The classical Hadamard three-circle theorem is generalized to complete Kähler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three-circle theorem. Two sharp monotonicity formulae are derived as corollaries. Among applications, we obtain sharp dimension estimates (with rigidity) of holomorphic functions with polynomial growth when the holomorphic sectional curvature is nonnegative. When the bisectional curvature is nonnegative, the sharp dimension estimate was due to Ni.
Gang Liu. "Three-circle theorem and dimension estimate for holomorphic functions on Kähler manifolds." Duke Math. J. 165 (15) 2899 - 2919, 15 October 2016. https://doi.org/10.1215/00127094-3645009