Optimal geometric inequalities and fully nonlinear conformal flows

Abstract

We establish sharp Sobolev-type geometric inequalities on ${\mathbb{S}}^n$ involving the total ${\sigma }_k$-curvatures ${\int }_{{\mathbb{S}}^n}{\sigma }_k(g)d{v}_g$. These results extend the optimal inequalities of Guan--Wang~\cite{GWDuke} from the cone ${\mathcal{C}}_k$ to the strictly larger cone ${\mathcal{C}}_{k-1}$, thereby enlarging the range of admissible conformal metrics. Our approach is variational and is implemented through a fully nonlinear conformal flow. Working in ${\mathcal{C}}_{k-1}$ 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 ${\mathcal{C}}_{k-2}$.

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