We introduce a new curvature flow which matches with the Ricci on metrics and preserves almost Hermitian condition. This enables us to use study manifolds.

We analyze the Ricci flow of a noncompact metric that describes a two-dimensional black hole. We consider entanglement entropy of a 2d black hole which is due to the quantum correlations between two subsystems: one is inside and the other is outside the black hole horizon. It is demonstrated that the entanglement entropy is monotonic along the Ricci flow. 1 [email protected]

In this paper, we study the evolution of L p-forms under Ricci flow with bounded curvature on a complete non-compact or a compact Riemannian manifold. We show that under curvature pinching conditions on such a manifold, the L norm of a smooth p-form is non-increasing along the Ricci flow. The L∞ norm is showed to have monotonicity property too.

In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let g(t) be a smooth complete solution to the Ricci flow on R, with the canonical Euclidean metric E as initial data, then g(t) is trivial, i.e. g(t) ≡ E.

In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman’s Harnack inequality for the fundamental solution of the conjugate heat equatio...

In this paper, we define a reduced distance function based at a point at the singular time T < ∞ of a Ricci flow. We also show the monotonicity of the corresponding reduced volume based at time T, with equality iff the Ricci flow is a gradient shrinking soliton. Our curvature bound assumption is more general than the type I condition.

In [2], the weak Kähler-Ricci flow was introduced for various geometric motivations. In the current work, we take further consideration on setting up the weak flow. Namely, the initial class is allowed to be no longer Kähler. The convergence as t → 0 is of great importance to study for this topic. 1 Motivation and Set-up Kähler-Ricci flow, the complex version of Ricci flow, has been under inten...