Compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature

Abstract

Motivated by a recent work of Chen–Zheng (J Geom Anal 32:141, 2022) on Strominger space forms, we prove that a compact Hermitian surface with pointwise constant holomorphic sectional curvature with respect to a Gauduchon connection \(\nabla ^t \) is either Kähler, or an isosceles Hopf surface with an admissible metric and \(t=-1\) or \(t=3\). In particular, a compact Hermitian surface with pointwise constant Lichnerowicz holomorphic sectional curvature is Kähler. We further generalize the result to the case for the two-parameter canonical connections introduced by Zhao–Zheng (On Gauduchon Kähler-like manifolds. ArXiv: 2108.08181), which extends a previous result by Apostolov–Davidov–Muškarov (Trans Am Math Soc 348:3051–3063, 1996).