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Let $(M,g(t))$ be a compact Riemannian manifold and the metric $g(t)$ evolve by the Yamabe flow. In the paper we derive the evolution for the first eigenvalue of geometric operator $-\Delta_{\phi}+\frac{R}{2}$ under the Yamabe flow, where $\Delta_{\phi}$ is the Witten-Laplacian operator, $\phi\in C^2(M)$, and $R$ is the scalar curvature with respect to the metric $g(t)$. As a consequence, we construct some monotonic quantities under the Yamabe flow.
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참고문헌 (17)
참고문헌 신청
X. D. Cao / 2007 / Eigenvalues of (−Δ+ R/2 ) on manifolds with nonnegative curvature operator / Math. Ann 337(2):435~441
X. D. Cao / 2008 / First eigenvalues of geometric operators under the Ricci flow / Proc. Amer. Math. Soc 136(11):4075~4078
S. W. Fang / 2015 / Evolution and monotonicity of eigenvalues under the Ricci flow / Sci. China Math 58(8):1737~1744
S. W. Fang / Eigenvalues of geometric operators related to the Witten-Laplacian under the Ricci flow / preprint
H. X. Guo / 2013 / Entropy and lowest eigenvalue on evolving manifolds / Pacific J. Math 264(1):61~81
X. D. Cao / 2008 / First eigenvalues of geometric operators under the Ricci flow / Proc. Amer. Math. Soc 136(11):4075~4078
S. W. Fang / 2015 / Evolution and monotonicity of eigenvalues under the Ricci flow / Sci. China Math 58(8):1737~1744
S. W. Fang / Eigenvalues of geometric operators related to the Witten-Laplacian under the Ricci flow / preprint
H. X. Guo / 2013 / Entropy and lowest eigenvalue on evolving manifolds / Pacific J. Math 264(1):61~81
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