A unified flow approach to smooth, even Lp-Minkowski problems

Abstract

We study long-time existence and asymptotic behavior for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of smooth, even Lp-Minkowski problems in R n+1 for p > −n − 1.

Deforming a hypersurface by principal radii of curvature and support function

“…Since the solution to (4.1) depends continuously on the initial data, the nearby smooth, closed strictly convex hypersurfaces will lose smoothness. Since σ n (0, 0) = 0, the argument fails if k = n, as expected in view of the results in [3] and also Theorem 1.2 here. to be non-negative definite.…”

“…Later in [24], using the deformation lemma, Hu-Ma-Shen proved that if p ≥ k + 1, k < n and ϕ ∈ C 2 (S n ) is (p + k − 1)-convex, then (1.5) admits a positive strictly convex solution. 3 Recently, for 1 < p < k + 1 and for even prescribed data, under the (p + k − 1)-convexity of ϕ, an existence result was proved by Guan and Xia in [21] using a refined gradient estimate and the constant rank theorem.…”

“…2 Firey also explains in [14, page 11] how Pogorelov's condition connects to his. 3 The statement of Hu-Ma-Shen's theorem is erroneous and in item (i) it should be read "if…”

“…For p > 2, ϕ ≡ 1, the C 1 convergence follows from the work of Chow-Gulliver [9] (up-to showing convexity is preserved). For p = −n − 1, k = n, ϕ ≡ 1, the flow was studied in [12,13] and for p > −n − 1, k = n, ϕ ≡ 1 in [3]. Acknowledgment M.I.…”

A flow of conformally balanced metrics with Kähler fixed points

“…Also closely related is another fundamental question in non-Kähler geometry and algebraic-geometric stability conditions [16,47,77,78], namely when does a positive (p, p) cohomology class admit as representative the p-th power of a Kähler form. Finally, on Kähler manifolds the flow (1.1) can be viewed as a complex version of the inverse Gauss curvature flow studied extensively in convex geometry, see for example [2,5,12,13,25,75], and is in itself quite interesting as a fully non-linear partial differential equation. More details on all these motivations will be provided in Section §2 below.…”