Parabolic Frequency on Manifolds
Name
2002.11015.pdf
Description
Submitted version
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131.85 KB
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Adobe PDF
Checksum (MD5)
1a0986da4a1cfd8b41d27e645c9d4969
Author(s)
Holck Colding, Tobias
Minicozzi II, William P
Date Issued
2021
Journal
International Mathematics Research Notices
Publisher
Oxford University Press (OUP)
Citation
Holck Colding, Tobias and Minicozzi II, William P. 2021. "Parabolic Frequency on Manifolds." International Mathematics Research Notices, 2022 (15).
Version
Original manuscript
Abstract
Abstract
We prove monotonicity of a parabolic frequency on static and evolving manifolds without any curvature or other assumptions. These are parabolic analogs of Almgren’s frequency function. When the static manifold is Euclidean space and the drift operator is the Ornstein–Uhlenbeck operator, this can been seen to imply Poon’s frequency monotonicity for the ordinary heat equation. When the manifold is self-similarly evolving by the Ricci flow, we prove a parabolic frequency monotonicity for solutions of the heat equation. For the self-similarly evolving Gaussian soliton, this gives directly Poon’s monotonicity. Monotonicity of frequency is a parabolic analog of the 19th century Hadamard three-circle theorem about log convexity of holomorphic functions on C. From the monotonicity, we get parabolic unique continuation and backward uniqueness.
We prove monotonicity of a parabolic frequency on static and evolving manifolds without any curvature or other assumptions. These are parabolic analogs of Almgren’s frequency function. When the static manifold is Euclidean space and the drift operator is the Ornstein–Uhlenbeck operator, this can been seen to imply Poon’s frequency monotonicity for the ordinary heat equation. When the manifold is self-similarly evolving by the Ricci flow, we prove a parabolic frequency monotonicity for solutions of the heat equation. For the self-similarly evolving Gaussian soliton, this gives directly Poon’s monotonicity. Monotonicity of frequency is a parabolic analog of the 19th century Hadamard three-circle theorem about log convexity of holomorphic functions on C. From the monotonicity, we get parabolic unique continuation and backward uniqueness.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1093/IMRN/RNAB052
