scholarly journals On the equivalence between noncollapsing and bounded entropy for ancient solutions to the Ricci flow

2020 ◽  
Vol 2020 (762) ◽  
pp. 35-51
Author(s):  
Yongjia Zhang
Keyword(s):  
Ricci Flow ◽  
Nonnegative Curvature ◽  
Curvature Operator ◽  
Type I ◽  
Curvature Bound ◽  
Reduced Volume ◽  
Ancient Solutions ◽  
Asymptotic Entropy ◽  
Uniformly Bounded

AbstractAs a continuation of a previous paper, we prove Perelman’s assertion, that is, for ancient solutions to the Ricci flow with bounded nonnegative curvature operator, uniformly bounded entropy is equivalent to κ-noncollapsing on all scales. We also establish an equality between the asymptotic entropy and the asymptotic reduced volume, which is a result similar to a paper by Xu (2017), where he assumes the Type I curvature bound.

scholarly journals Lower bounds on Ricci flow invariant curvatures and geometric applications

2015 ◽  
Vol 2015 (703) ◽  
Author(s):  
Thomas Richard
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Lower Bounds ◽  
Ricci Flow ◽  
Nonnegative Curvature ◽  
Curvature Operator ◽  
Nonnegative Ricci Curvature ◽  
Invariant Cones

AbstractWe consider Ricci flow invariant cones 𝒞 in the space of curvature operators lying between the cones “nonnegative Ricci curvature” and “nonnegative curvature operator”. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to the Ricci flow has its curvature operator which satisfies

scholarly journals BOUNDING SECTIONAL CURVATURE ALONG THE KÄHLER–RICCI FLOW

2009 ◽  
Vol 11 (06) ◽  
pp. 1067-1077 ◽  
Author(s):  
WEI-DONG RUAN ◽  
YUGUANG ZHANG ◽  
ZHENLEI ZHANG
Keyword(s):  
Sectional Curvature ◽  
Chern Class ◽  
Ricci Flow ◽  
Kähler Manifold ◽  
Ricci Soliton ◽  
Curvature Operator ◽  
Kahler Manifold ◽  
First Chern Class ◽  
Compact Kähler Manifold ◽  
Uniformly Bounded

If a normalized Kähler–Ricci flow g(t), t ∈ [0,∞), on a compact Kähler manifold M, dim ℂ 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.

scholarly journals Ancient solutions of the homogeneous Ricci flow on flag manifolds

2021 ◽  
Vol 36 (1) ◽  
pp. 99-145
Author(s):  
S. Anastassiou ◽  
I. Chrysikos
Keyword(s):  
Fixed Points ◽  
Ricci Flow ◽  
Flag Manifold ◽  
Einstein Metric ◽  
Flag Manifolds ◽  
Type I ◽  
Poincaré Compactification ◽  
The Past ◽  
Ancient Solutions ◽  
Pass Through

For any flag manifold M=G/K of a compact simple Lie group G we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions pass through an invariant Einstein metric on M, and by [13] they must develop a Type I singularity in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag manifold M=G/K with second Betti number b2(M) = 1, for a generic initial invariant metric. We describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose α-limit set consists of fixed points at infinity of MG. Based on the Poincaré compactification method, we show that these fixed points correspond to invariant Einstein metrics and we study their stability properties, illuminating thus the structure of the system’s phase space.

scholarly journals Ancient solutions for Andrews’ hypersurface flow

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Peng Lu ◽  
Jiuru Zhou
Keyword(s):  
Ricci Flow ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Euclidean Spaces ◽  
Round Cylinder ◽  
Ancient Solutions ◽  
Singular Time

AbstractWe construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994.As time {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time. But as {t\rightarrow-\infty} the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions {S^{J}\times\mathbb{R}^{n-J}}, {1\leq J\leq n-1}. These results are the analog of the corresponding results in Ricci flow ({J=n-1}) and mean curvature flow.

scholarly journals Ancient solutions of the Ricci flow on bundles

2017 ◽  
Vol 318 ◽  
pp. 411-456 ◽  
Author(s):  
Peng Lu ◽  
Y.K. Wang
Keyword(s):  
Ricci Flow ◽  
Ancient Solutions

scholarly journals Ricci flow on manifolds with boundary with arbitrary initial metric

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tsz-Kiu Aaron Chow
Keyword(s):  
Ricci Flow ◽  
Positive Curvature ◽  
Curvature Operator ◽  
Manifolds With Boundary ◽  
Convex Boundary ◽  
Uniqueness Of The Solution ◽  
Positive Time ◽  
Natural Boundary Conditions ◽  
Short Time ◽  
Time Existence

Abstract In this paper, we study the Ricci flow on manifolds with boundary. In the paper, we substantially improve Shen’s result [Y. Shen, On Ricci deformation of a Riemannian metric on manifold with boundary, Pacific J. Math. 173 1996, 1, 203–221] to manifolds with arbitrary initial metric. We prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously totally geodesic for positive time. Moreover, we prove that the flow we constructed preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition.

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