THE QUERMASSINTEGRAL INEQUALITIES FOR k-CONVEX STARSHAPED DOMAINS

We give a proof of the isoperimetric inequality for quermassintegrals of non-convex starshaped domains, using a result of Gerhardt [6] and Urbas [13] on an expanding geometric curvature flow. The Alexandrov-Fenchel inequalities [1, 2] for the quermassintegrals of convex domains are fundamental in classical geometry. For a bounded domain Ω ⊂ Rn+1, we denote M = ∂Ω the boundary of Ω. We will assume M smooth in this paper. Let κ(x) = (κ1(x), · · · , κn(x)) be the principal curvatures of x ∈ M , and let σk(λ) the kth elementary function in λ = (λ1, · · · , λn) ∈ R n (with σ0(λ) ≡ 1). There are several equivalent definitions of the quermassintegral V(n+1)−k(Ω). For positive integer k, we will take the following (1) V(n+1)−k(Ω) = Cn,k ∫ M σk−1(κ)dμM , where where σk is the kth elementary symmetric function, (2) Cn,k = σk(I) σk−1(I) , with I = (1, · · · , 1). One may also recover Vn+1(Ω) by the Minkowski type formula, (3) V(n+1)−k(Ω) = ∫ M uσk(κ)dμM , where u = 〈X, ν〉, X is the position function of M , and ν is the outer-normal of M at X. Vn+1(Ω) is a multiple of the volume of Ω by a dimensional constant, Vn(Ω) is a multiple of the surface area of M = ∂Ω by another dimensional constant. If Ω is convex, the celebrated Alexandrov-Fenchel quermassintegral inequality states that, for 0 ≤ m ≤ n, (4) (V(n+1)−m(Ω) V(n+1)−m(B) ) 1 n+1−m ≤ (Vn−m(Ω) Vn−m(B) ) 1 n−m , Research of the first author was supported in part by an NSERC Discovery Grant, and research of the second author is supported in part by a CRM fellowship. 1 2 PENGFEI GUAN AND JUNFANG LI where B is the standard ball in Rn+1. The equality holds if and only if Ω is a ball. The case m = 0 is the classical isoperimetric inequality. There have been some interests in extending the original AlexandrovFenchel inequality to non-convex domains (e.g., [12], [7]). In this short paper, we extend this inequality to starshaped domains in Rn+1. These domains are special type of domains where fully nonlinear partial differential equations were studied in pioneer work by Caffarelli-Nirenberg-Spruck [4, 5]. We follow the notations in [12] as below. Definition 1. For Ω ⊂ Rn+1, we say Ω is k-convex if κ(x) ∈ Γ̄k for all x ∈ M , where Γk is the Garding’s cone Γk = {λ ∈ R n | σm(λ) > 0, ∀m ≤ k}. We say Ω is strictly k-convex if κ(x) ∈ Γk for all x ∈ M . n-convex is convex in usual sense, 1-convex is sometimes referred as mean convex. Theorem 2. Suppose Ω is a k-convex starshaped domain in Rn+1, then inequality (4) is true for 0 ≤ m ≤ k. The equality holds if and only if Ω is a ball. In [12], Trudinger considered inequality (4) for general the k-convex domains. He proposed an elliptic method by reducing the problem to a Hessian type equation in the domain, but the reduction argument there is incomplete. Our proof here is a parabolic one, using the flow studied by Gerhardt [6] and Urbas [13]. For our purpose, we will only use a special case of their result for the following evolution equation on a hypersurface M0 in R n+1, (5) Xt = σk−1 σk (κ)ν. Theorem 3. (Gerhardt [6], Urbas [13]) If Ω0 is a starshaped strictly kconvex domain, then solution for flow (5) exists for all time t > 0 and it converges to a round sphere after a proper rescaling. We define (6) Ik(Ω) = V 1 n+1−k (n+1)−k(Ω) V 1 n−k n−k (Ω) . The key observation is that the isoperimetric ratio Ik(Ω) of the quermassintegrals are monotone along expanding flow of (5). From what follows, we will denote M(t) the solution of flow (5) at time t. If there is no confusion, we will just write M(t) = M . To simplify notation, we will also write σm for σm(κ) unless specified otherwise. To prepare our proof of Theorem 2, we first list the evolution equations of various geometric quantities under the following general evolution equation. (7) ∂tX = Fν, THE QUERMASSINTEGRAL INEQUALITIES 3 where F = F (κ,X, t). Proposition 4. Under flow (7), we have the following evolution equations. (8) ∂tgij = 2Fhij ∂tν = −∇F ∂t(dμg) = Fσ1dμg ∂thij = −∇i∇jF + F (h )ij ∂th i j = −∇ ∇jF − F (h )j ∂tσm = −∇j([Tm−1] i j∇iF )− Fσm−1,1(h i j ; (h )j) where we denote hj ≡ g hkj , (h )ij ≡ g hikhlj and (h )j ≡ g ghskhlj , [Tl] i j is the l-th Newton transformation, and σm−1,1(A;B) = ∂σm ∂Aij (A)Bij is a polarization of σm. Proof. The proof follows from straightforward computations using the Codazzi property of the second fundamental form. The last identity follows from the divergent free property of [Tm−1] i j. Lemma 5. Under flow (5), (9) ∂t ∫