A class of anisotropic expanding curvature flows

We consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_k^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_k is the k-th symmetric polynomial of the principle curvature radii of the hypersurface, k is an integer and 1<= k<= n. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of a elliptic equation, when the constants alpha, beta belong to a suitable range, and the function f satisfies a strictly spherical convexity condition. When beta=1, the soliton equation is just the equation of Lp Christoffel-Minkowski problem. Thus our argument provides a proof to the well-known L_p Christoffel-Minkowski problem for the case p>= k+1 where p=2-alpha, which is identify with Ivaki's recent result.


Introduction
Flows of convex hypersurfaces in R n+1 by a class of speed functions which are homogeneous and symmetric in principal curvatures have been studied by many authors. Firey [15] first introduced the Gauss curvature flow as a model for the shape change of tumbling stones. In [23] Huisken considered the mean curvature flow. He showed that the flow has a unique smooth solution and the hypersurface converges to a round sphere if the initial hypersurface is closed and convex. Later, a range of flows with the speed of homogeneous of degree one in principal curvatures were established, see [12,13,3] etc. for example. For the curvature flow at the speed of α-power of the Gauss-Knonecker curvature, it was conjectured that the solution will converge to a round point along the flow for α > 1 n+2 . Chow [12], Andrews [4], Choi and Daskalopoulos [9] gave the partial answers respectively. In [8], Brendle et al. finally resolved the conjecture for all α > 1 n+2 in all dimensions recently. 1 For the problem on the existence of the prescribed polynomial of the principal curvature radii of the hypersurface, Urbas [29], Chow and Tsai [14], Gerhardt [16], Xia [31] studied the convergence for the flow with the speed of F (λ 1 , ...λ n ), where F is symmetric polynomial of the principal curvature radii λ i of the hypersurface. Guan and Ma [18], Hu et al. [21], and Guan-Xia [20] etc. gave the proofs for a class of L p Christoffel-Minkowski problems. On the other hand, as a nature extension, anisotropic flows usually provide alternative proofs and smooth category approach of the existence of solutions to elliptic PDEs arising in convex body geometry, see [30,10,19,26,24] etc.. In this paper, we consider two expanding flows of the convex hypersurfaces at the speeds of u α σ β k (λ 1 , ...λ n ) and f u α σ β k (λ 1 , ...λ n ) respectively, where u is the support function, f is a smooth positive function on S n , α, β ∈ R 1 , β > 1 k and σ k (λ 1 , ..., λ n ) is the k-th symmetric polynomial of the principal curvature radii of the hypersurface, k is an integer and 1 ≤ k ≤ n. Generally, for the flow with the speed of high powers of curvatures, it is required that the initial hypersurface is uniformly convex and satisfies a suitable pinching conditions, so as to preserve the uniformly convexity and converge to a sphere ( [1], [6]). Here we prove the same result without any pinching conditions for the flow at the speed u α σ β k . Moreover, for α ≤ 1 − kβ, and β > 1 k we prove that the solution to the flow which is moving at the speed f u α σ β k exists for all time and converges smoothly after normalisation to a soliton which is a solution of f u α−1 σ β k = c if f is a smooth positive function on S n and satisfies the condition that (∇ i ∇ j f 1 1+kβ−α + δ ij f 1 1+kβ−α ) is positive definite. We have the same result for k = n without any constraint on positive smooth function f , which recovers the result of Chou and Wang [11]. When β = 1, the flow has been studied by Ivaki recently [24]. In this case the self-similar solution of the flow ∂ t u = f u α σ β k is the solution to f u α−1 σ k = c which is just the L p Christoffel-Minkowski problem for p = 2 − α.
Let M 0 be a smooth, closed and uniformly convex hypersurface in R n+1 , and M 0 encloses the origin. We study the following anisotropic expanding curvature flow where σ k is the k-th elementary symmetric function for principal curvature radii, i.e σ k (., t) = λ i is the principal curvature radii of hypersurface M t , parametrized by X(., t) : S n → R n+1 , and ν is the unit outer normal at X(., t).
