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Optimal geometric inequalities and fully nonlinear conformal flows

  • Yuxin Ge
  • Guofang Wang
  • Wei Wei
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Abstract

We establish sharp Sobolev-type geometric inequalities on π•Šn involving the total Οƒk-curvatures βˆ«π•ŠnΟƒk(g) dvg. These results extend the optimal inequalities of Guan--Wang~\cite{GWDuke} from the cone π’žk to the strictly larger cone π’žkβˆ’1, thereby enlarging the range of admissible conformal metrics. Our approach is variational and is implemented through a fully nonlinear conformal flow. Working in π’žkβˆ’1 introduces substantial analytic difficulties; in particular, one must obtain C2 a priori estimates while simultaneously verifying that the flow remains parabolic. We resolve these issues via a carefully designed test function and by applying the maximum principle to the maximal eigenvalue of the Hessian matrix. As applications, we solve two open problems in dimensions 3 and 4. Finally, we give examples to show that these inequalities cannot be extended to π’žkβˆ’2.

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Cite

BibTeXRIS
BibTeX
@misc{ge2026optimal,
  title = {{Optimal geometric inequalities and fully nonlinear conformal flows}},
  author = {Yuxin Ge and Guofang Wang and Wei Wei},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11421},
  eprint = {2609.11421},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11421},
}
RIS
TY  - UNPB
TI  - Optimal geometric inequalities and fully nonlinear conformal flows
AU  - Yuxin Ge
AU  - Guofang Wang
AU  - Wei Wei
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11421
UR  - https://arxiv.org/abs/2609.11421
AB  - We establish sharp Sobolev-type geometric inequalities on π•Š^n involving the total Οƒ_k-curvatures ∫_π•Š^nΟƒ_k(g) dv_g. These results extend the optimal inequalities of Guan--Wang~\cite{GWDuke} from the cone π’ž_k to the strictly larger cone π’ž_kβˆ’1, thereby enlarging the range of admissible conformal metrics. Our approach is variational and is implemented through a fully nonlinear conformal flow. Working in π’ž_kβˆ’1 introduces substantial analytic difficulties; in particular, one must obtain C^2 a priori estimates while simultaneously verifying that the flow remains parabolic. We resolve these issues via a carefully designed test function and by applying the maximum principle to the maximal eigenvalue of the Hessian matrix. As applications, we solve two open problems in dimensions 3 and 4. Finally, we give examples to show that these inequalities cannot be extended to π’ž_kβˆ’2.
ER  -