An application of dual convex bodies to the inverse Gauss curvature flow

By means of dual convex bodies, we obtain regularity of solutions to the expanding Gauss curvature flows with homogeneity degrees $-p$, $0<p<1$. At the end, we remark that our method can also be used to obtain regularity of solutions to the shrinking Gauss curvature flows with homogeneity degrees less than one.


Introduction
The setting of this paper is the n-dimensional Euclidean space, R n . A compact convex subset of R n with non-empty interior is called a convex body.
Let K be a convex body. The support function of K, denoted by s K , is defined as where z, y denotes the standard inner product of z and y.
Let K be a strictly convex body, having the origin in its interior such that its boundary, denoted by ∂K, is smoothly embedded in R n by Let n K (x) be the outward unit normal vector of K for every x ∈ ∂K. The support function of K has the following simple form s K (z) := n −1 K (z), z , for each z ∈ S n−1 , where n −1 K : S n−1 → ∂K is the inverse of the Gauss map n K . We denote the standard metric on S n−1 byḡ ij and the standard Levi-Civita connection of S n−1 by∇. We denote the Gauss curvature of ∂K by K and remark that, as a function on ∂K, it is related to the support function of the convex body by S n−1 := 1 K • n −1 K := det g (∇ i∇j s +ḡ ij s).
We denote the matrix of the radii of curvature of ∂K by r = [r ij ] 1≤i,j≤n−1 , the entries of r are considered as functions on the unit sphere. They are related to the support function by the identity r ij :=∇ i∇j s + sḡ ij . We denote the eigenvalues of [r ij ] 1≤i,j≤n−1 with respect to the metricḡ ij by λ i for 1 ≤ i ≤ n − 1, and we assume that λ 1 ≤ λ 2 ≤ · · · ≤ λ n−1 .
In this paper, we present a new method to study the long-time behavior of flow by certain powers of the Gauss curvature by linking expanding Gauss curvature flows to shrinking Gauss curvature flows, see section 6 for the latter.
For a given smooth, strictly convex embedding x K , we consider a family of smooth, strictly convex bodies {K t } t , given by the smooth embeddings x : ∂K × [0, T ) → R n , which are evolving according to the anisotropic expanding Gauss curvature flow (1.1) ∂ t x(·, t) := Φ(n Kt (·)) K p n−1 (·, t) n Kt (·), x(·, 0) = x K (·), where Φ : S n−1 → R is a positive smooth function and p ∈ (0, ∞). In this equation for each time t we have x(∂K, t) = ∂K t . Throughout this paper we assume that K 0 , the initial smooth convex body, encloses the origin of R n . The flow (1.1) has been studied by Andrews [1] in dimension two and it was proved that properly rescaled flows will evolve solutions to the unit circle if Φ ≡ 1. For Φ ≡ 1, the inverse Gauss curvature flow has been investigated by Chow and Tsai [11,12], by Schnürer [29] for p = 2 in R 3 , and by Q. R. Li [24] for 1 < p ≤ 2 in R 3 . When Φ ≡ 1, the flow (1.1) is an interesting case of expanding flows ∂ t x = F − p n−1 n, where F is a positive, strictly monotone, concave function of principal radii of curvature with the homogeneity degree n − 1, and F is zero on the boundary of the positive cone Γ = {(α 1 , · · · , α n−1 ) : α i > 0}. For p = 1, expanding flows have first been studied by Gerhardt and Urbas [14,32,33]. In particular, they proved that starting the flow with a smooth, strictly convex hypersurface the solution remains smooth and strictly convex. Moreover, the solution exists on (0, ∞) and becomes spherical as it expands. In addition, in [33] this result was extended to cover the case 0 < p < 1. A special class of expanding flows, the inverse mean curvature flow, has been employed by Huisken and Ilmanen to prove the Riemannian Penrose inequality [18]. Recently, in [14], Gerhardt thoroughly studied flow of closed star-shaped hypersurfaces and strictly convex hypersurfaces under the expanding flows ∂ t x = F − p n−1 n. He demonstrated after a proper rescaling, the rescaled flow will evolve a closed star-shaped hypersurface if 0 < p < 1, and a strictly convex hypersurface if p > 1, to a sphere.
To study the flow (1.1), one can study the evolution equation of the support functions of {K t } t : The short time existence and the uniqueness of solutions to (1.2) with an initial smooth and strictly convex body follow from the strict parabolicity of the equation, see property 2 below. Moreover, it is proved in [11] that if Φ ≡ 1, then the convexity is preserved. Our main theorem, which is stated at the end of this section, bears the restriction Φ ≡ 1, but some of the calculations are for an arbitrary Φ.
Let B R denote the origin-centered ball of R n with radius R > 0. For simplicity, B 1 is denoted by B. The following is the main result of the paper.
We remark that the C 1 convergence and the C ∞ convergence to the unit ball were proved in [11] and [33], respectively. Our main contribution is to give a new proof based by employing dual convex bodies. In the rest of this paper, we will focus on the case 0 < p < 1.