In this paper, we prove the following Theorem 1.1. Let M 0 be a smooth, closed and uniformly convex hypersurface in R n+1 , n ≥ 2, enclosing the origin. If f ≡ 1, α ∈ R, β > 1 k and α ≤ 1 − kβ, k is an integer and 1 ≤ k ≤ n. Then the flow (1.1) has a unique smooth and uniformly convex solution M t for all time t > 0. After a proper rescaling X → φ −1 (t)X, where .., 1) the hypersurface M t = φ −1 M t converges exponentially to a sphere centered at the origin in the C ∞ topology.
We denote by u =< X, ν > the support function of M t at X. When f ≡ 1, by a direct calculation, we obtain from the flow (1.1) that the support function u satisfies Rescaling the hypersurface M t in the way of Theorem 1.1, employing a new parameter τ , we get the normalised flow For general function f , under the flow (1.1), the support function the support function u satisfies where the definition of V k+1 (u, u, ..., u) see Section 2. Considering the normalised flow of (1.5) given by ds.
In (1.6), we still use u instead ofũ. Consider the following functional which was introduced by Andrews in [5] When α ≥ 1 − kβ, we will prove in Lemma 2.3 that J (u) is strictly decreasing along the flow(1.6) unless u solves the elliptic equation Theorem 1.2. Let M 0 be a smooth, closed and uniformly convex hypersurface in R n+1 , n ≥ 2, enclosing the origin. Suppose α, β ∈ R 1 , β > 1 k and α ≤ 1 − kβ, k is an integer and 1 ≤ k < n, f is a smooth positive function on S n and ∇ i ∇ j f For k = n, we obtain the same result without constrait on f as follows. Theorem 1.3. Let M 0 be a smooth, closed and uniformly convex hypersurface in R n+1 , n ≥ 2, enclosing the origin. Suppose α ∈ R 1 , β > 1 n and α ≤ 1 − nβ. Then for any f smooth positive function on S n , the flow (1.1) has a unique smooth and uniformly convex solution M t for all time t > 0. After normalisation, the rescaled hypersurfaces M t converge smoothly to a smooth solution of (1.8), which is a minimiser of the functional (1.7). Remark 1.1. When β = 1, our second flow (1.1) is just the one that Ivaki has studied recently [24]. In [24], Ivaki employed the functional in [20] to prove that the flow ∂ t u = f u α σ k has a unique smooth solution, and the rescaled flow converges smoothly to a homothetic self-similar solution which is a solution f u α−1 σ k = c for k < n, α ≤ 1 − k and positive function f satisfies that (∇ i ∇ j f 1 1+k−α +δ ij f 1 1+k−α ) is positive definite. When k = n, he obtained the same result without imposing any condition on f . It has been obtained by Chou and Wang in [11]. Before Ivaki [24], Hu et al. [21] proved the existence result for the L p Christoffel Minkowski problem for p = 2 − α: f u α−1 σ k = 1, α ≤ 1 − k, 1 ≤ k ≤ n−1 for any positive function f satisfying Under the condition that f is positive even fuction and satisfies Guan and Xia in [20] obtained the existence result for 1 − k < α < 1.
We still denote by u =< X, ν > the support function of M t at X. We show the condition α ≤ 1 − kβ is necessary. In fact, by use of the method of [26,27], we show Theorem 1.4. Suppose α > 1 − kβ, α, β ∈ R and β > 0, k is an integer and 1 ≤ k ≤ n. There exist a smooth, closed, uniformly convex hypersurface M 0 , such that under the flow (1.1), This paper is organised as follows. In Section 2, we recall some properties of convex hypersurfaces and show that the functional (1.7) is strictly decreasing along the normalised flow (1.6) unless u satisfies the elliptic equation (1.8). In Section 3, we establish the a priori estimates, which ensure the long time existence of the normalized flows. In Section 4, we show that the flow (1.1) converge to the unit sphere (i.e. Theorem 1.1) and complete the proofs of Theorem 1.2 and Theorem 1.3. Finally, in Section 5, we prove Theorem 1.4.

