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A Curvature Flow Approach to \(L^p\) Christoffel-Minkowski Problem for \(1<p<k+1\)

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Abstract

In this paper, we study an anisotropic expanding flow of smooth, closed, uniformly convex hypersurfaces in \(\mathbb {R}^{n+1}\) with speed \(\psi \sigma _k^{\alpha }(\lambda )\), where \(\alpha >\frac{1}{k}\) is a constant, \(\sigma _k(\lambda )\) is the k-th elementary symmetric polynomial of the principal radii of curvature and \(\psi \) is a preassigned positive smooth function defined on \(\mathbb {S}^n\). We prove that under some assumptions of \(\psi \), the solution to the flow after normalisation exists for all time and converges smoothly to a solution of the well-known \(L^p\) Christoffel-Minkowski problem \(u^{1-p}(x) \sigma _k (\nabla ^2u+uI)=c\psi (x)\) for \(1<p<k+1\).

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Acknowledgements

The author would like to thank Professor Haizhong Li for suggesting the problem and many valuable comments. The author also thanks Professor Yuguang Shi for his constant encouragement and support.

Funding

The author was supported by the National Key R &D Program of China 2020YFA0712800 and the China Postdoctoral Science Foundation No. 2023M740108.

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Authors and Affiliations

  1. Key Laboratory of Pure and Applied Mathematics, School of Mathematical SciencesPeking University, Beijing, 100871, P. R. China

    Ruijia Zhang

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  1. Ruijia Zhang
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Zhang, R. A Curvature Flow Approach to \(L^p\) Christoffel-Minkowski Problem for \(1<p<k+1\). Results Math 79, 53 (2024). https://doi.org/10.1007/s00025-023-02069-0

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