Ricci curvature, the convexity of volume and minimal Lagrangian submanifolds

We show that, in toric Kähler geometry, the sign of the Ricci curvature corresponds exactly to convexity properties of the volume functional.We also discuss analogous relationships in the more general context of quasi-homogeneous manifolds, and existence results for minimal Lagrangian submanifolds.

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Received 6 November 2021

Received revised 28 February 2023

Accepted 13 May 2023

Published 6 June 2024