Extension to general inverse quotient curvature flows

Extend the long-time existence theory and corresponding counterexample analysis from the inverse quotient curvature flow considered in the paper to general inverse quotient curvature flows.

Background

The paper establishes long-time existence and convergence for the inverse σk\sigma_k curvature flow of entire spacelike strictly convex hypersurfaces under specified asymptotic subsolution and curvature lower-bound conditions. It also constructs counterexamples for the inverse quotient curvature flow σn−1/σn\sigma_{n-1}/\sigma_n, showing finite-time interior singularity formation and the existence of quotient self-shrinkers with interior curvature singularities.

The authors explicitly note that their examples cannot currently be extended to general inverse quotient curvature flows. Determining whether analogous constructions or a broader theory can be obtained for general quotient flows remains unresolved.

References

Note that we still cannot extend the above examples to the general inverse quotient curvature flows.

— Inverse Hessian Curvature Flow in Minkowski Space II: The Dirichlet problem at infinity  (2608.20037 - Li et al., 20 Aug 2026) in Section 1, immediately after Theorem 1.3 (Theorem \ref{Counterexample})

However, for conformal metrics lying only in $\mathcal C_{k-1}$, neither of these flows is known to be suitable (for example, admissibility or parabolicity may fail).

— Optimal geometric inequalities and fully nonlinear conformal flows  (2609.11421 - Ge et al., 10 Sep 2026) in Section 1, paragraph introducing the perturbed conformal flow