A class of inverse curvature flows and 𝐿^{𝑝} dual Christoffel-Minkowski problem

Abstract

In this paper, we consider a large class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space R n + 1 \mathbb {R}^{n+1} with speed ψ u α ρ δ f − β \psi u^\alpha \rho ^\delta f^{-\beta } , where ψ \psi is a smooth positive function on unit sphere, u u is the support function of the hypersurface, ρ \rho is the radial function, f 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 \psi =1 , we prove that the flow exists for all time and converges to infinity if α + δ + β ⩽ 1 \alpha +\delta +\beta \leqslant 1 , and α ⩽ 0 > β \alpha \leqslant 0>\beta , while in case α + δ + β > 1 \alpha +\delta +\beta >1 , α , δ ⩽ 0 > β \alpha ,\delta \leqslant 0>\beta , 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 at the origin. In particular, the results of Gerhardt [J. Differential Geom. 32 (1990), pp. 299–314; Calc. Var. Partial Differential Equations 49 (2014), pp. 471–489] and Urbas [Math. Z. 205 (1990), pp. 355–372] can be recovered by putting α = δ = 0 \alpha =\delta =0 . Our previous works [Proc. Amer. Math. Soc. 148 (2020), pp. 5331–5341; J. Funct. Anal. 282 (2022), p. 38] and Hu, Mao, Tu and Wu [J. Korean Math. Soc. 57 (2020), pp. 1299–1322] can be recovered by putting δ = 0 \delta =0 and α = 0 \alpha =0 respectively. By the convergence of these flows, we can give a new proof of uniqueness theorems for solutions to L p L^p -Minkowski problem and L p L^p -Christoffel-Minkowski problem with constant prescribed data. Similarly, we consider the L p L^p dual Christoffel-Minkowski problem and prove a uniqueness theorem for solutions to L p L^p dual Minkowski problem and L p L^p dual Christoffel-Minkowski problem with constant prescribed data. At last, we focus on the long time existence and convergence of a class of anisotropic flows (i.e. for general function ψ \psi ). The final result not only gives a new proof of many previously known solutions to L p L^p dual Minkowski problem, L p L^p -Christoffel-Minkowski problem, etc. by such anisotropic flows, but also provides solutions to L p L^p dual Christoffel-Minkowski problem with some conditions.