Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

  • Qi-Rui Li

    The Australian National University, Canberra, Australia and Zhejiang University, Hangzhou, China
  • Weimin Sheng

    Zhejiang University, Hangzhou, China and The Australian National University, Canberra, Australia
  • Xu-Jia Wang

    The Australian National University, Canberra, Australia
Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems cover
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Abstract

In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space Rn+1\mathbb R^{n+1} with speed frαKf r^{\alpha} K, where KK is the Gauss curvature, rr is the distance from the hypersurface to the origin, and ff is a positive and smooth function. If αn+1\alpha \ge n+1, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere if f1f \equiv 1. Our argument provides a new proof in the smooth category for the classical Aleksandrov problem, and resolves the dual qq-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang [30] for q<0q < 0. If α<n+1\alpha < n+1, corresponding to the case q>0q > 0, we also establish the same results for even function ff and origin-symmetric initial condition, but for non-symmetric ff, counterexample is given for the above smooth convergence.

Cite this article

Qi-Rui Li, Weimin Sheng, Xu-Jia Wang, Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems. J. Eur. Math. Soc. 22 (2020), no. 3, pp. 893–923

DOI 10.4171/JEMS/936