Deforming a hypersurface by principal radii of curvature and support function

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 kth 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 $$1\le k\le n-1, p\ge k+1$$ , 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 and Ma (Invent Math 151:553–577, 2003) and thus we give a partial answer to a question raised in Guan and Xia (Calc Var Part Differ Equ 57:69, 2018. https://doi.org/10.1007/s00526-018-1341-y . Moreover, for $$k=n, p\ge n+1$$ , we prove the $$C^{\infty }$$ convergence of the normalized flow to a homothetic self-similar solution without imposing any condition on $$\varphi .$$ In the final section of the paper, for $$1\le k<n$$ , 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.