We show that Perelman’s W functional on Kahler manifolds has a natural counterpart on Sasaki manifolds. We prove, using this functional, that Perelman’s results on Kahler-Ricci flow (the first Chern class is positive) can be generalized to Sasaki-Ricci flow, including the uniform bound on the diameter and the scalar curvature along the flow. We also show that positivity of transverse bisectiona...

In this paper, we investigate the geometrical axioms of Riemannian submersions in context η-Ricci–Yamabe soliton (η-RY soliton) with a potential field. We give categorization each fiber submersion as an η-RY soliton, η-Ricci and η-Yamabe soliton. Additionally, consider many circumstances under which target manifold is or quasi-Yamabe deduce Poisson equation on specific scenario if vector field ...

This paper studies the normalized Ricci flow on surfaces with conical singularities. It’s proved that the normalized Ricci flow has a solution for a short time for initial metrics with conical singularities. Moreover, the solution makes good geometric sense. For some simple surfaces of this kind, for example, the tear drop and the football, it’s shown that they admit Ricci soliton metric. MSC 2...

In this paper we prove a conjecture by Feldman-IlmanenKnopf in [14] that the gradient shrinking soliton metric they constructed on the tautological line bundle over CP is the uniform limit of blowups of a type I Ricci flow singularity on a closed manifold. We use this result to show that limits of blow-ups of Ricci flow singularities on closed four dimensional manifolds do not necessarily have ...

In this paper, we show that an n-dimensional connected noncompact Ricci soliton isometrically immersed in the ‡at complex space form (C n+1 2 ; J; h; i), with potential vector …eld of the Ricci soliton is the characteristic vector …eld of the real hypersurface is an Einstein manifold. We classify connected Hopf hypersurfaces in the ‡at complex space form (C n+1 2 ; J; h; i) and also obtain a ch...

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota’s argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct ...

In this paper we prove a conjecture by Feldman–Ilmanen–Knopf (2003) that the gradient shrinking soliton metric they constructed on the tautological line bundle over CP is the uniform limit of blow-ups of a type I Ricci flow singularity on a closed manifold. We use this result to show that limits of blow-ups of Ricci flow singularities on closed four-dimensional manifolds do not necessarily have...

We prove the following results: (i) A Sasakian metric as a nontrivial Ricci soliton is null η-Einstein, and expanding. Such a characterization permits to identify the Sasakian metric on the Heisenberg group H as an explicit example of (non-trivial) Ricci soliton of such type. (ii) If an ηEinstein contact metric manifold M has a vector field V leaving the structure tensor and the scalar curvatur...

The Ricci bracket flow is a geometric evolution on Lie algebras which is related to the Ricci flow on the corresponding Lie group. For nilpotent Lie groups, these two flows are equivalent. In the solvable case, it is not known whether they are equivalent. We examine a family of solvable Lie algebras and identify various elements of that family which are solitons under the Ricci bracket flow. We...

If a normalized Kähler-Ricci flow g(t), t ∈ [0,∞), on a compact Kähler manifold M , dimC M = n ≥ 3, with positive first Chern class satisfies g(t) ∈ 2πc1(M) and has curvature operator uniformly bounded in Ln-norm, the curvature operator will also be uniformly bounded along the flow. Consequently the flow will converge along a subsequence to a Kähler-Ricci soliton.