Compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature

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 del(t) is either Kahler, 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 Kahler. We further generalize the result to the case for the two-parameter canonical connections introduced by Zhao-Zheng (On Gauduchon Kahler-like manifolds. ArXiv: 2108.08181), which extends a previous result by Apostolov-Davidov-Mugkarov (Trans Am Math Soc 348:3051-3063, 1996).