Deforming a Convex Hypersurface by Anisotropic Curvature Flows

1 Introduction

Let M0 be a smooth, closed, strictly convex hypersurface in the Euclidean space Rn, which encloses the origin and is given by a smooth embedding X0:Sn-1→Rn. Consider a family of closed hypersurfaces {Mt} with Mt=X⁢(Sn-1,t), where X:Sn-1×[0,T)→Rn is a smooth map satisfying the initial value problem(1.1)

Geometric flows with speed of symmetric polynomial of the principal curvature radii of the hypersurface have been extensively studied; see e.g. [34, 8, 11, 36].

On the other hand, anisotropic curvature flows provide alternative methods to prove the existences of elliptic PDEs arising from convex geometry; see e.g. [3, 4, 5, 6, 21, 26, 27, 28, 33].

A positive homothetic self-similar solution of (1.1), if it exists, is a solution to the fully nonlinear equation

for some positive constant 𝑐. Here ℎ is the support function defined on Sn-1. We are concerned with the existence of smooth solutions for equation (1.2).

When k=n-1, equation (1.2) is just the smooth case of the Orlicz–Minkowski problem. The Orlicz–Minkowski problem is a basic problem in the Orlicz–Brunn–Minkowski theory in convex geometry. It is a generalization of the classical Minkowski problem which asks what are the necessary and sufficient conditions for a Borel measure on the unit sphere Sn-1 to be a multiple of the Orlicz surface area measure of a convex body in Rn. In [14], Haberl, Lutwak, Yang and Zhang studied the even case of the Orlicz–Minkowski problem. After that, the Orlicz–Minkowski problem attracted great attention from many scholars; see for example [9, 10, 19, 22, 37, 35, 39].

When φ⁢(s)=s1-p, k=n-1, equation (1.2) reduces to the Lp-Minkowski problem, which has been extensively studied; see e.g. [2, 7, 16, 18, 20, 23, 24, 29, 30, 31, 38].

When 1≤k<n-1, equation (1.2) is the so-called Orlicz–Christoffel–Minkowski problem. For φ⁢(s)=s1-p, 1≤k<n-1, (1.2) is known as the Lp-Christoffel–Minkowski problem and is the classical Christoffel–Minkowski problem for p=1. Under a sufficient condition on the prescribed function, existence of the solution for the classical Christoffel–Minkowski problem was given in [12].

The Lp-Christoffel–Minkowski problem is related to the problem of prescribing the 𝑘-th 𝑝-area measures. Hu, Ma and Shen [17] proved the existence of convex solutions to the Lp-Christoffel–Minkowski problem for p≥k+1 under appropriate conditions. Using the methods of geometric flows, Ivaki [21] and then Sheng and Yi [33] also gave the existence of smooth convex solutions to the Lp-Christoffel–Minkowski problem for p≥k+1. In case 1<p<k+1, Guan and Xia [13] established the existence of convex body with the prescribed 𝑘-th even 𝑝-area measures.

In this paper, we study the long-time existence and convergence of flow (1.1) for strictly convex hypersurfaces and the existence of smooth solutions to the Orlicz–Christoffel–Minkowski problem (1.2).

The scalar function η⁢(t) in (1.1) is usually used to keep Mt normalized in a certain sense; see for example [4, 21, 33]. In this paper, 𝜂 is given by

where h⁢(⋅,t) is the support function of the convex hypersurface Mt. It will be proved in Section 2 that ∫Sn-1h⁢σk⁢dx is non-decreasing along the flow under this choice of 𝜂.

To obtain the long-time existence of flow (1.1), we need some constraints on 𝜑.

1. φ⁢(s) is a positive and continuous function defined in (0,+∞) such that φ>α⁢s-k-ε for some positive constants 𝜀 and 𝛼 for 𝑠 near 0 and ϕ⁢(s)=∫0s1φ⁢(τ)⁢dτ is unbounded as s→+∞. Here 𝑘 is the order of σk.

When f≡1, we have the following result.

As an application, we have the following corollary.

From the proof of Lemma 7 in Section 3, we will see if φ′⁢(s)⁢sφ⁢(s)=a0 for some negative constant a0, then the convexity condition on 𝑓 reduces to f-1k-a0⁢ei⁢j+∇j⁡∇j⁡(f-1k-a0) being positive definite. Hence, when φ⁢(s)=s1-p for p≥k+1 with the above condition on 𝑓, our conclusion recovers the existence results to the Lp-Christoffel–Minkowski problem which have been obtained in [17, 21, 33].

