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On a complete |$p$| -nonparabolic |$3$| -dimensional manifold with non-negative scalar curvature and vanishing second homology, we establish a sharp monotonicity formula for the proper |$p$| -Green function along its level sets for |$1

It is conjectured by Song–Tian that the normalized Kähler–Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle is uniformly smooth away from the singular fibers. In this work, we adapt the method in the work of Hein–Tosatti on collapsing Calabi–Yau metrics and obtain a uniform spatial Hölder estimate along the normalized Kähler–Ricci flow for all time away from the singular fiber. [ABSTRACT FROM AUTHOR]

We show the existence of complete negative Kähler–Einstein metric on Stein manifolds with holomorphic sectional curvature bounded from above by a negative constant. We prove that any Kähler metrics on such manifolds can be deformed to the complete negative Kähler–Einstein metric using the normalized Kähler–Ricci flow. [ABSTRACT FROM AUTHOR]

We prove that compact complex manifolds admitting metrics with negative Chern curvature operator either admit a |$d d^c$| -exact positive |$(1,1)$| current or are Kähler with ample canonical bundle. In the case of complex surfaces we obtain a complete classification. The proofs rely on a global existence and convergence result for the pluriclosed flow. [ABSTRACT FROM AUTHOR]