Mean Curvature of Hypersurfaces in Killing Submersions with Bounded Shadow

Given a complete hypersurface isometrically immersed in an ambient manifold, in this paper we provide a lower bound for the norm of the mean curvature vector field of the immersion assuming that: 1) The ambient manifold admits a Killing submersion with unit-length Killing vector field. 2)The projection of the image of the immersion is bounded in the base manifold. 3)The hypersurface is stochastically complete, or the immersion is proper.


Introduction
Lower bounds for the norm of the mean curvature of an isometric immersion of bounded image in the Euclidean space has been largely studied in order to understand the Calabi problem.It is well known that there is no gap on the norm of the mean curvature of a bounded immersion.More precisely, it is known that a complete isometric immersion in the Euclidean space with bounded image can be a minimal immersion.For instance, in his celebrated paper [14], Nadirashvili constructed a complete minimal surface inside a round ball in R 3 .Later on the construction of minimal immersions inside of bounded domains of the Euclidean space was carried out by Martin, Morales, Tokuomaru, Alarcón, Ferrer and Meeks among others (see [12,13,20,2,7]).More recently, Alarcón and Forstnerič have proved in [1] that every bordered Riemann surface carries a conformal complete minimal immersion into R 3 with bounded image.
Despite of this freedom in the norm of the mean curvature we want stress here that all this previous examples are geodesically complete but stochastically incomplete and are non-properly immersed in the Euclidean space.A Riemmanian manifold it is said stochastically complete if e t△ 1 = 1 for all t ≥ 0 where △ = div∇ is the Laplacian and {P t = e t△ } t≥0 is the heat semi-group of the Laplacian (see section 3.4 for a more detailed description of stochastic completeness).
Historically, the first attempt to construct a minimal immersion with bounded image dealt with the construction of an example with bounded projection.In 1980, (before the example of Nadirashvili), Jorge and Xavier in [9] exhibit a nonflat complete minimal surface lying between two parallel planes.This example is stochastically complete (see [5]) but is non-properly immersed.
The relevant point here is that stochastic completeness or the properness of the isometric immersion implies lower bounds for the norm of the mean curvature of bounded immersions.In fact, in [3], Alías, Bessa and Dajczer proved that given a complete isometric immersion α : Σ n−1 → R n with Σ stochastically complete and with bounded projection of the image of α in a 2-plane, namely, there exists a geodesic ball B R 2 R of radius R and a projection π : R n → R 2 such that π(ϕ(Σ)) ⊂ B R 2 R , the norm of the mean curvature H of the immersion is bounded by sup where h(R) is the norm of the mean curvature of the generalized cylinder π −1 (∂B R 2 R ) in R n .More generally, in [3], it is proved that Theorem 1.1 (See [3]).Let ϕ : Σ m → N n−l × R l be an isometric immersion of a complete Riemannian manifold Σ of dimension m > l + 1.Let B N R (p) be the geodesic ball of N n−l centered at p with radius R. Given q ∈ Σ, assume that the radial sectional curvature K rad N along the radial geodesics issuing from p = π N (ϕ(q)) ∈ N n−1 is bounded as √ κ}, where we replace π/2 √ κ by +∞ if κ ≤ 0.Then, if Σ is stochastically complete or ϕ : Σ → N n−l × R l , the supremum of the norm of the mean curvature vector field is bounded by where In this paper we are interested in a similar case but when we have an isometric immersion ϕ : Σ n → M n+1 and the ambient manifold M admits a Killing submersion π : M n+1 → B n (see sections 2 and 3).Our goal is to obtain lower bounds for the mean curvature of the immersion ϕ when the projection of the image ϕ(Σ) is bounded, i.e., there exist a geodesic ball In our main results, we prove that if we assume that Σ is stochastically complete, or the immersion is proper, then the norm of the mean curvature is bounded from below by a function that depends on R, on an upper bound for the sectional curvatures of B B R , and on the bundle curvature of the Killing submersion π : M n+1 → B. Outline of the paper.In §2 are stated the main results of this paper; theorem 2.1 and theorem 2.5.In §3 are developed every necessary lemma, proposition, and theorems in order to prove in §4 theorem 2.1 and in §5 theorem 2.5.

Main Results
Let (M, g M ) be (n + 1)-dimensional Riemannian manifold.It is said that M admits a Killing submersion with an unit-length Killing vector field if there exist a Riemmanian submersion π : (M, g M ) → (B, g B ) into a n-dimensional base manifold B, such that the fibers of the submersion are integral curves of an unit-length Killing vector field ξ ∈ X(M ).The tangent space T p M at any point p ∈ M can be decomposed T p M = V(p) ⊕ H(p) in its vertical V(p) = ker(dπ p ) and horizontal part H(p) = (ker(dπ p )) ⊥ respectively.
