The Euler characteristic and signature of four-dimensional closed manifolds and the normalized Ricci flow equation

2015 ◽  
Vol 180 (1) ◽  
pp. 229-239
Author(s):  
Murat Limoncu
Keyword(s):  
Euler Characteristic ◽  
Ricci Flow ◽  
Flow Equation ◽  
Normalized Ricci Flow

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.

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.

scholarly journals Deformed $$\sigma $$-models, Ricci flow and Toda field theories

2021 ◽  
Vol 111 (6) ◽  
Author(s):  
Dmitri Bykov ◽  
Dieter Lüst
Keyword(s):  
Sigma Models ◽  
Ricci Flow ◽  
Flag Manifold ◽  
Flow Equation ◽  
Target Space ◽  
Field Theories ◽  
Two Dimensional

AbstractIt is shown that the Pohlmeyer map of a $$\sigma $$ σ -model with a toric two-dimensional target space naturally leads to the ‘sausage’ metric. We then elaborate the trigonometric deformation of the $$\mathbb {CP}^{n-1}$$ CP n - 1 -model, proving that its T-dual metric is Kähler and solves the Ricci flow equation. Finally, we discuss a relation between flag manifold $$\sigma $$ σ -models and Toda field theories.

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.

RICCI FLOW EQUATION ON C-REDUCIBLE METRICS

2011 ◽  
Vol 08 (04) ◽  
pp. 773-781 ◽  
Author(s):  
NASRIN SADEGHZADEH ◽  
ASSADOLLAH RAZAVI
Keyword(s):  
Fixed Point ◽  
Ricci Flow ◽  
Einstein Metrics ◽  
Flow Equation ◽  
Finsler Metrics

This paper focuses on the study of a deformation of Finsler metrics satisfying Ricci flow equation. We will prove that every deformation of Randers (Kropina)-metrics satisfying Ricci flow equation is Einstein, we will also show that a deformation of Einstein metrics with initial Randers (Kropina)-metrics remains Randers (Kropina). In other words the deformation of Randers (or Kropina)-metrics is exactly the fixed point of (un-normal and normal) Ricci flow equation.

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 An analytic approach to the normalized Ricci flow-like equation: Revisited

2015 ◽  
Vol 44 ◽  
pp. 30-33
Author(s):  
Nikos I. Kavallaris ◽  
Takashi Suzuki
Keyword(s):  
Ricci Flow ◽  
Analytic Approach ◽  
Normalized Ricci Flow
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