Optimal geometric inequalities and fully nonlinear conformal flows

Abstract: We establish sharp Sobolev-type geometric inequalities on $\mathbb{S}<sup>n$ involving the total $σ<em>k$-curvatures $\int</em>{\mathbb{S}<sup>n}σ_k(g)\,dv_g$. These results extend the optimal inequalities of Guan--Wang~\cite{GWDuke} from the cone $\mathcal{C}<em>k$ to the strictly larger cone $\mathcal{C}</em>{k-1}$, thereby enlarging the range of admissible conformal metrics. Our approach is variational and is implemented through a fully nonlinear conformal flow. Working in $\mathcal{C}<em>{k-1}$ introduces substantial analytic difficulties; in particular, one must obtain $C<sup>2$ 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 $\mathcal{C}</em>{k-2}$.