This paper is organized as follows. In Section 2, we give some basic knowledge about flow (1.1) and evolution equations of some geometric quantities. In Section 3, the long-time existence of flow (1.1) will be obtained. First, under assumption (A), uniform positive upper and lower bounds for support functions of {Mt} are derived. Based on the bounds of support functions, we obtain the uniform bounds of principal curvatures by constructing proper auxiliary functions. The long-time existence of flow (1.1) then follows by standard arguments. In Section 4, by considering a related geometric functional, we prove that a subsequence of {Mt} converges to a smooth solution to equation (1.2), completing the proofs of Theorem 1 and Theorem 2.

2 Preliminaries

Let Rn be the 𝑛-dimensional Euclidean space, and let Sn-1 be the unit sphere in Rn. Assume 𝑀 is a smooth closed strictly convex hypersurface in Rn. Without loss of generality, we may assume that 𝑀 encloses the origin. The support function ℎ of 𝑀 is defined as h⁢(x):=maxy∈M⁡⟨y,x⟩ for all x∈Sn-1, where ⟨⋅,⋅⟩ is the standard inner product in Rn.

Denote the Gauss map of 𝑀 by νM. Then 𝑀 can be parametrized by the inverse Gauss map X:Sn-1→M with X⁢(x)=νM-1⁢(x). The support function ℎ of 𝑀 can be computed by

Note that 𝑥 is just the unit outer normal of 𝑀 at X⁢(x). Differentiating (2.1), we have

Since ∇i⁡X⁢(x) is tangent to 𝑀 at X⁢(x), we have ∇i⁡h=⟨∇i⁡x,X⁢(x)⟩. It follows that

By differentiating (2.1) twice, the second fundamental form Ai⁢j of 𝑀 can be computed in terms of the support function (see for example [34]),

where ∇i⁢j=∇i⁡∇j denotes the second-order covariant derivative with respect to ei⁢j. The induced metric matrix gi⁢j of 𝑀 can be derived by Weingarten’s formula,

The principal radii of curvature are the eigenvalues of the matrix bi⁢j=Ai⁢k⁢gj⁢k. When considering a smooth local orthonormal frame on Sn-1, by virtue of (2.3) and (2.4), we have

We will use bi⁢j to denote the inverse matrix of bi⁢j.

From the evolution equation of X⁢(x,t) in flow (1.1), we derive the evolution equation of the corresponding support function h⁢(x,t),

The radial function 𝜌 of 𝑀 is given by ρ⁢(u):=max⁡{λ>0:λ⁢u∈M} for all u∈Sn-1. Note that ρ⁢(u)⁢u∈M. From (2.2), 𝑢 and 𝑥 are related by ρ⁢(u)⁢u=∇⁡h⁢(x)+h⁢(x)⁢x and ρ2=|∇⁡h|2+h2.

In the rest of the paper, we take a local orthonormal frame {e1,…,en-1} on Sn-1 such that the standard metric on Sn-1 is {δi⁢j}. Double indices always mean to sum from 1 to n-1. We denote partial derivatives ∂⁡σk∂⁡bi⁢j and ∂2⁡σk∂⁡bp⁢q⁢∂⁡bm⁢n by σki⁢j and σkp⁢q,m⁢n respectively. For convenience, we also write N=1f⁢(x)⁢φ⁢(h)⁢h, F=N⁢σk⁢η⁢(t).

Now, we can prove that the mixed volume ∫Sn-1h⁢(x,t)⁢σk⁢(x,t)⁢dx is non-decreasing along flow (1.1).

By flow equation (1.1), we can derive evolution equations of some geometric quantities.

3 The Long-Time Existence of the Flow

In this section, we will give a priori estimates about support functions and curvatures to obtain the long-time existence of flow (1.1) under assumptions of Theorem 1 and Theorem 2.

In the rest of this paper, we assume that M0 is a smooth, closed, strictly convex hypersurface in Rn and h:Sn-1×[0,T)→R is a smooth solution to the evolution equation (2.6) with the initial h⁢(⋅,0) the support function of M0. Here 𝑇 is the maximal time for which the solution exists. Let Mt be the convex hypersurface determined by h⁢(⋅,t), and let ρ⁢(⋅,t) be the corresponding radial function.

We first give the uniform positive upper and lower bounds of h⁢(⋅,t) and ρ⁢(u,t) for t∈[0,T).

By the equality ρ2=h2+|∇⁡h|2, we can obtain the gradient estimate of the support function from Lemma 3.

The uniform bounds of η⁢(t) can be derived from Lemmas 1 and 3.

To obtain the long-time existence of flow (1.1), we need to establish the uniform bounds on principal curvatures. By Lemma 3, for any t∈[0,T), h⁢(⋅,t) always ranges within a bounded interval I′=[1C,C], where 𝐶 is the constant in Lemma 3. First, we give the estimates of σk.