Stability of Kähler-Ricci flow on a Fano manifold

Abstract

Let $$(M,J)$$ be a Fano manifold which admits a Kahler-Einstein metric $$g_{KE}$$ (or a Kahler-Ricci soliton $$g_{KS}$$ ). Then we prove that Kahler-Ricci flow on $$(M,J)$$ converges to $$g_{KE}$$ (or $$g_{KS}$$ ) in $$C^\infty $$ in the sense of Kahler potentials modulo holomorphisms transformation as long as an initial Kahler metric of flow is very closed to $$g_{KE}$$ (or $$g_{KS}$$ ). The result improves Main Theorem in [14] in the sense of stability of Kahler-Ricci flow.