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An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow

  • Hongxin Guo
  • Guozhi Lin
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Abstract

For a rotationally symmetric Ricci flow g(t)=ds2+f(s,t)2g𝕊n−1, n≥3, we study the scale-invariant curvature anisotropy A=1−fs2+ffss introduced by Angenent--Knopf. At the critical power p*=(n−1)/(n−2), with Φ=sgn(A)|A|p*/2, we prove the distributional identity (∂t−Δ)|A|p* =−4/(n−1) |Φs+(n−1)fs/fΦ|2. Integration against a conjugate heat kernel gives an anisotropic entropy formula. As an application, we obtain an entropy threshold for Type-I singularities on 𝕊n: if the entropy associated with a singular-time kernel is at most $1$ at an earlier regular time, every tangent flow at the corresponding singular point is a round shrinking sphere. This criterion follows from the 0/1 quantization of the singular-time anisotropic density.

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Cite

BibTeXRIS
BibTeX
@misc{guo2026anisotropic,
  title = {{An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow}},
  author = {Hongxin Guo and Guozhi Lin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37154},
  eprint = {2610.37154},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37154},
}
RIS
TY  - UNPB
TI  - An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow
AU  - Hongxin Guo
AU  - Guozhi Lin
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37154
UR  - https://arxiv.org/abs/2610.37154
AB  - For a rotationally symmetric Ricci flow g(t)=ds^2+f(s,t)^2g_𝕊^n−1, n≥3, we study the scale-invariant curvature anisotropy A=1−f_s^2+ff_ss introduced by Angenent--Knopf. At the critical power p_*=(n−1)/(n−2), with Φ=sgn(A)|A|^p_*/2, we prove the distributional identity (∂_t−Δ)|A|^p_* =−4/(n−1) |Φ_s+(n−1)f_s/fΦ|^2. Integration against a conjugate heat kernel gives an anisotropic entropy formula. As an application, we obtain an entropy threshold for Type-I singularities on 𝕊^n: if the entropy associated with a singular-time kernel is at most $1$ at an earlier regular time, every tangent flow at the corresponding singular point is a round shrinking sphere. This criterion follows from the 0/1 quantization of the singular-time anisotropic density.
ER  -