On compact kähler manifolds of nonnegative bisectional curvature, I

Uniqueness theorems of K�hler metrics of semipositive bisectional curvature on compact Hermitian symmetric spaces

“…Our principal theorem, together with the splitting theorem of Howard et al [5] on compact K/ihler manifolds of semipositive bisectional curvature, yields a structure theorem of K~ihler metrics of semipositive bisectional curvature on possibly reducible compact Hermitian symmetric manifolds. Combined with the works of Howard and Smyth [4], Mori [11], Siu and Yau [15], Sin [14], and Bando [1], we obtain a satisfactory classification of compact K/ihler manifolds of semipositive bisectional curvature and positive Ricci curvature in dimensions =< 3.…”

“…Theorem 1, coupled with the splitting theorem of Howard et al [5] on compact K~ihler manifolds of semipositive bisectional curvature, yields immediately the following metric rigidity theorem on possibly reducible compact Hermitian symmetric manifold. Remark.…”

A uniqueness theorem in Kähler geometry

“…We start with the following proposition (a complex version of Berger's theorem, e.g., Theorem in [2]), which was implied in Lemma 1 in [10]. Since the proof is simple, we provide it here for completeness.…”

“…Using the notation in [10,12], we say a multi-linear form on a manifold is quasi-positive if it is nonnegative everywhere and strictly positive at least at one point. If n = 2, we only need ellipticity condition (1.2) on F. More specifically, the result can be strengthened as follow.…”

“…Our results can be interpreted as a nonlinear version of Wu's result for scalar curvature in [10]. The method we adopt to establish the main theorem is the constant theorem for fully nonlinear elliptic partial differential equations.…”