evolution equation of the dual convex bodies
In this section, we calculate the evolution equation of dual convex bodies, which as we will see later on, it can be employed as a useful tool in obtaining regularity of solutions. To my knowledge, evolution equation of dual convex bodies was first introduced by Stancu in [30] in the context of centro-affine normal flows. The method used in [30] for obtaining the evolution equation of dual convex bodies relied on a certain duality property of L p -affine surface areas, therefore that method cannot be used here. In this section, we give an argument that is applicable to flow by powers of the Gauss curvature. It is worth noting that the idea of 'duality'(different from the one we consider here) has been used in [4,5]. In [5], an upper curvature bound is obtained for the curve shortening flow by studying the isoperimetric profile and to obtain a uniform lower curvature bound, the isoperimetric profile of the 'dual' (in this case, the complement of the region enclosed by a simple closed curve) is required. Similarly in [4], an upper curvature bound is obtained from interior noncollapsing and a lower curvature bound is obtained from exterior non-collapsing which is not directly interior non-collapsing of the 'dual' (complement) but it is closely related.
Let K be a convex body having the origin in its interior. The dual convex body of K with respect to the origin, denoted by K • , is defined as If K is smooth, then K • is also smooth, see [16]. We furnish all the geometric quantities associated with K • by a superscript • . Next, we find the evolution equation of the support function of K • t as the support function of K t evolves by (1.2). To this aim, we parameterize ∂K t over the unit sphere where r(z(·, t), t) is the radial function of K t in direction z(·, t). We then recall that how the geometric quantities that we need can be expressed in this parametrization.
Let K be a smooth convex body whose boundary is parameterized over the unit sphere with the radial function r. The metric [g ij ] 1≤i,j≤n−1 , unit normal n, support function s, and the second fundamental form h ij of ∂K can be be written in terms of r and whose spatial derivatives as follows: A good reference for these identities is [34].
Lemma 2.1. As K t evolve according to the flow (1.2), then its radial function evolves as follows where from the third line to fourth line we used (2). By comparing terms on the second line with those on the fourth line, we get Replacing ∂ t z in (2.2) by its equivalent expression from equation (2.1) completes the proof.
As 1 r is the support function of ∂K • (see, [28]), we can find the entries of the Thus, we have We will use this identity in the proof of the next theorem.
Proof. By means of the identities equation (3), equation (2.3), and Lemma 2.1 calculate where on the last line, we replaced r by 1 s • .

Bounding the isoperimetric ratio
In this section, we assume that Φ ≡ 1. We recall the following result of Chow and Gulliver from [10].
There exists a positive constant C depending only on the initial condition s(·, 0) such that, as long as the solution to (1.2) exists and r ij > 0 on S n−1 × [0, T ), the support function s(·, t) satisfies the uniform gradient estimate Next is an immediate corollary of this proposition.
Then there is a finite positive constant C such that, the family of convex bodies Proof. The origin belongs to the interior of K 0 , so does to the interior of K t and On the other hand, the ratio (max

