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In this paper we consider the monotonicity of the lowest constant $\lambda^{b}_{a}(g)$ under the Ricci-Bourguignon flow and the normalized Ricci-Bourguignon flow such that the equation \begin{equation*} -{\rm \Delta} u + au\log u + bRu= \lambda^{b}_{a}(g) u \end{equation*} with $\int_M u^2\, {\rm dV}=1,$ has positive solutions, where $a$ and $b$ are two real constants. We also construct various monotonic quantities under the Ricci-Bourguignon flow and the normalized Ricci-Bourguignon flow. Moreover, we prove that a compact steady breather which evolves under the Ricci-Bourguignon flow should be Ricci-flat.
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참고문헌 (11)
참고문헌 신청
Xiaodong Cao / 2006 / Eigenvalues of (-∆ + R/2 ) on manifolds with nonnegative curvature operator / Mathematische Annalen 337(2):435~441
Xiaodong Cao / 2008 / First eigenvalues of geometric operators under the Ricci flow / Proceedings of the American Mathematical Society 136(11):4075~4078
Bin Chen / 2018 / Monotonicity of eigenvalues of geometric operators along the Ricci–Bourguignon flow / Pacific Journal of Mathematics 296(1):1~20
B. Chow / 2007 / The Ricci ow: techniques and applications. Part I / American Mathematical Society
L. Cremaschi / 2016 / Some variations on Ricci ow
Xiaodong Cao / 2008 / First eigenvalues of geometric operators under the Ricci flow / Proceedings of the American Mathematical Society 136(11):4075~4078
Bin Chen / 2018 / Monotonicity of eigenvalues of geometric operators along the Ricci–Bourguignon flow / Pacific Journal of Mathematics 296(1):1~20
B. Chow / 2007 / The Ricci ow: techniques and applications. Part I / American Mathematical Society
L. Cremaschi / 2016 / Some variations on Ricci ow
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