A geometric flow associated to Q-curvature
Abstract
Branson's Q-curvature is a natural fourth-order counterpart of scalar curvature in conformal geometry. In analogy with the classical recovery of the Ricci tensor from the linearized scalar curvature, Lin and Yuan introduced a symmetric $2$-tensor Jg through the formal adjoint of the linearized Q-curvature and established its explicit decomposition in terms of the Bach tensor and auxiliary curvature terms. In this paper, we study the associated fourth-order geometric flow ∂t g = −2Jg on closed Riemannian manifolds of dimension n ≥ 4. We establish short-time existence and uniqueness using a biharmonic DeTurck gauge and derive global and local integral smoothing estimates under curvature bounds. In dimension four, we prove long-time existence assuming a uniform Sobolev inequality and sufficiently small total L2 curvature energy throughout the maximal interval of existence. Under uniform curvature and Sobolev bounds, time integrability of the L2-norm of J implies smooth convergence in fixed coordinates to a J-flat metric. Explicit Einstein and product solutions illustrate the evolution and show that small curvature energy alone does not prevent degeneration. Positive initial Yamabe constant and positive total Q-curvature give a uniform positive Yamabe lower bound throughout the smooth four-dimensional flow, whereas in higher dimensions total Q-curvature can decrease even near flat metrics. Finally, we prove that every closed four-dimensional J-soliton is trivial, its metric is Bach-flat with constant Q-curvature, and its soliton vector field is Killing.
Related
- Geometric C1-stability of geodesic spheres under volume-preserving mean curvature flow
- Eigenvectors of p-Curvature for Geometric Difference Equations
- Jacobian estimates and geometric inequalities under intermediate Ricci curvature
- The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow
- Singularities of Mean Curvature Flow with Bounded Total Curvature
BibTeX
@misc{chen2026geometric,
title = {{A geometric flow associated to $Q$-curvature}},
author = {Jiaqi Chen and Xudong Kang and Wei Yuan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02993},
eprint = {2610.02993},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02993},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A geometric flow associated to Q-curvature
AU - Jiaqi Chen
AU - Xudong Kang
AU - Wei Yuan
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02993
UR - https://arxiv.org/abs/2610.02993
AB - Branson's Q-curvature is a natural fourth-order counterpart of scalar curvature in conformal geometry. In analogy with the classical recovery of the Ricci tensor from the linearized scalar curvature, Lin and Yuan introduced a symmetric $2$-tensor J_g through the formal adjoint of the linearized Q-curvature and established its explicit decomposition in terms of the Bach tensor and auxiliary curvature terms. In this paper, we study the associated fourth-order geometric flow ∂_t g = −2J_g on closed Riemannian manifolds of dimension n ≥ 4. We establish short-time existence and uniqueness using a biharmonic DeTurck gauge and derive global and local integral smoothing estimates under curvature bounds. In dimension four, we prove long-time existence assuming a uniform Sobolev inequality and sufficiently small total L^2 curvature energy throughout the maximal interval of existence. Under uniform curvature and Sobolev bounds, time integrability of the L^2-norm of J implies smooth convergence in fixed coordinates to a J-flat metric. Explicit Einstein and product solutions illustrate the evolution and show that small curvature energy alone does not prevent degeneration. Positive initial Yamabe constant and positive total Q-curvature give a uniform positive Yamabe lower bound throughout the smooth four-dimensional flow, whereas in higher dimensions total Q-curvature can decrease even near flat metrics. Finally, we prove that every closed four-dimensional J-soliton is trivial, its metric is Bach-flat with constant Q-curvature, and its soliton vector field is Killing.
ER -