The Killing vector field ξ induces a smooth (1, 1)-tensor field ∇ξ, see section 3, given by ∇ξ(p) : where ∇ denotes the Levi-Civita connection in M .With the Hilbert-Schmidt norm ∇ξ(p)2 of the linear map ∇ξ(p) we can define the τ -function as For any point p ∈ M and any v ∈ H(p), the sectional curvature of the plane v ∧ ξ spanned by v and ξ is non-negative and bounded from above by (see section 3) The above inequality is an equality if dim(B) = 2 and in such a case τ is also known1 as the bundle curvature of the submersion and τ 2 coincides with the sectional curvature of the vertical planes.
In this paper we are interested in hypersurfaces ϕ : Σ → M isometrically immersed in M , and such that π • ϕ(Σ) is bounded in B. Namely, there exists a geodesic ball In this case the restriction τ • ϕ to the hypersurface of the τ -function is absolutely bounded and we denote by In the statement of theorem 2.1, we compare the norm of the mean curvature of Σ with the norm of the mean curvature of the inclusion map of the cylinder ∂B where M n (κ) is the n-dimensional simply -connected space form of sectional curvature κ, i.e., Our main theorem is the following Theorem 2.1.Let Σ be a complete and non-compact Riemannian manifold.Let ϕ : Σ → M be an isometric immersion.Suppose that M admits a Killing submersion π : M → B with unit-length Killing vector field, suppose moreover that ϕ where inj(o) is the injectivity radius of o and we replace π/2 √ κ by +∞ if κ < 0.Then, if Σ is stochastically complete, the supremum of the norm of the mean curvature vector field of Σ satisfies where h n κ (R) is the norm of the mean curvature of the generalized cylinder ∂B Because lim t→0 h n κ (t) = +∞, as an immediate corollary of the main theorem we can state Corollary 2.2.Let π : M → B be a Killing submersion with a Killing vector field of unit-length, suppose that B has bounded geometry, i.e., (1) The injectivity radius inj(B) of B is positive, r inj := inj(B) > 0.
(2) The sectional curvatures of B are bounded from above by a positive constant, sec(B) ≤ κ < 0.
Remark 2.3.From the main theorem we can give an estimate for R c (r inj , κ, τ M ) as min r inj , sup We must remark here that we do not claim that this estimate is sharp.
Example 2.4 (Application of the theorem 2.1 to the E(κ, τ ) spaces).A simplyconnected homogeneous 3-dimensional space with 4-dimensional isometry group is always a Riemmanian fibration over to a simply-connected 2-dimensional real space form M 2 (κ) and the fibers are integral curves of a unit Killing field, see [6,19].Namely, every simply-connected homogeneous 3-dimensional space with 4-dimensional isometry group admits a Killing submersion with unit-length Killing vector field.In fact, this spaces can be classified, up to isometries, by their values κ and τ and can be denoted as E(κ, τ ) spaces.When the bundle curvature vanishes τ = 0 and κ = 0 we obtain the Riemannian products S 2 (κ) × R for κ > 0, and H 2 (κ) × R for κ < 0. In the case of τ = 0 we have Berger spheres for κ > 0, the Heisenberg group Nil 3 for κ = 0 and the universal cover P SL 2 (R) of P SL 2 (R) for κ < 0.
If we have a complete and stochastically complete surface Σ immersed in ), with R < π/2 √ κ when κ > 0, then by the main theorem the norm of the mean curvature vector field satisfies sup In the particular case when κ < 0, sup And therefore if −κ ≥ τ 2 ≥ 0, any minimal surface immersed in π −1 (B In the case when the Killing submersion admits a smooth section such that the normal exponential map is a diffeomorphism, the lower bound for the norm of the mean curvature vector field can be improved replacing the hypothesis on the stochastic completeness of Σ in theorem 2.1 by the properness of the immersion as it is stated in the following theorem Theorem 2.5.Let Σ be a complete and non-compact Riemannian manifold.Let ϕ : Σ → M be a proper isometric immersion.Suppose that M admits a Killing submersion π : M → B with unit-length Killing vector field, suppose moreover that where inj(o) is the injectivity radius of o and we replace π/2 √ κ by +∞ if κ < 0. Suppose moreover, that π admits a smooth section s : Then, the supremum of the norm of the mean curvature vector field of Σ satisfies sup where h n κ (R) is the norm of the mean curvature of the generalized cylinder ∂B In the case of E(κ, τ ) spaces with κ ≤ 0 we can use the model for the E(κ, τ ) as (see [11]) the space endowed with the Riemannian metric such that the following three vector fields whose fibers are the integral curves of the unit-length Killing vector field E 3 .Moreover, s(x, y) → (x, y, 0) constitutes a smooth global section from M 2 (κ) to E(κ, τ ).The normal exponential map satisfies exp((x, y, 0), t) = exp (x,y,0) (tE 3 ) = (x, y, t).
