Communications in Analysis and Geometry

Volume 28 (2020)

Number 7

Integrability theorems and conformally constant Chern scalar curvature metrics in almost Hermitian geometry

Pages: 1603 – 1645



Mehdi Lejmi (Department of Mathematics, Bronx Community College of the City University of New York, Bronx, N.Y., U.S.A.)

Markus Upmeier (Institut für Mathematik, Universität Augsburg, Germany)


The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the Kähler case. Our main question is the existence of almost Kähler metrics with conformally constant Chern scalar curvature. This problem is completely solved for ruled manifolds and in a complementary case where methods from the Chern–Yamabe problem are adapted to the non-integrable case. Also a moment map interpretation of the problem is given, leading to a Futaki invariant and the usual picture from geometric invariant theory.

The second-named author was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – UP 85/2-1, UP 85/3-1.

Received 29 June 2017

Accepted 11 March 2018

Published 7 December 2020