Parabolic frequency and backward uniqueness on compact Finsler metric measure spaces
Abstract
We establish monotonicity formulas for parabolic frequency functions on closed Finsler metric measure spaces with fixed or evolving metrics and measures. When the reference measure is fixed, an integral bound on the metric variation yields monotonicity for the Finsler heat equation. For coupled metric measure flows, we construct positive global supersolutions and subsolutions of a linearized conjugate heat equation from local radial barriers. These weights define a frequency function whose monotonicity follows from an integral bound involving the metric variation and the change in weighted Ricci curvature. For parabolic equations with a source controlled by the solution and its gradient, we obtain a differential inequality for the frequency and prove that a solution vanishing at the terminal time must vanish throughout the preceding interval. The proof controls the comparison of the two weights through an energy quotient computed with the subsolution measure.
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BibTeX
@misc{shen2026parabolic,
title = {{Parabolic frequency and backward uniqueness on compact Finsler metric measure spaces}},
author = {Bin Shen and Zisu Zhao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.09969},
eprint = {2611.09969},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.09969},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Parabolic frequency and backward uniqueness on compact Finsler metric measure spaces
AU - Bin Shen
AU - Zisu Zhao
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.09969
UR - https://arxiv.org/abs/2611.09969
AB - We establish monotonicity formulas for parabolic frequency functions on closed Finsler metric measure spaces with fixed or evolving metrics and measures. When the reference measure is fixed, an integral bound on the metric variation yields monotonicity for the Finsler heat equation. For coupled metric measure flows, we construct positive global supersolutions and subsolutions of a linearized conjugate heat equation from local radial barriers. These weights define a frequency function whose monotonicity follows from an integral bound involving the metric variation and the change in weighted Ricci curvature. For parabolic equations with a source controlled by the solution and its gradient, we obtain a differential inequality for the frequency and prove that a solution vanishing at the terminal time must vanish throughout the preceding interval. The proof controls the comparison of the two weights through an energy quotient computed with the subsolution measure.
ER -