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A class of curvature flows expanded by support function and curvature function


Authors: Shanwei Ding and Guanghan Li
Journal: Proc. Amer. Math. Soc. 148 (2020), 5331-5341
MSC (2010): Primary 53C44
DOI: https://doi.org/10.1090/proc/15189
Published electronically: September 18, 2020
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Abstract: In this paper, we consider a class of expanding flows of closed, smooth, uniformly convex hypersurfaces in Euclidean $ \mathbb{R}^{n+1}$ with speed $ u^\alpha f^\beta $ ( $ \alpha , \beta \in \mathbb{R}^1$), where $ u$ is the support function of the hypersurface, $ f$ is a smooth, symmetric, homogenous of degree one, positive function of the principal curvature radii of the hypersurface. If $ \alpha \leqslant 0<\beta \leqslant 1-\alpha $, we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalization, to a round sphere centered at the origin.


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Additional Information

Shanwei Ding
Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China
ORCID: 0000-0002-8383-5219

Guanghan Li
Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China

DOI: https://doi.org/10.1090/proc/15189
Keywords: Expanding flow, asymptotic behaviour, support function, curvature function
Received by editor(s): March 20, 2020
Received by editor(s) in revised form: May 14, 2020
Published electronically: September 18, 2020
Additional Notes: This research was partially supported by NSFC (Nos. 11761080 and 11871053).
Communicated by: Jia-Ping Wang
Article copyright: © Copyright 2020 American Mathematical Society