Ricci flow with metric torsion on surfaces of positive Euler characteristic
Abstract
We study an adapted Ricci flow of connections with metric torsion on surfaces with positive Euler characteristic. We first prove that there do not exist any nontrivial solitons of the flow on the $2$-sphere thus confirming a conjecture of Branding--Kröncke (J. Geom. Anal. 27.3 (2017), arXiv:1606.09121). We give an explicit family of torsion data for which the corresponding global solutions fail to converge on 𝕊2. Nevertheless, we provide several sufficient conditions for the convergence of the flow to a stationary point. We first prove that the normalized adapted Ricci flow always converges on ℝℙ2, which completely answers a question in the paper of Branding and Kröncke. Using this, we deduce that the flow converges on 𝕊2 whenever the initial metric and the torsion one-form are antipodally symmetric. We also prove a Łojasiewicz--Simon gradient inequality for the flow and use it to prove convergence to a stationary point provided the solution is close to an arbitrary stationary point.
Related
- K3 surfaces with finite automorphism group in positive characteristic
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- Kaplansky's conjectures fail for torsion-free hyperbolic groups in positive characteristic
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BibTeX
@misc{dwivedi2026ricci,
title = {{Ricci flow with metric torsion on surfaces of positive Euler characteristic}},
author = {Shubham Dwivedi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15880},
eprint = {2609.15880},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15880},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ricci flow with metric torsion on surfaces of positive Euler characteristic
AU - Shubham Dwivedi
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15880
UR - https://arxiv.org/abs/2609.15880
AB - We study an adapted Ricci flow of connections with metric torsion on surfaces with positive Euler characteristic. We first prove that there do not exist any nontrivial solitons of the flow on the $2$-sphere thus confirming a conjecture of Branding--Kröncke (J. Geom. Anal. 27.3 (2017), arXiv:1606.09121). We give an explicit family of torsion data for which the corresponding global solutions fail to converge on 𝕊^2. Nevertheless, we provide several sufficient conditions for the convergence of the flow to a stationary point. We first prove that the normalized adapted Ricci flow always converges on ℝℙ^2, which completely answers a question in the paper of Branding and Kröncke. Using this, we deduce that the flow converges on 𝕊^2 whenever the initial metric and the torsion one-form are antipodally symmetric. We also prove a Łojasiewicz--Simon gradient inequality for the flow and use it to prove convergence to a stationary point provided the solution is close to an arbitrary stationary point.
ER -