Preliminary
We recall some basic notations at first. Let M be a smooth, closed, uniformly convex hypersurface in R n+1 . Assume that M is parametrized by the inverse Gauss map X : S n → M ⊂ R n+1 and encloses origin. The support function u : The supremum is attained at a point y = X(x), x is the outer normal of M at y. Hence Let e 1 , · · · , e n be a smooth local orthonormal frame field on S n , and ∇ the covariant derivative with respect to the standard metric e ij on S n . Denote by g ij , g ij , h ij the metric, the inverse of the metric and the second fundamental form of M, respectively. Then the second fundamental form of M is given by (see e. g. [29]) and h ij is symmetric and satisfies the Codazzi equation By the Gauss-Weingarten formula Since M is uniformly convex, h ij is invertible. Hence the principal curvature radii are the eigenvalues of the matrix For a function u ∈ C 2 (S n ), we denote by W u the matrix In the case W u is positive definite, the eigenvalue of W u is the principal radii of a strictly convex hypersurface with support function u. Let u i ∈ C 2 (S n ), i = 1, · · · , n + 1. Set Therefore we define the k + 1-th volume by Next, we state the well-known Alexandrov-Fenchel inequality.
Then for any v ∈ C 2 (S n ), the Alexandrov-Fenchel inequality holds: the equality holds if and only if v = au 1 + n+1 l=1 a l x l for some constants a, a 1 , · · · , a n+1 .
We consider the flow (1.6). For convenience we still use t instead of τ to denote the time variable if no confusions arise, and we set We mention the fact that S n uσ k dµ = |S n | here which comes from the scaling ofũ. Hence (1.6) can be written as ∂u ∂t = ρu − u Z 1 |S n | . By a direct calculation, we have Since h ij satisfies the Codazzi equations, we have i ∇ i σ ij = 0 (see [2],[5]), and By the Alexandrov-Fenchel inequality in Lemma 2.1, we have where α ≤ 1 − kβ < 0 and the Hölder inequality shows that Z 2 ≥ Hence we obtain the uniform upper bound on η(t). Next we prove the uniform lower bound. Set θ ≤ − 1 in the Alexandrov-Fenchel inequality (2.1), we obtain |S n | . Hence, Z θ (u) ≤ Z θ (u 0 ). By the Hölder inequality again, we have Therefore we get the uniform bound on η(t). Proof. Since α ≤ 1 − kβ, β > 1 k , from the above calculation process, when p = − 1 β , we have along the normalised flow (1.6) The last inequality holds from the Hölder inequality, and the equality holds if and only if f u α−1 σ β k = c, where c is a constant.
Lemma 3.2. Let n ≥ 2, 1 ≤ k ≤ n, α ≤ 1 − kβ, β > 1 k , and X(., t) be the solution to the normalised flow (1.6) which encloses the origin for t ∈ [0, T ). Then there is a positive constant C depending on the initial hypersurface and f , α, β, such that Proof. Consider the auxiliary function If α + kβ − 1 ≤ 0, the sign of the coefficient of the highest order term Q 2 is negative. The sign of the coefficient of the lower order term Q is positive. So it is easy to see C −1 ≤ Q ≤ C, where C is the positive constant depending on f , α, β, min S n ×[0,T ) u and max S n ×[0,T ) u.
When f = 1, by use of (3.1), we can get C −1 ≤ Q ≤ C for the flow (1.4). Then by Lemma 3.1, we have C −1 ≤ σ k ≤ C. That is, k and X(., t) be the solution to the normalised flow (1.4) which encloses the origin for t ∈ [0, T ). Then there is a positive constant C depending on the initial hypersurface and α, β, such that Lemma 3.4. Let n ≥ 2, 1 ≤ k ≤ n, α ≤ 1 − kβ, β > 1 k and X(., t) be the solution to the normalised flow (1.6) which encloses the origin for t ∈ [0, T ). Then there is a positive constant C depending on the initial hypersurface and f , α, β, such that |∇ log u| ≤ C.
