Asymptotic behavior of flows by powers of the Gaussian curvature

By S. Brendle, Kyeongsu Choi, P. Daskalopoulos. 2016

199 citations recorded by the source corpus.

We consider a one-parameter family of strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ moving with speed $- K^\alpha \nu$, where $\nu$ denotes the outward-pointing unit normal vector and $\alpha \geq \frac{1}{n+2}$. For $\alpha > \frac{1}{n+2}$, we show that the flow converges to a round sphere after rescaling. In the affine invariant case $\alpha=\frac{1}{n+2}$, our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bd1fd827b92b85bd4b015b2de5bc3ecc0f705deb

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

In Geometry and Topology

Citations in the corpus

The Lp dual Minkowski problem and related parabolic flows (2021, methodology): “When q = n+ 1, this is to ask if the sphere is the only soliton to the α-Gauss curvature flow [4, 12], and an affirmative answer was obtained for p ∈ (−n− 1, 1) [12] and for p ≥ 1 [36].”
A flow method for the dual Orlicz–Minkowski problem (2020, unknown): “[2, 3, 9, 10, 13, 16, 19, 24, 25, 37, 61, 62] and references therein.”
A unified flow approach to smooth, even Lp-Minkowski problems (2016, background): “Moreover, note that when φ ≡ 1 and −n− 1 < p < 1, by the result of [15], the limit is the unit sphere.”
Variations of a class of Monge–Ampère-type functionals and their applications (2021, background): “4) when α > 1 n+2 [1, 2, 8], which makes our paper self-contained.”