We consider Ricci flow invariant cones C in the space of curvature operators lying between nonnegative Ricci curvature and nonnegative curvature operator. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to Ricci flow has its curvature operator which satsisfies R+ε I ∈ C at the initial time, then it satisfies R+Kε I ∈ C on some time interval depen...

In this paper, we prove that the first eigenvalues of −∆+ cR (c ≥ 1 4 ) are nondecreasing under the Ricci flow. We also prove the monotonicity under the normalized Ricci flow for the cases c = 1/4 and r ≤ 0. 1. First eigenvalue of −∆+ cR Let M be a closed Riemannian manifold, and (M,g(t)) be a smooth solution to the Ricci flow equation ∂ ∂t gij = −2Rij on 0 ≤ t < T . In [Cao07], we prove that a...

Geometric monotone properties of the first nonzero eigenvalue of Laplacian form operator under the action of the Ricci flow in a compact nmanifold ( ) 2 ≥ n are studied. We introduce certain energy functional which proves to be monotonically non-decreasing, as an application, we show that all steady breathers are gradient steady solitons, which are Ricci flat metric. The results are also extend...

a r t i c l e i n f o a b s t r a c t Keywords: The Laplace operator for graphs The Harnack inequalities Eigenvalues Diameter We establish a Harnack inequality for finite connected graphs with non-negative Ricci curvature. As a consequence, we derive an eigenvalue lower bound, extending previous results for Ricci flat graphs.

The Ricci curvature at an isolated singularity of an immersed hypersurface exhibits a local behaviour echoing a global property of the Ricci curvature on a complete hypersurface in euclidean space (i.e., the Efimov theorem [12], that sup Ric ≥ 0). This local behaviour takes the form of a near universal bound on the decay of the Ricci curvature at a simple singularity (eg. a cone singularity) in...

We study the quantum sphere Cq [S] as a quantum Riemannian manifold in the quantum frame bundle approach. We exhibit its 2-dimensional cotangent bundle as a direct sum Ω ⊕ Ω in a double complex. We find the natural metric, volume form, Hodge * operator, Laplace and Maxwell operators. We show that the q-monopole as spin connection induces a natural Levi-Civita type connection and find its Ricci ...

In this paper, we consider the characterization of eigenfunctions for Laplacian operators on some Riemannian manifolds. Firstly we prove that for the space form (M K , gK) with the constant sectional curvature K, the first eigenvalue of Laplacian operator λ1 (M K) is greater than the limit of the first Dirichlet eigenvalue of Laplacian operator λ1 (BK (p, r)). Based on this, we then present a c...

Spectrum of kinematic fast dynamo operators in Ricci compressible flows in Einstein 2-manifolds is investigated. A similar expression, to the one obtained by Chicone, Latushkin and Montgomery-Smith (Comm Math Phys (1995)) is given, for the fast dynamo operator. The operator eigenvalue is obtained in a highly conducting media, in terms of linear and nonlinear orders of Ricci scalar. Eigenvalue s...

Rjk = R̄jk + ∇̄mD jk − ∇̄jD mk −Dn jkD nm −Dn mkD nj . (To see why this is true, just compute in local normal coordinates for ḡ!) From now on all components are tensorial (as opposed to local coordinate components.) The terms involving second covariant derivatives of the metric are: 1 2 g(−∇̄m∇̄lgjk +∇̄m∇̄kgjl +∇̄m∇̄jglk +∇̄j∇̄lgmk−∇̄j∇̄kgml−∇̄j∇̄mglk). The first term is a kind of Laplacian. The third and six...