# An Anisotropic shrinking flow and L_p Minkowski problem

## TL;DR

This paper studies a geometric flow of convex hypersurfaces driven by a speed involving support functions and curvature, proving convergence to solutions of elliptic equations, and applying results to the L_p Minkowski problem.

## Contribution

It introduces a new anisotropic shrinking flow framework and establishes convergence results, including solutions to the L_p Minkowski problem for a broad range of parameters.

## Key findings

- Flow converges to a smooth soliton under certain conditions.
- Unique solutions to associated elliptic equations are obtained.
- Provides a uniform proof for the existence of solutions to the L_p Minkowski problem.

## Abstract

We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_n^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_n is the n-th symmetric polynomial of the principle curvature radii of the hypersurface. 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 an elliptic equation, when the constants alpha, beta belong to a suitable range, provided the initial hypersuface is origin-symmetric and f is a smooth positive even function on S^n. For the case alpha>= 1+n*beta, beta>0, we prove that the flow converges smoothly after normalisation to a unique smooth solution of an elliptic equation without any constraint on the initial hypersuface and smooth positive function f. When beta=1, our argument provides a uniform proof to the existence of the solutions to the equation of L_p Minkowski problem for p belongs to (-n-1,+infty).