Upper and lower bounds on the principal curvatures
In this section, we present a technique to obtain lower and upper bounds on the principal curvatures of the evolving convex bodies supposing that there are uniform lower and upper bounds on the evolving support functions. Our approach to derive lower and upper bounds consists of three steps. We first obtain an upper bound on the Gauss curvature using Tso's trick [31]. We then continue to derive a lower bound on the Gauss curvature using the evolution equation of dual convex bodies (2.4). In the last step, we apply the parabolic maximum principle to the evolution equation of r ij ; to be defined later in this section.
Proof. Define β = p n−1 . We consider the function Differentiating Φ at the point where the minimum of Ψ occurs yields: When we differentiate Φ with respect to t we conclude where (S n−1 ) ′ ij := ∂S n−1 ∂r ij is the derivative of the S n−1 with respect to the entry r ij of the radii of curvature matrix. Applying inequality (4.1) to the preceding identity we deduce Now, we estimate the mean curvature H =ḡ ij r ij = n−1 i=1 1 λi from below by a negative power of Ψ. We obtain Consequently, inequality (4.2) can be rewritten as follows for positive constants C(p) and C ′ (p, R − , R + , Φ). Hence From this, it follows that 1 Ψ ≤ Ct p p−1 for a new constant C. Equivalently, we have a bound for Ψ from below. Therefore, we have bounded K from above in terms of p, R − , R + , Φ and time.
Before we can obtain a lower bound on the Gauss curvature, we need a preparatory lemma.
Using the maximum principle, we are going to prove that Ψ is bounded from above by a function of n, p, R − , R + , Φ and time. At the point where the maximum of Ψ occurs, we have Hence, we obtain and consequently, We calculate It is not difficult to see that Indeed, we only need to take into account that where for the second equation we used (4.3), and for the third inequality we considered the fact that ||x|| 2 = s 2 + ||∇s|| 2 ≤ R 2 + (In fact, x = sz +∇s and z,∇s = 0, so ||x|| 2 = s 2 + ||∇s|| 2 .). Using this and inequality (4.4) we infer that, at the point where the maximum of Ψ is reached, we have We can control the mean curvature H = n−1 i=1 Therefore, we can rewrite the inequality (4.5) as follows for positive constants C 1 and C 2 . Hence, for new constants C 3 and C 4 . The corresponding claim for K follows.
for constants depending on n, p, R − , R + , and Φ.
Proof. We apply the previous lemma to the evolution equation (2.4): Observe that Now, using (4.6) with β = − p n−1 , we obtain that . Therefore, we have bounded from above K • in terms of n, p, R − , R + , Φ, and time. To complete the proof, we recall Kaltenbach's identity: for every x ∈ ∂K, there exists an where x and x • are related by x, x • = 1 (A proof of this identity can be found in [22], see also [16,26].). By the above identity, we conclude that K is bounded from below by constants depending on n, p, R − , R + , Φ, and time.
Denote the inverse matrix of [r ij ] 1≤i,j≤n−1 by [r ij ] 1≤i,j≤n−1 . We recall the following evolution equation of r ij from [11] as {K t } t evolves under (1.2).
Proof. We estimate the terms on each line of (4.7). a: Estimating the terms on the first line: The first term is an essential good term regarded as an elliptic operator which is non-positive at the point and direction where a maximum eigenvalue occurs. The sum of the second and the third term is less than equal to −(r 11 ) 2 ΦF . b: Estimating the terms on the second line: Notice that by the fourth property 1≤i,j≤n−1 ) . Therefore, by [33] page 112, this term can be estimated from above by −2Φ(r 11 ) 2 (∇1F ) 2 F . c: Estimating the terms on the third line: By the arithmetic mean-geometric mean inequality we have Thus, combining these estimates all together we have Consequently, by the assumption Φ ≡ 1, we have An ODE comparison completes the proof.
Denote the principal curvatures by κ i = 1 λi for 1 ≤ i ≤ n − 1. Now we have all necessary prerequisites to obtain lower and upper bounds on the principal curvatures.
. Then there exists constants C and C ′ depending on C 1 and C 2 such that, ∀t ∈ [0, t 0 ], Proof. In fact, the upper bound on the principal curvatures has been established in Lemma 4.4. So, we need to verify the claimed lower bound on the principal curvatures. As κ 1 ≥ κ 2 ≥ . . . ≥ κ n−1 , from the assumption and Lemma 4.4 we get

