite-dimensional Riemannian manifold. We address the question whether points in $\mathcal{H}_0$ can be joined by a geodesic, and strengthening the finding of Lempert and Vivas, we show that this cannot always be done even with a certain type of generalized geodesics. As in Lempert and Vivas, the result is obtained through the analysis of a Monge–Ampère equation.

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