On gradient shrinking and expanding Kähler-Ricci solitons (Q2063716)

In this paper, the author proves three theorems about gradient shrinking and expanding Kähler-Ricci solitons. The first theorem says that a compact gradient shrinking Kähler-Ricci soliton with subharmonic scalar curvature is Kähler-Einstein. The proof goes as follows: since it is known that the scalar curvature \(R\) of a gradient shrinking soliton is always non-negative, together with the subharmonic property, one can show that the integral of \(|\nabla R|^2\) should be zero and hence \(R\) is a constant. Then the result follows from basic relations between various curvatures on a gradient Kähler-Ricci soliton. The remaining two theorems are about complete non-compact gradient Kähler-Ricci solitons. The author proves that a complete noncompact gradient shrinking Kähler-Ricci soliton with constant scalar curvature and nonnegative fourth-order divergence curvature (or Bochner) tensor is rigid. Also, a classification theorem for complete noncompact gradient expanding Kähler-Ricci solitons is derived. The proofs of these two theorems are quite similar, besides integration by parts, the main ingredient of the proof is an equation on fourth-order covariant derivatives of the curvature tensor on a gradient Kähler-Ricci soliton proved by the author.