Analogue of the Gauss-curvature variational functional for intermediate curvature flows

Develop an analogue, for the intermediate-curvature flows involving $\sigma_k$ with $k<n$, of the variational functional available in the Gauss-curvature case, in order to provide a corresponding convergence tool for general data.

Background

The paper studies normalized contracting flows with speed $f(\nu)r^{\alpha}\sigma_k(\kappa)$ in the supercritical range $\alpha>k+1$, where $f$ is an arbitrary positive smooth angular function. Its convergence proof does not rely on a variational structure; instead, it uses the pointwise relative residual $p=\partial_t\log u$ and establishes exponential decay through a maximum-principle argument.

For the Gauss-curvature case $k=n$, previous work used a variational functional to analyze convergence. The authors state that no corresponding functional is known for the intermediate-curvature cases $k<n$. Constructing such a functional would provide an alternative analytical framework for convergence and could clarify the variational structure of these anisotropic curvature flows.