Some remarks on a class of logarithmic curvature flow

TL;DR

This paper introduces a new class of logarithmic curvature flows aimed at solving the weighted Christoffel-Minkowski problem, emphasizing the importance of curvature bounds and a priori estimates for convergence.

Contribution

It proposes a novel logarithmic curvature flow framework and highlights the key challenge of establishing curvature bounds for convergence analysis.

Findings

Established a priori estimates for the flow

Identified the critical role of curvature bounds

Outlined the main obstacle in proving convergence

Abstract

In this paper, we introduce a class of new logarithmic curvature flow. The flows are designed to embrace the monotonicity of the related functional, and the convergence of this flow would tackle the solvability of the weighted Christoffel-Minkowski problem, but a full proof scheme is missing, the key factor of forming this phenomenon lies in the establishment of the upper bound of the principal curvature, which essentially depends on finding a clean condition on smooth positive function defined on the unit sphere $\sn$. Except for obtaining this tricky estimate, we get all the other a priori estimates and hope that this note can attract wide attention to this interesting issue.