Deforming a hypersurface by principal radii of curvature and support function

TL;DR

This paper investigates the evolution of convex hypersurfaces in Euclidean space driven by curvature-dependent speeds, establishing convergence to self-similar solutions under certain conditions and providing examples of loss of smoothness.

Contribution

It proves convergence of the normalized flow to self-similar solutions without relying on the constant rank theorem, addressing a question in prior research.

Findings

Flow converges smoothly to self-similar solutions under specified conditions.

Convergence results hold for various ranges of parameters p and k.

An example shows loss of smoothness when certain convexity conditions are violated.

Abstract

We study the motion of smooth, closed, strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ expanding in the direction of their normal vector field with speed depending on the $k$th elementary symmetric polynomial of the principal radii of curvature $\sigma_k$ and support function $h$. A homothetic self-similar solution to the flow that we will consider in this paper, if exists, is a solution of the well-known $L_p$-Christoffel-Minkowski problem $\varphi h^{1-p}\sigma_k=c$. Here $\varphi$ is a preassigned positive smooth function defined on the unit sphere, and $c$ is a positive constant. For $1\leq k\leq n-1, p\geq k+1$, assuming the spherical hessian of $\varphi^{\frac{1}{p+k-1}}$ is positive definite, we prove the $C^{\infty}$ convergence of the normalized flow to a homothetic self-similar solution. One of the highlights of our arguments is that we do not need the constant rank theorem/deformation lemma of Guan-Ma and thus we give a partial answer to a question raised in Guan-Xia. Moreover, for $k=n, p\geq n+1$, we prove the $C^{\infty}$ convergence of the normalized flow to a homothetic self-similar solution without imposing any further condition on $\varphi.$ In the final section of the paper, for $1\leq k<n$, we will give an example that spherical hessian of $\varphi^{\frac{1}{p+k-1}}$ is negative definite at some point and the solution to the flow loses its smoothness.