In this spaces the hypothesis of theorem 2.5 are therefore fulfilled and hence we can state Corollary 2.6.Let ϕ : Σ → E(κ, τ ) be a proper isometric immersion from the complete and non-compact surface to , then the supremum of the norm of the mean curvature vector field of Σ satisfies where h 2 κ (R) is the norm of the mean curvature of the generalized cylinder ∂B In this cases the above corollary is a direct application of [3].For the case of τ = 0 we have the Heisengerb group Nil 3 = E(κ = 0, τ = 0) for κ = 0 and in the case of negative curvature κ < 0 we can assume up to scaling that we are in universal cover of P SL 2 (R) , namely P SL 2 (R) = E(κ = −1, τ = 0).By using the above corollary, any properly immersed non-compact surface ϕ : Σ → Nil 3 with bounded projection π(ϕ(Σ) has bounded from below the supremum of the norm of the mean curvature vector field by (1) sup In the case of negative curvature if we have a complete and non-compact surface Σ properly immersed in , then the surface has bounded from below the norm of the mean curvature vector field by (2) sup Observe that inequalities (1) and ( 2) are optimal because the right side coincides with the norm of the mean curvature of the cylinders 3. Preliminaries 3.1.Killing Submersions.Let M and B two manifolds.A submersion π : M → B is a mapping of M onto B such that its derivative dπ p : T p M → T π(p) B has maximal rank (it is onto) for any p ∈ M .Then, the distribution p → V(p) = ker(dπ p ) called the vertical distribution is an involutive distribution and hence If (M, g) is moreover a Riemannian manifold, an other distribution called the horizontal distribution can be constructed as p → H(p) = (ker(dπ p )) ⊥ .Likewise, a vector field X ∈ X(M ) is called horizontal if it belongs to H.Then, for any p ∈ M we can decompose the tangent space T p M as A Riemannian submersion π : (M, g M ) → (B, g B ) is a submersion such that dπ preserves the lengths of horizontal vectors.Namely dπ p is a local isometry from A Riemannian submersion π : M → B is a Killing submersion if the fibers π −1 (x) for any x ∈ B are integral curves of a Killing vector field ξ ∈ X(M ).Along this paper is assumed that the Killing vector field ξ is an unit-length vector field ( ξ = 1).See [10] for the general discussion of a Killing submersion with a Killing vector field of non-constant norm.
Recall that a vector field ξ ∈ X(M ) is a Killing vector field of (M, g) (see [16]) if its Lie derivative of the metric tensor vanishes identically, L ξ (g) = 0.If ξ is a Killing vector field, the metric tensor does not change under the flow of ξ and ξ generates local isometries.
The following proposition of a Killing vector field will be used along this paper in order to characterize a Killing vector field Proposition 3.1 (See [16]).Let (M, g) be a Riemannian manifold.Then, the following conditions are equivalents for a vector field ξ ∈ X(M ) (1) ξ is Killing; that is, L ξ g = 0.
(2) ∇ξ is skew-adjoint relative to g; that is, If π : M → B is a Killing submersion, for any p ∈ M , by using the vertical vector field ξ, the following linear map ∇ξ : Since ξ is a unit-length Killing vector field, by proposition 3.1 is the integral curve of ξ, and by using that ξ = 1 and equality (3), we conclude for any v ∈ T p M .Therefore, ∇ ξ ξ = 0, and as we have stated π −1 (x) is a geodesic in M .Moreover, from equation (4) we deduce that ∇ v ξ is perpendicular to ξ, and hence horizontal.The restriction of ∇ξ to H(p) induces therefore a linear map ∇ξ(p) : H(p) → H(p).
In the following proposition it is summarized the properties of the (1, 1)-tensor field ∇ξ and of τ = ∇ξ Proof.Given a point p ∈ M and an horizontal vector v ∈ H(p) with unit-length, v = 1, in order to obtain the sectional curvature sec(v ∧ξ) let us consider a vector field X ∈ X(B) defined in a neighborhood U ∋ π(p), such that X(π(p)) = dπ(v) and with vanishing covariant derivative ∇ B X X = 0 in B, i.e., a geodesic vector field.Then the lift X ∈ X(M ) of X defined in π −1 (U ) ∋ p satisfies Where here and in what follows the superscript H denotes the horizontal part of a vector.Then, In order to simplify the expression let us define the following vector fields Y := ∇ X ξ and Z := [X, ξ].Observe that both X, Y are horizontal vector fields.Since Y = ∇ξ(X) and where we have used that In order to obtain item (1) and ( 2) of the proposition we only need to relate ∇ξ(v) 2 with ξ 2 .When we focus on p ∈ M and consider an orthonormal basis where we have used that ∇ Ei ξ, E j 2 is symmetric in i, j and ∇ Ei ξ, E i 2 = 0 because ξ is a Killing vector field .We now, need to relate ∇ v ξ with the Hilbert-Schmidt norm ∇ξ .Recall that for the linear map ∇ξ : H(p) → H(p) the Hilbert-Schmidt norm is given by ( 7) In the particular case when n = 2, by using inequalities ( 6) and ( 7), when n > 2, taking into account that for any i and j, and item (2) of the proposition follows.By using sec(v ∧ξ) = ∇ v ξ 2 with v 2 = 1 item (1) of the proposition follows as well.
