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 $\mathbb {R}^{n+1}$ with speed $\psi u^\alpha \rho ^\delta f^{-\beta }$, where $\psi$ is a smooth positive function on unit sphere, $u$ is the support function of the hypersurface, $\rho$ 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 $\psi =1$, we prove that the flow exists for all time and converges to infinity if $\alpha +\delta +\beta \leqslant 1$, and $\alpha \leqslant 0<\beta$, while in case $\alpha +\delta +\beta >1$, $\alpha ,\delta \leqslant 0<\beta$, 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 $\alpha =\delta =0$. 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 $\delta =0$ and $\alpha =0$ respectively. By the convergence of these flows, we can give a new proof of uniqueness theorems for solutions to $L^p$-Minkowski problem and $L^p$-Christoffel-Minkowski problem with constant prescribed data. Similarly, we consider the $L^p$ dual Christoffel-Minkowski problem and prove a uniqueness theorem for solutions to $L^p$ dual Minkowski problem and $L^p$ 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 $\psi$). The final result not only gives a new proof of many previously known solutions to $L^p$ dual Minkowski problem, $L^p$-Christoffel-Minkowski problem, etc. by such anisotropic flows, but also provides solutions to $L^p$ dual Christoffel-Minkowski problem with some conditions.