Pairwise dissimilarity representations are frequently used as an alternative to feature vectors in pattern recognition. One of the problems encountered in the analysis of such data, is that the dissimilarities are rarely Euclidean, while statistical learning algorithms often rely on Euclidean dissimilarities. Such non-Euclidean dissimilarities are often corrected or a consistent Euclidean geome...

In the present paper, we study the L2−preserved Schrödinger heat flow under the Ricci flow on closed manifolds. First, we establish the global existence and the uniqueness of the solution to the heat flow under the Ricci flow. Next, we prove an elliptic type gradient estimate of smooth positive solutions to the heat flow and get a Harnack inequality. M.S.C. 2010: 53C21.

In [26], Perelman established a differential Li-YauHamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds (also see [23]). As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these res...

If g(t) is a three-dimensional Ricci flow solution, with sectional curvatures that are O(t) and diameter that is O(t 1 2 ), then the pullback Ricci flow solution on the universal cover approaches a homogeneous expanding soliton.

In this paper, we consider a generalized Ricci flow and establish the higher derivatives estimates for compact manifolds. As an application, we prove the compactness theorem for this generalized Ricci flow.

In this paper, we study the behavior of Ricci flows on compact orbifolds with finite singularities. We show that Perelman’s pseudolocality theorem also holds on orbifold Ricci flow. Using this property, we obtain a weak compactness theorem of Ricci flows on orbifolds under some natural technical conditions. This generalizes the corresponding theorem on manifolds. As an application, we can use K...

We study some estimates along the Kähler Ricci flow on Fano manifolds. Using these estimates, we show the convergence of Kähler Ricci flow directly if the α-invariant of the canonical class is greater than n n+1 . Applying these convergence theorems, we can give a Kähler Ricci flow proof of Calabi conjecture on such Fano manifolds. In particular, the existence of KE metrics on a lot of Fano sur...

We investigate the properties of the combinatorial Ricci flow for surfaces, both forward and backward – existence, uniqueness and singularities formation. We show that the positive results that exist for the smooth Ricci flow also hold for the combinatorial one and that, moreover, the same results hold for a more general, metric notion of curvature. Furthermore, using the metric curvature appro...