scholarly journals Normalized Ricci Flow on Riemann Surfaces and Determinant of Laplacian

2005 ◽  
Vol 71 (3) ◽  
pp. 241-242 ◽  
Author(s):  
A. Kokotov ◽  
D. Korotkin
Keyword(s):  
Ricci Flow ◽  
Riemann Surfaces ◽  
Normalized Ricci Flow ◽  
Determinant Of Laplacian

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 Kähler-Ricci mean curvature flow of a strictly area decreasing map Between Riemann Surfaces

2020 ◽  
Vol 31 (08) ◽  
pp. 2050061
Author(s):  
Shujing Pan
Keyword(s):  
Scalar Curvature ◽  
Ricci Flow ◽  
Mean Curvature ◽  
Riemann Surfaces ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Positive Scalar Curvature ◽  
Curvature Decay ◽  
The Mean ◽  
Compact Riemann Surfaces

Suppose that [Formula: see text] is a product of compact Riemann surfaces [Formula: see text],[Formula: see text], i.e. [Formula: see text], and [Formula: see text] is a graph in [Formula: see text] of a strictly area dereasing map [Formula: see text]. Let [Formula: see text] evolve along the Kähler–Ricci flow, and [Formula: see text] in [Formula: see text] evolve along the mean curvature flow. We show that [Formula: see text] remains to be a graph of a strictly area decreasing map along the Kähler–Ricci mean curvature flow and exists for all time. In the positive scalar curvature case, we prove the convergence of the flow and the curvature decay along the flow at infinity.

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 A note on uniformization of Riemann surfaces by Ricci flow

2006 ◽  
Vol 134 (11) ◽  
pp. 3391-3393 ◽  
Author(s):  
Xiuxiong Chen ◽  
Peng Lu ◽  
Gang Tian
Keyword(s):  
Ricci Flow ◽  
Riemann Surfaces

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
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