Two-parametric bifurcations of singular points of the normalized Ricci flow on generalized Wallach spaces

2015 ◽  
Author(s):  
Nurlan Abiev
Keyword(s):  
Ricci Flow ◽  
Singular Points ◽  
Normalized Ricci Flow

scholarly journals О топологической структуре некоторых множеств, получаемых из нормализованных потоков Риччи на обобщенных пространствах Уоллаха

2015 ◽  
Author(s):  
N.A. Abiev
Keyword(s):  
Dynamical Systems ◽  
Ricci Flow ◽  
Algebraic Surface ◽  
Symmetric Polynomial ◽  
Singular Points ◽  
Connected Components ◽  
Topological Structures ◽  
General Properties ◽  
Normalized Ricci Flow

We study topological structures of the sets (0,1/2)3∩Ω and (0,1/2)3∖Ω, where Ω is one special algebraic surface defined by a symmetric polynomial of degree 12. These problems arise in studying of general properties of degenerate singular points of dynamical systems obtained from the~normalized Ricci flow on generalized Wallach spaces. Our main goal is to prove the connectedness of (0,1/2)3∩Ω and to determine the number of connected components of (0,1/2)3∖Ω.

EIGENVALUES OF GEOMETRIC OPERATORS RELATED TO THE WITTEN LAPLACIAN UNDER THE RICCI FLOW

2017 ◽  
Vol 59 (3) ◽  
pp. 743-751
Author(s):  
SHOUWEN FANG ◽  
FEI YANG ◽  
PENG ZHU
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Ricci Flow ◽  
Compact Riemannian Manifold ◽  
Laplacian Operator ◽  
Witten Laplacian ◽  
Normalized Ricci Flow ◽  
Geometric Operator

AbstractLet (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. In the paper, we prove that the eigenvalues of geometric operator −Δφ + $\frac{R}{2}$ are non-decreasing under the Ricci flow for manifold M with some curvature conditions, where Δφ is the Witten Laplacian operator, φ ∈ C2(M), and R is the scalar curvature with respect to the metric g(t). We also derive the evolution of eigenvalues under the normalized Ricci flow. As a consequence, we show that compact steady Ricci breather with these curvature conditions must be trivial.

The normalized Ricci flow and homology in Lagrangian submanifolds of generalized complex space forms

2020 ◽  
Vol 17 (06) ◽  
pp. 2050094 ◽  
Author(s):  
Fatemah Mofarreh ◽  
Akram Ali ◽  
Wan Ainun Mior Othman
Keyword(s):  
Fundamental Form ◽  
Ricci Flow ◽  
Complex Space ◽  
Space Form ◽  
Second Fundamental Form ◽  
Complex Space Form ◽  
Standard Sphere ◽  
The Second Fundamental Form ◽  
Generalized Complex ◽  
Normalized Ricci Flow

In this paper, we prove that a simply connected Lagrangian submanifold in the generalized complex space form is diffeomorphic to standard sphere [Formula: see text] and the normalized Ricci flow converges to a constant curvature metric, provided its squared norm of the second fundamental form satisfies some upper bound depending only on the squared norm of the mean curvature vector field, the constant sectional curvature, and the dimension of the Lagrangian immersion of the ambient space. Next, we conclude that stable currents do not exist and homology groups vanish in a compact real submanifold of the general complex space form, provided that the second fundamental form satisfies some extrinsic conditions. We show that our results improve some previous results.

scholarly journals On the Maximal Rate of Convergence Under the Ricci Flow

2020 ◽  
Author(s):  
Brett Kotschwar
Keyword(s):  
Rate Of Convergence ◽  
Ricci Flow ◽  
Closed Manifold ◽  
Maximal Rate ◽  
Exponential Rate ◽  
Normalized Ricci Flow ◽  
Self Similar

Abstract We estimate from above the rate at which a solution to the normalized Ricci flow on a closed manifold may converge to a limit soliton. Our main result implies that any solution that converges modulo diffeomorphisms to a soliton faster than any fixed exponential rate must itself be self-similar.

scholarly journals Non-singular solutions to the normalized Ricci flow equation

2007 ◽  
Vol 340 (3) ◽  
pp. 647-674 ◽  
Author(s):  
Fuquan Fang ◽  
Yuguang Zhang ◽  
Zhenlei Zhang
Keyword(s):  
Ricci Flow ◽  
Flow Equation ◽  
Singular Solutions ◽  
Normalized Ricci Flow

NORMALIZED RICCI FLOW, SURGERY, AND SEIBERG–WITTEN INVARIANTS

2014 ◽  
Vol 25 (02) ◽  
pp. 1450005
Author(s):  
MASASHI ISHIDA
Keyword(s):  
Existence Theorem ◽  
Euler Characteristic ◽  
Ricci Flow ◽  
Einstein Metrics ◽  
Nonsingular Solution ◽  
Ricci Flows ◽  
Normalized Ricci Flow ◽  
Four Manifolds ◽  
Special Case

We investigate the behavior of solutions of the normalized Ricci flow under surgeries of four-manifolds along circles by using Seiberg–Witten invariants. As a by-product, we prove that any pair (α, β) of integers satisfying α + β ≡ 0 (mod 2) can be realized as the Euler characteristic χ and signature τ of infinitely many closed smooth 4-manifolds with negative Perelman's [Formula: see text] invariants and on which there is no nonsingular solution to the normalized Ricci flows for any initial metric. In particular, this includes the existence theorem of non-Einstein 4-manifolds due to Sambusetti [An obstruction to the existence of Einstein metrics on 4-manifolds, Math. Ann.311 (1998) 533–547] as a special case.

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