2024/11/12 by Fong, Frederick Tsz-Ho
#53E10 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2411.08198
In this article, we establish some uniqueness and symmetry results of self-similar solutions to curvature flows by some homogeneous speed functions of principal curvatures in some warped product spaces. In particular, we proved that any compact star-shaped self-similar solution to any parabolic flow with homogeneous degree -1 (including the inverse mean curvature flow) in warped product spaces I ×ϕ Mn, where Mn is a compact homogeneous manifold and ϕ'' ≥ 0, must be a slice. The same result holds for compact self-expanders when the degree of the speed function is greater than -1 and with an extra assumption ϕ' ≥ 0. Furthermore, we also show that any complete non-compact star-shaped, asymptotically concial expanding self-similar solutions to the flow by positive power of mean curvature in hyperbolic and anti-deSitter-Schwarzschild spaces are rotationally symmetric.