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 Optimal Curvature Estimates for Homogeneous Ricci Flows

2017 ◽  
Vol 2019 (14) ◽  
pp. 4431-4468 ◽  
Author(s):  
Christoph Böhm ◽  
Ramiro Lafuente ◽  
Miles Simon
Keyword(s):  
Homogeneous Space ◽  
Homogeneous Spaces ◽  
Convergence Result ◽  
Einstein Metric ◽  
Type I ◽  
Extinction Time ◽  
Ricci Flows ◽  
Gap Theorem ◽  
Ancient Solutions ◽  
Curvature Estimates

AbstractWe prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n)({\mathrm{scal}}(g(t)) - {\mathrm{scal}}(g(0)) )$. This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, with constants depending only on the dimension $n$. A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on that space. The above curvature estimates follow from a gap theorem for Ricci-flatness on homogeneous spaces. This theorem is proved by contradiction, using a local $W^{2,p}$ convergence result which holds without symmetry assumptions.

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.

An old friend is better than two new ones? Approaches to market research in the context of digital transformation for the antitrust laws enforcement

2020 ◽  
pp. 37-55 ◽  
Author(s):  
A. E. Shastitko ◽  
O. A. Markova
Keyword(s):  
Business Models ◽  
Rapid Development ◽  
Digital Transformation ◽  
Market Definition ◽  
Type I ◽  
Type Ii ◽  
Digital Platforms ◽  
Advantages And Disadvantages ◽  
Pass Through ◽  
Type Ii Errors

Digital transformation has led to changes in business models of traditional players in the existing markets. What is more, new entrants and new markets appeared, in particular platforms and multisided markets. The emergence and rapid development of platforms are caused primarily by the existence of so called indirect network externalities. Regarding to this, a question arises of whether the existing instruments of competition law enforcement and market analysis are still relevant when analyzing markets with digital platforms? This paper aims at discussing advantages and disadvantages of using various tools to define markets with platforms. In particular, we define the features of the SSNIP test when being applyed to markets with platforms. Furthermore, we analyze adjustment in tests for platform market definition in terms of possible type I and type II errors. All in all, it turns out that to reduce the likelihood of type I and type II errors while applying market definition technique to markets with platforms one should consider the type of platform analyzed: transaction platforms without pass-through and non-transaction matching platforms should be tackled as players in a multisided market, whereas non-transaction platforms should be analyzed as players in several interrelated markets. However, if the platform is allowed to adjust prices, there emerges additional challenge that the regulator and companies may manipulate the results of SSNIP test by applying different models of competition.

South Asian Developmental Crisis (Review Article)

1973 ◽  
Vol 12 (2) ◽  
pp. 181-188
Author(s):  
Rafiq Ahmad
Keyword(s):  
Second World War ◽  
World War ◽  
The Past ◽  
The Great Depression ◽  
Political Independence ◽  
Integrated Theory ◽  
Pass Through ◽  
Developed World ◽  
Second World ◽  
Made In

Like nations and civilizations, sciences also pass through period of crises when established theories are overthrown by the unpredictable behaviour of events. Economics is passing through such a crisis. The challenge thrown by the Great Depression of early 1930s took a decade before Keynes re-established the supremacy of economics. But this supremacy has again been upset by the crisis of poverty in the vast under-developed world which attained political independence after the Second World War. Poverty had always existed but never before had it been of such concern to economists as during the past twenty five years or so. Economic literature dealing with this problem has piled up but so have the agonies of poverty. No plausible and well-integrated theory of economic development or under-development has emerged so far, though brilliant advances have been made in isolated directions.

“Ambiguist” Statesmanship

Worldview ◽  
1960 ◽  
Vol 3 (9) ◽  
pp. 7-8
Author(s):  
Will Herberg
Keyword(s):  
Foreign Policy ◽  
Fixed Points ◽  
Natural Law ◽  
The Other ◽  
Moral Decision ◽  
Actual Situation ◽  
The Past ◽  
Mixed Feelings ◽  
Rational Mind ◽  
The One

John Courtney Murray's writing cannot fail to be profound and instructive, and I have profited greatly from it in the course of the past decade. But I must confess that his article, "Morality and Foreign Policy" (Worldview, May), leaves me in a strange confusion of mixed feelings. On the one hand, I can sympathize with what I might call the historical intention of the natural law philosophy he espouses, which I take to be the effort to establish enduring structures of meaning and value to serve as fixed points of moral decision in the complexities of the actual situation. On the other hand, I am rather put off by the calm assurance he exhibits when he deals with these matters, as though everything were at bottom unequivocally rational and unequivocally accessible to the rational mind. And I am really distressed at what seems to 3ie to be his woefully inadequate appreciation of the position of the "ambiguists," among whom I cannot deny I count myself.

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.

Winter Navigation through the Strait of Belle Isle

1964 ◽  
Vol 17 (4) ◽  
pp. 364-375
Author(s):  
R. E. G. Simmons
Keyword(s):  
The Third ◽  
The Past ◽  
Shortest Route ◽  
Pack Ice ◽  
Normal Period ◽  
Ice Conditions ◽  
Short Season ◽  
Pass Through ◽  
Alternative Routes ◽  
St Lawrence River

The shortest route from the Great Lakes and St. Lawrence River to Europe passes through the Strait of Belle Isle. The alternative routes pass through the Cabot Strait and are between 100 and 400 miles longer according to the European port of destination. The Strait of Belle Isle is, however, normally closed to navigation from the end of December until the middle of July due to the presence of pack ice and icebergs.Air reconnaissance patrols flown over the Labrador, Belle Isle and East Newfoundland areas seem to indicate that, for the past few years at least, ice conditions have not been so severe as to hamper navigation throughout the normal period of closure. Consolidated ice is only present from the third week of January to mid-February and clears in mid-April to mid-May; only icebergs present a problem in May and June. It is hoped t o show that with proper air reconnaisance at the beginning and end of the ice season, navigation through the Strait could be extended to eight or nine months of the year, or even longer, instead of the present short season of only 5½ months.

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
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