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# Evolution of Eigenvalues along Rescaled Ricci Flow

Published:2011-08-24
Printed: Mar 2013
• Junfang Li,
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294
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## Abstract

In this paper, we discuss monotonicity formulae of various entropy functionals under various rescaled versions of Ricci flow. As an application, we prove that the lowest eigenvalue of a family of geometric operators $-4\Delta + kR$ is monotonic along the normalized Ricci flow for all $k\ge 1$ provided the initial manifold has nonpositive total scalar curvature.
 Keywords: monotonicity formulas, Ricci flow
 MSC Classifications: 58C40 - Spectral theory; eigenvalue problems [See also 47J10, 58E07] 53C44 - Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

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