Asymptotic convergence for modified scalar curvature flow

In this paper, we study the flow of closed, star-shaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^\alpha \sigma^{1/2}_{2}$, where $\sigma^{1/2}_{2}$ is the normalized square root of the scalar curvature, $\alpha \geq 2$, and $r$ is the distance from points on the hypersurface to the origin. We prove that the flow exists for all time and the star-shapedness is preserved. Moreover, after normalization, we show that the flow converges exponentially fast to a sphere centered at origin. When $\alpha \lt 2$, a counter-example is given for the above convergence.

Full Text (PDF format)

Received 19 May 2019

Accepted 12 August 2020

Published 21 September 2023