# A class of anisotropic expanding curvature flows

## TL;DR

This paper studies a class of expanding curvature flows for convex hypersurfaces in Euclidean space, proving long-time existence, convergence to solitons, and connecting to the Lp Christoffel-Minkowski problem.

## Contribution

It introduces a new class of anisotropic expanding curvature flows and establishes their long-term behavior and convergence, linking to the Lp Christoffel-Minkowski problem.

## Key findings

- Flow has a unique smooth, convex solution for all time.
- Flow converges smoothly to a soliton solution.
- Provides a proof for the Lp Christoffel-Minkowski problem for p >= k+1.

## Abstract

We consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_k^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_k is the k-th symmetric polynomial of the principle curvature radii of the hypersurface, k is an integer and 1<= k<= n. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of a elliptic equation, when the constants alpha, beta belong to a suitable range, and the function f satisfies a strictly spherical convexity condition. When beta=1, the soliton equation is just the equation of Lp Christoffel-Minkowski problem. Thus our argument provides a proof to the well-known L_p Christoffel-Minkowski problem for the case p>= k+1 where p=2-alpha, which is identify with Ivaki's recent result.