Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

By Qi-Rui Li, Weimin Sheng, Xu-jia Wang. Journal of the European Mathematical Society (Print), 2017

157 citations recorded by the source corpus.

In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with speed $f r^{\alpha} K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and $f$ is a positive and smooth function. If $\alpha \ge n+1$, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if $f \equiv 1$. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang (Acta Math. 216 (2016): 325-388), for the case $q 0$, we also establish the same results for even function $f$ and origin-symmetric initial condition, but for non-symmetric $f$, counterexample is given for the above smooth convergence.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/2017aa89ac49b981845884c01eb75beddaa0de2d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Yiming Zhao
2. Yong Huang
3. Yan-nan Liu
4. Jian Lu
5. Dongmeng Xi
6. Qi-Rui Li
7. Haodi Chen
8. Shibing Chen

In Astronomy and Astrophysics

1. Jian Lu
2. Yan-nan Liu
3. Yong Huang
4. Yiming Zhao
5. Haodi Chen
6. Qi-Rui Li

Citations in the corpus

• A flow method for the dual Orlicz–Minkowski problem (2020, methodology): “Our flow (6) is inspired by [46] in which a flow was used to study the dual Minkowski problem.”
• The Minkowski problem in Gaussian probability space (2020, background): “The dual Minkowski problem, while still largely open, has been solved in the o-symmetric case when the associated index q satisfies q ∈ [0, n], see, for example [8, 11, 15, 35, 38, 49, 74, 75].”