scholarly journals On Type-I singularities in Ricci flow

2011 ◽  
Vol 19 (5) ◽  
pp. 905-922 ◽  
Author(s):  
Joerg Enders ◽  
Reto Müller ◽  
Peter M. Topping
Keyword(s):  
Ricci Flow ◽  
Type I

scholarly journals The Size of the Singular Set of a Type I Ricci Flow

2017 ◽  
Vol 27 (4) ◽  
pp. 3099-3119 ◽  
Author(s):  
Panagiotis Gianniotis
Keyword(s):  
Ricci Flow ◽  
Type I ◽  
Singular Set

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 Ricci flow of warped Berger metrics on $${\mathbb {R}}^{4}$$

2020 ◽  
Vol 59 (5) ◽  
Author(s):  
Francesco Di Giovanni
Keyword(s):  
Ricci Flow ◽  
Type I ◽  
Type Ii ◽  
Rotationally Symmetric ◽  
Global Type ◽  
Maximal Time

Abstract We study the Ricci flow on $${\mathbb {R}}^{4}$$ R 4 starting at an SU(2)-cohomogeneity 1 metric $$g_{0}$$ g 0 whose restriction to any hypersphere is a Berger metric. We prove that if $$g_{0}$$ g 0 has no necks and is bounded by a cylinder, then the solution develops a global Type-II singularity and converges to the Bryant soliton when suitably dilated at the origin. This is the first example in dimension $$n > 3$$ n > 3 of a non-rotationally symmetric Type-II flow converging to a rotationally symmetric singularity model. Next, we show that if instead $$g_{0}$$ g 0 has no necks, its curvature decays and the Hopf fibres are not collapsed, then the solution is immortal. Finally, we prove that if the flow is Type-I, then there exist minimal 3-spheres for times close to the maximal time.

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.

Nonexistence of noncompact Type-I ancient three-dimensional κ-solutions of Ricci flow with positive curvature

2019 ◽  
Vol 21 (06) ◽  
pp. 1850049 ◽  
Author(s):  
Max Hallgren
Keyword(s):  
Ricci Flow ◽  
Positive Curvature ◽  
Three Dimensional ◽  
Short Paper ◽  
Type I ◽  
Noncompact Type

In this short paper, we show that there does not exist a noncompact Type-I [Formula: see text]-solution of the Ricci flow with positive curvature in dimension 3.

Heat-Etching with a Bullivant Type II Simple Freeze-Cleave Device

1970 ◽  
Vol 28 ◽  
pp. 106-107 ◽  
Author(s):  
Ronald S. Weinstein ◽  
N. Scott McNutt
Keyword(s):  
New Zealand ◽  
General Hospital ◽  
Massachusetts General Hospital ◽  
Type I ◽  
Type Ii ◽  
Collaborative Effort ◽  
Temperature And Pressure ◽  
The University

The Type I simple cold block device was described by Bullivant and Ames in 1966 and represented the product of the first successful effort to simplify the equipment required to do sophisticated freeze-cleave techniques. Bullivant, Weinstein and Someda described the Type II device which is a modification of the Type I device and was developed as a collaborative effort at the Massachusetts General Hospital and the University of Auckland, New Zealand. The modifications reduced specimen contamination and provided controlled specimen warming for heat-etching of fracture faces. We have now tested the Mass. General Hospital version of the Type II device (called the “Type II-MGH device”) on a wide variety of biological specimens and have established temperature and pressure curves for routine heat-etching with the device.

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