Proof. Let w = log u. Then we have Assume the auxiliary function |∇u| 2 − Au 2 < 0, for a positive constant A > 0 along the flow. Otherwise there is a point (x t 0 , t 0 ) where t 0 is the first time, such that |∇u| 2 −Au 2 = 0, A > 0 is a constant to be decided later. Hence at the point (x t 0 , t 0 ), ∇ i |∇w| 2 = 0. Choosing an orthonormal frame and rotating the the coordinate, such that w 1 = |∇w|, w i = 0 for i = 2, · · · , n, and (w ij ) is diagonal. Then we get (a ij ) := (w ij + w i w j + δ ij ) = diag(1 + w 2 1 , 1 + w 22 , · · · , 1 + w nn ), 10 and Substituting u i = uw i and w 2 1 = A into the above inequality, and denote σ k = σ k (a ij ), we have by the classic Newton-MacLaurin inequality [17], and the fact that σ k (a ij ) is bounded by Lemma 3.2 for the Case 2. Let A be large enough we then get a contradiction. This completes the proof.
When f = 1, by use of the same argument in Lemma 3.4 and the result of Lemma 3.1. We have Corollary 3.5. Let n ≥ 2, 1 ≤ k ≤ n, α ≤ 1 − kβ, β > 1 k and X(., t) be the solution to the normalised flow (1.4) which encloses the origin for t ∈ [0, T ). Then there is a positive constant C depending on the initial hypersurface and α, β, such that |∇u| ≤ C. Lemma 3.6. Let u(., t), t ∈ [0, T ), be a smooth, uniformly convex solution to (1.6). If 1 ≤ k ≤ n, α ≤ 1 − kβ, β > 1 k , then there is a positive constant C 1 and C 2 depending only on f , α, β and the lower and upper bounds of u(·, 0) such that Proof. Since for the normalised flow (1.6), S n uσ k dµ = |S n | is constant. From Lemma 3.2, there is a positive constant C, such that C −1 ≤ u α−1 σ β k ≤ C. Hence we have Hence we obtain the uniform lower and upper bounds on u from Lemma 3.4. Then by Lemma 3.2, we get the uniform lower and upper bounds on σ k . Now we are going to estimate the upper and lower bounds of the principle curvature radii of the hypersurface M t . We rewrite the equation (1.6) in the following form Lemma 3.7. Let 1 ≤ k < n, α, β ∈ R 1 , β > 1 k and α ≤ 1 − kβ, X(., t) be the solution to the normalised flow (1.6) for t ∈ [0, T ), which encloses the origin. Assume f is a smooth positive function on S n and (∇ i ∇ j f 1 1+kβ−α + δ ij f 1 1+kβ−α ) is positive definite. Then there is a constant C depending only on f , α, β, min S n ×[0,T ) u and max S n ×[0,T ) u, such that the principal curvature radii of X(·, t) are bounded from above and below for all t ∈ [0, T ) and i = 1, ..., n.
Proof. Suppose the maximum eigenvalue of the matrix [ h ij u ] at time t is attained at the point x t with unit eigenvector ξ t ∈ T xt S n . By a rotating the frame e 1 , · · · , e n at x t , assume that at x t we have ξ t = e 1 . At (x t , t), we have Since F = σ 1 k k is concave and homogeneous of degree one, from [29] we have Since ∇ i h 11 1+kβ−α > 0, substituting it into the above inequality, we have that is, h 11 ≤ C, where C depends on the initial hypersurface, the minimum eigenvalue of (∇ i ∇ j f 1 1+kβ−α + δ ij f 1 1+kβ−α ), f , α, and β. Now together with Lemma 3.6, we get We therefore complete the proof.
