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Eigenvalues on Manifolds with Nonnegative Curvature Operator

Eigenvalues on Manifolds with Nonnegative Curvature...

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Eigenvalues of ( - Δ + R 2 ) on Manifolds with Nonnegative Curvature Operator Xiaodong Cao [email protected] May 12, 2006 Abstract In this paper, we show that the eigenvalues of ( -4 + R 2 ) are nondecreas- ing under the Ricci flow for manifolds with nonnegative curvature operator. Then we show that the only steady Ricci breather with nonnegative curvature operator is the trivial one which is Ricci-flat. 1 Introduction Let ( M, g ( t )) be a compact Riemannian Manifold. Studying the eigenvalues of geo- metric operators is a very powerful tool for the understanding of Riemannian mani- folds. In [6], Perelman shows that the functional F = Z M ( R + |∇ h | 2 ) e - h dv is nondecreasing along the Ricci flow coupled to a backward heat-type equation. More precisely, if ( M, g ( t )) is a solution to the Ricci flow: ∂t g ij = - 2 R ij , (1.1) and h satisfies the following evolution equation: ∂t h = -4 h + |∇ h | 2 - R , then we have d dt F = 2 Z | R ij + h ij | 2 e - h dv . If we define μ ( g ij ) = inf F ( g ij , h ) , 1
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where the infimum is taken over all smooth h which satisfies Z M e - h dv = 1 , then μ ( g ij ) is the lowest eigenvalue of the operator - 4 4 + R , and the nondecreasing of the functional F implies the nondecreasing of μ ( g ij ). As an application, Perelman was able to show that there are no nontrivial steady or expanding breathers on compact manifolds. In this paper, we consider the eigenvalues of the operator -4 + R 2 on manifolds with nonnegative curvature operator. In Section Two, we first derive the evolution equation of those eigenvalues under the Ricci flow. In Section Three, we prove the following theorem: Theorem 1 On a Riemannian manifold with nonnegative curvature operator, the eigenvalues of the operator -4 + R 2 are nondecreasing under the Ricci flow. Then in Section Four, we rule out the non-trivial steady Ricci breather with nonnegative curvature operator as an application of our theorem.
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