2022/03/04 by Shanwei Ding, Ding, Shanwei, Guanghan Li +1
Mathematics · #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2203.02165
In this paper, we consider a large class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space ℝn+1 with speed ψuαρδf-β, where ψ is a smooth positive function on unit sphere, u is the support function of the hypersurface, ρ is the radial function, f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. When ψ=1, we prove that the flow exists for all time and converges to infinity if α+δ+β≤1, β>0 and α≤0, while in case α+δ+β>1,α,δ≤0, the flow blows up in finite time, and where we assume the initial hypersurface to be strictly convex. In both cases the properly rescaled flows converge to a sphere centered the origin. In particular, the results of Gerhardt \citeGC,GC3 and Urbas \citeUJ2 can be recovered by putting α=δ=0. Our previous works \citeDL,DL2 can be recovered by putting δ=0. By the convergence of these flows, we can give a new proof of uniqueness theorems for solutions to Lp-Minkowski problem and Lp-Christoffel-Minkowski problem with constant prescribed data. Similarly, we pose the Lp dual Christoffel-Minkowski problem and prove a uniqueness theorem for solutions to Lp dual Minkowski problem and Lp dual Christoffel-Minkowski problem with constant prescribed data. At last, we focus on the longtime existence and convergence of a class of anisotropic flows (i.e. for general function ψ). The final result not only gives a new proof of many previously known solutions to Lp dual Minkowski problem, Lp-Christoffel-Minkowski problem, etc. by such anisotropic flows, but also provides solutions to Lp dual Christoffel-Minkowski problem with some conditions.