Proof of the main theorem
First, we remark that lim t→T max S n−1 s = ∞. This result follows from Lemmas 4.1, 4.3, and 4.5 combined with a standard argument that has been used in Corollary I1.9 of [1]. Second, we point out that the lifespan of a solution to the inverse Gauss curvature flow (1.2) is [0, ∞), for 0 < p < 1 and Φ ≡ 1. Indeed, if K 0 ⊆ B R , for R large enough, then from the comparison principle it follows that where R(t) := max S n−1 s(·, 0) 1−p + (1 − p)t τ has uniform C k bounds for all τ ∈ [δ/2, δ] and these bounds are independent of t and τ . Furthermore, the volume of the convex body corresponding tos(·, τ ) is bounded from above by V (B 2C ). Thus, for all t ∈ [0, ∞) and τ ∈ [δ/2, δ]. This implies that we have uniform C k bounds for for all t ∈ [0, ∞) and τ ∈ [δ/2, δ], in particular for τ = δ/2. Next, we show that for . This is a continuous function. We have f (∞) = ∞ −t > 0. On the other hand, we have The claim follows. So, we have proved that: As min converges in the C ∞ topology to the origin-centered unit ball, as t approaches ∞.
1. An alternative approach to prove Theorem 1.1 is to combine Harnack inequality of Y. Li [25] with displacement bounds, almost in an identical manner as in [1].

Further discussion: Shrinking Gauss curvature flow
In the preceding sections, using the evolution equation of dual convex bodies we studied long time behavior of the expanding Gauss curvature flows in the case Φ ≡ 1. Here, we would like to point out that this method can also be used to obtain C 2 regularity of solutions under certain powers of Gauss curvature flow to prove the known Theorem 6.1, which was proved by Andrews in [3], including the case p = 1. The contribution of this section is mainly on the uniform lower bound on the Gauss curvature for the normalized solution.
For a given smooth, strictly convex embedding x K , consider a family of smooth convex bodies {K t } t given by the smooth embeddings x : ∂K × [0, T ) → R n , which are evolving according to the anisotropic shrinking Gauss curvature flow, namely, where Φ : S n−1 → R is a positive smooth function, 0 < T < ∞, and p ∈ (0, ∞) . In this equation x(∂K, t) = ∂K t .
Theorem 6.1. [3] Assume that p ∈ n−1 n+1 , 1 . Let x K be a smooth, strictly convex embedding of ∂K. Then there exists a smooth, unique solution of (6.1). Furthermore, there is a point q ∈ R n such that the rescaled convex bodies given by converge in the C ∞ topology to a homothetic solution of (6.1).
Additionally, if p ∈ 0, n−1 n+1 and the isoperimetric ratio remains bounded, then the same conclusion holds as in the case of p ∈ n−1 n+1 , 1 .
On the contrary to the statement of Theorem 1.1, we do not need to confine ourselves to the case Φ ≡ 1. The reason is that Theorem 10 of [3] is available for any positive smooth function Φ. Remark 6.2. For a convex body K, denote the maximum width by ω + (K) := max S n−1 (s(z) + s(−z)) and the minimum width by ω − (K) := min S n−1 (s(z) + s(−z)). A convex body with centroid b, satisfies ω− n+1 B ⊆ K − b ⊆ nω+ n+1 B; this result goes back to Minkowski (1897), see Schneider [28], page 308. This in particular implies that n+1 nω+ B ⊆ (K − b) • ⊆ n+1 ω− B. In connection to the flow (6.1), after obtaining the evolution equation of the corresponding dual convex bodies, one can acquire a uniform lower bound on the Gauss curvature in terms of inradius, circumradius, and time; likewise done in Lemma 2.3 in [21]: In [3], it was proved that the ratio ω + (K t )/ω − (K t ) remains bounded along the flow (6.1), if p ∈ n−1 n+1 , 1 . Therefore, the rescaled solution given is also a solution to (6.1) with the initial datas(z, 0) = V (B) here b t is the centroid of K t . So, from Remark 6.2 we getc n+1 ≤s(z, 0) ≤ nC n+1 and n+1 nC ≤s • (z, 0) ≤ n+1 c . Now we can use the argument of [21] (similar to the argument in the proof of Lemma 4.3) to obtain a uniform lower bound on the Gauss curvature (which is invariant under Euclidian translations) for the normalized solution,K t , in terms ofc,C, n, and p. We can then continue as in [3] to prove Theorem 6.1. The advantage is that we avoid using the Haranck inequality and the displacement bounds of [3]. However, we mention that Haranck inequalities and displacement bounds are fundamental ingredients to prove existence of solutions to some flows with non-smooth initial data, see [3,7]. Moreover, they can also be used to obtain stability of some geometric inequalities that come from geometric flows [20]. Therefore, by no means our method here substitutes those in [1,3], but abridges the regularity arguments. For some powers p > 1 and Φ ≡ 1, the flow (6.1) has been treated in [2,6,9,15].