On the evolution of invariant Riemannian metrics on one class of generalized Wallach spaces under the influence of the normalized Ricci flow

2017 ◽  
Vol 27 (4) ◽  
pp. 227-238 ◽  
Author(s):  
N. A. Abiev
Keyword(s):  
Ricci Flow ◽  
Riemannian Metrics ◽  
Normalized Ricci Flow ◽  
Invariant Riemannian Metrics

scholarly journals Some aspects of Ricci flow on the 4-sphere

10.53733/152 ◽  
2021 ◽  
Vol 52 ◽  
pp. 381-402
Author(s):  
Sun-Yung Alice Chang ◽  
Eric Chen
Keyword(s):  
Scalar Curvature ◽  
Curvature Tensor ◽  
Ricci Flow ◽  
Weyl Tensor ◽  
Riemannian Metrics ◽  
Positive Scalar Curvature ◽  
Conformal Invariants ◽  
Positive Scalar ◽  
Gap Theorem ◽  
Normalized Ricci Flow

In this paper, on 4-spheres equipped with Riemannian metrics we study some integral conformal invariants, the sign and size of which under Ricci flow characterize the standard 4-sphere. We obtain a conformal gap theorem, and for Yamabe metrics of positive scalar curvature with L^2 norm of the Weyl tensor of the metric suitably small, we establish the monotonic decay of the L^p norm for certain p>2 of the reduced curvature tensor along the normalized Ricci flow, with the metric converging exponentially to the standard 4-sphere.

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 Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces

2021 ◽  
Vol 25 (2) ◽  
pp. 913-948
Author(s):  
Miles Simon ◽  
Peter M Topping
Keyword(s):  
Ricci Flow ◽  
Riemannian Metrics

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.

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