Lemma 3.8. Let 1 ≤ k ≤ n, α, β ∈ R 1 , β > 1 k and α ≤ 1 − kβ. Let X(., t) be the solution to the normalised flow (1.4) for t ∈ [0, T ), which encloses the origin. Then there is a constant C depending only on the initial hypersurface and α, β, such that the principal curvature radii of X(·, t) are bounded from above and below for all t ∈ [0, T ) and i = 1, ..., n.
Proof. We prove the lemma just as Lamma 3.7. Suppose the maximum eigenvalue of the matrix [h ij ] at time t is attained at the point x t with unit eigenvector ξ t ∈ T xt S n . By rotating the frame e 1 , · · · , e n at x t , assume that at x t we have ξ t = e 1 . At (x t , t), we have where C 0 and C 1 are two positive constants which depend only on the initial hypersurface and α, β. Therefore we have where C 3 is also a positive constants depending only on the initial hypersurface and α, β. Now together with Lemma 3.3, we get 14 Now we show Lemma 3.7 holds for any positive smooth function f when k = n, β > 1 n and α ≤ 1 − nβ.
Lemma 3.9. Let n ≥ 2, k = n. If α ≤ 1 − nβ, β > 1 n and X(·, t) be the solution to the normalised flow (1.6) which encloses the origin for t ∈ [0, T ). Then there is a constant C depending only on the initial hypersurface and f , α, β, such that the principal curvature radii of X(·, t) are bounded from above and below for all t ∈ [0, T ) and i = 1, ..., n.
Proof. Consider the auxiliary function where τ is a unit vector in the tangential space of S n , while ǫ and M are large constants to be decided. Assume w achieve its maximum at (x 0 , t 0 ) in the direction τ = (1, 0, · · · , 0). By a coordinate rotation, h ij and h ij are diagonal at this point. Then at the point (x 0 , t 0 ).
From the estimates obtained in Lemmata 3.8, 3.7 and 3.9, we know that the equations (1.4) and (1.6) are uniformly parabolic. By the C 0 estimates (Lemmata 3.1 and 3.6), the gradient estimates (Lemma 3.5 and Corollary 3.4) and the C 2 estimates Lammata 3.8, 3.7 and 3.9, and the Krylov's theory [25], we get the Hölder continuity of ∇ 2 u and u t . Then we can get higher order derivation estimates by the regularity theory of the uniformly parabolic equations. Therefore we get the long time existence and the uniqueness of the smooth solution to the normalized flows (1.4) and (1.6), respectively. In this section we give the proof of Theorems 1.1 at first. In order to prove the convergence of the normalized flow (1.4), we require the following better gradient estimate. Proof. Let w = log u. Then we have h ij = u ij + uδ ij = e w (w ij + w i w j + δ ij ) and w t = (e w ) α+kβ−1 σ β k ([w ij + w i w j + δ ij ]) − γ. Consider the auxiliary function At the point where Q attains its spatial maximum, we have 0 = ∇ i Q = w l w li , 0 ≥ ∇ ij Q = w li w lj + w l w lij , (i) if 0 ≤ ρ ≤ |t| θ , then r α σ β k ([r −1 δ ij + 2r −3 r i r j − r −2 r ij ]) ≥ c|t| θ α |t| kβθ(µ−1) = c|t| θ−1 , where p = (x, ψ(|x|, t)) is a point on the graph of ψ.
On the other hand, let X(., t) be the solution to ∂X ∂t = −ba r α σ β k ν with initial condition X(., τ ) = ∂B 1 (z), where b = 2 α sup{|p| α : p ∈ M t , τ < t < t 0 } and r = |X − z|. We can choose τ so small that the ball B 1 2 (z) is contained in the interior of X(., t) for all t ∈ (τ, t 0 ). By the comparison principle, we know that the ball B 1 2 (z) is contained in the interior of M t for all t ∈ (τ, t 0 ). Hence, as t → t 0 , we have min r(.t) → 0 and max r(., t) > |z| = 10. Hence (5.7) is proved for M t .
For a large constant a > 0. Making the rescaling M t = a − 1 q M t , M t solve the flow (5.6). Hence we complete the proof.