Abstract
This work is devoted to the study of parabolic frequency for solutions of the heat equation on Riemannian manifolds. We show that the parabolic frequency functional is almost increasing on compact manifolds with nonnegative sectional curvature, which generalizes a monotonicity result proved by Poon (Commun Partial Differ Equ 21(3–4):521–539, 1996) and by Ni (Analysis, complex geometry, and mathematical physics: in honor of Duong H. Phong, 2015). The proof is based on a generalization of R. Hamilton’s matrix Harnack inequality (Hamilton in Commun Anal Geom 1(1):113–126, 1993) for small time. As applications, we obtain a unique continuation result. Monotonicity of a new quantity under two-dimensional Ricci flow, closely related to the parabolic frequency functional, is derived as well.
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Acknowledgements
The first author would like to thank Professor Richard Schoen for his interest and generous support, and Professor Yannick Sire for conversations on possible applications of Theorem 1.2. The second author was supported by NSF of China under Grant No. 11601359, NSF of Jiangsu Province No. BK20160301, and China Postdoctoral Foundation grants No. 2017T100394 and 2016M591900.
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Communicated by F.H. Lin.
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