First Eigenvalues of Geometric Operators under the Ricci Flow
نویسنده
چکیده
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 all eigenvalues λ(t) of the operator −∆+ R2 are nondecreasing under the Ricci flow on manifolds with a nonnegative curvature operator. Assume f = f(x, t) is the corresponding eigenfunction of λ(t); that is, (−∆+ R 2 )f(x, t) = λ(t)f(x, t)
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