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| Format: | Preprint |
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2018
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| Online Access: | https://arxiv.org/abs/1803.08470 |
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| _version_ | 1866912453827231744 |
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| author | Ivaki, Mohammad N. |
| author_facet | Ivaki, Mohammad N. |
| contents | 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 $σ_k$ 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 $φh^{1-p}σ_k=c$. Here $φ$ 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 $φ^{\frac{1}{p+k-1}}$ 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 $φ.$ In the final section of the paper, for $1\leq k<n$, we will give an example that spherical hessian of $φ^{\frac{1}{p+k-1}}$ is negative definite at some point and the solution to the flow loses its smoothness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1803_08470 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Deforming a hypersurface by principal radii of curvature and support function Ivaki, Mohammad N. Analysis of PDEs 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 $σ_k$ 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 $φh^{1-p}σ_k=c$. Here $φ$ 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 $φ^{\frac{1}{p+k-1}}$ 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 $φ.$ In the final section of the paper, for $1\leq k<n$, we will give an example that spherical hessian of $φ^{\frac{1}{p+k-1}}$ is negative definite at some point and the solution to the flow loses its smoothness. |
| title | Deforming a hypersurface by principal radii of curvature and support function |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1803.08470 |