Evolution of Eigenvalues of a Geometric Operator Under Ricci Flow on a Riemannian Manifold

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.

Download Full-text

Almost Euclidean Isoperimetric Inequalities in Spaces Satisfying Local Ricci Curvature Lower Bounds

International Mathematics Research Notices ◽

10.1093/imrn/rny070 ◽

2018 ◽

Vol 2020 (5) ◽

pp. 1481-1510 ◽

Cited By ~ 1

Author(s):

Fabio Cavalletti ◽

Andrea Mondino

Keyword(s):

Riemannian Manifold ◽

Isoperimetric Inequality ◽

Ricci Curvature ◽

Lower Bounds ◽

Ricci Flow ◽

General Framework ◽

Isoperimetric Inequalities ◽

Optimal Transportation ◽

Maximal Volume ◽

Bounded Below

Abstract Motivated by Perelman’s Pseudo-Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost euclidean isoperimetric inequality. The result is actually established in the more general framework of non-smooth spaces satisfying local Ricci curvature lower bounds in a synthetic sense via optimal transportation.

Download Full-text

First Eigenvalues of Geometric Operator under The Ricci-Bourguignon Flow

Journal of the Indonesian Mathematical Society ◽

10.22342/jims.24.1.434.51-60 ◽

2017 ◽

Vol 24 (1) ◽

pp. 51-60

Author(s):

Shahroud Azami

Keyword(s):

Riemannian Manifold ◽

Scalar Curvature ◽

Compact Riemannian Manifold ◽

Nonnegative Curvature ◽

Laplacian Operator ◽

Witten Laplacian ◽

Geometric Operator

Let $(M,g(t))$ be a compact Riemannian manifold and the metric $g(t)$ evolve by the Ricci-Bourguignon flow. We find the formula variation of the eigenvalues of geometric operator $-\Delta_{\phi}+cR$ under the Ricci-Bourguignon flow, where $\Delta_{\phi}$ is the Witten-Laplacian operator and $R$ is the scalar curvature. In the final we show that some quantities dependent to the eigenvalues of the geometric operator are nondecreasing along the Ricci-Bourguignon flow on closed manifolds with nonnegative curvature.

Download Full-text

Evolution of Eigenvalues of Geometric Operator Under the Rescaled List’s Extended Ricci Flow

Bulletin of the Iranian Mathematical Society ◽

10.1007/s41980-021-00580-0 ◽

2021 ◽

Author(s):

Shahroud Azami

Keyword(s):

Ricci Flow ◽

Geometric Operator

Download Full-text

Gradient estimate for the forward conjugate heat equation of forms under the Ricci flow

International Journal of Mathematics ◽

10.1142/s0129167x21500816 ◽

2021 ◽

pp. 2150081

Author(s):

Liangdi Zhang

Keyword(s):

Riemannian Manifold ◽

Heat Equation ◽

Ricci Flow ◽

Differential Forms ◽

Gradient Estimate ◽

Conjugate Heat Equation

We establish bounds for the gradient of solutions to the forward conjugate heat equation of differential forms on a Riemannian manifold with the metric evolves under the Ricci flow.

Download Full-text

The Cotton Tensor and the Ricci Flow

Geometric Flows ◽

10.1515/geofl-2017-0001 ◽

2017 ◽

Vol 2 (1) ◽

Author(s):

Carlo Mantegazza ◽

Samuele Mongodi ◽

Michele Rimoldi

Keyword(s):

Riemannian Manifold ◽

Evolution Equation ◽

Ricci Flow ◽

Three Dimensional ◽

Dimensional Case ◽

Bach Tensor ◽

Cotton Tensor

AbstractWe compute the evolution equation of the Cotton and the Bach tensor under the Ricci flow of a Riemannian manifold, with particular attention to the three dimensional case, and we discuss some applications.

Download Full-text