Finally, we are going to prove that τ is an basic function.Given any point p ∈ M with π(p) = y let us consider the integral curve γ ξ : R → M of ξ tangent to the fiber π −1 (y) with γ ξ (0) = p (and γξ (0) = ξ(p)).It is sufficient to prove that To obtain that let us consider a sufficient small tubular neighborhood of γ((−ǫ, ǫ)) and the following orthonormal basis {ξ(γ(t)), E 1 , • • • , E n } at γ(t), (n = dim(B) and {E i } are horizontal vectors).Then (8) But for any i, j Hessian and Laplacian in immersions and submersions.We are interested in the following setting with ϕ an isometric immersion and π a Killing submersion.Since in this paper we will assume that Σ is stochastically complete, Σ satisfies a weak maximum principle for the Laplacian of bounded functions f : Σ → R, see theorem 3.6.Our strategy will be to make use of an specific function f : B → R and to study the Laplacian of the function f • π • ϕ : Σ → R. In this section, we develop in proposition 3.3 the required relation between △(f • π • ϕ), the mean curvature of the immersion ϕ and the bundle curvature τ of the submersion π.
Let ϕ : Σ → M be an isometric immersion.For any point ϕ(p) ∈ M we can decompose the tangent space as T ϕ(p) M = dϕ(T p Σ) ⊕ (dϕ(T p Σ)) ⊥ .Let us denote by ∇ M and ∇ Σ the Levi-Civita connection on M and Σ.For any p ∈ Σ, x, y ∈ T p Σ and Y ∈ X(Σ) an extension of y to X(Σ), the second fundamental form II p (x, y) is given by x Y ) the second fundamental form II p (x, y) ∈ (dϕ(T p Σ) ⊥ ) and recall moreover that the mean curvature of the immersion ϕ : Σ → M in p is defined by Let f : M → R be a smooth function, the gradient of f and the gradient of the restricted function f • ϕ : Σ → R satisfy the following relation The Hessian of the restriction f • ϕ is given then by (10) If Σ is an hypersurface of M (i.e., dim(Σ) = n) there exists (at least locally) a vector field ν normal to Σ and such that (dϕ(T p Σ)) ⊥ = span{ν}.Given an orthonormal basis } is an orthonormal basis of T ϕ(p) M and hence But now we are interested in the particular case when f : then ∇f is an horizontal vector field in X(M ), π-related with ∇f ∈ X(B).Let us decompose ν = ν H + ν V in its horizontal and vertical part, then where we have used that dπ(∇ νH ν H ) = ∇ B dπ(ν) dπ(ν) see [15].Therefore, Hence, finally In order to simply the expression we will make us of the τ -mean curvature H τ of Σ defined in (12).Then ( 13) where here f = F • r o , t = r 0 (x) and Proof.By using the definition of the Hessian and the chain rule, Therefore, . But if the sectional curvatures of the base manifold are bounded as k ≤ sec ≤ κ, see Theorem 27 of [17], and Hence, finally if If we have a Killing submersion π : M → B we can lift the radial function f to f = f • π and using equation ( 11) of proposition 3.3 we obtain for F ′ > 0, ( 16) . This above inequality can be rewritten in the following corollary, Corollary 3.5.Let Σ be an hypersurface immersed in M by ϕ : Σ → M , let M admit a Killing submersion π : M → B with unit-length Killing vector field.Suppose that the sectional curvatures of B are bounded from above and below by Let F : R → R be a smooth and non-decreasing function, let provided that v 0 is a bounded continuous positive function.The manifold Σ is said to be stochastically complete, see [8], if for any x ∈ Σ and any t > 0. The main property of stochastic completeness which is used in this paper is that if a Riemannian manifold is stochastic complete a weak maximum principle is satisfied for bounded functions in C 2 .More precisely, if Σ is stochastically complete we can state the following theorem Theorem 3.6 (See [18]).Let Σ be a connected non-compact Riemannian manifold.Suppose that Σ is stochastically complete, then for every u ∈ C 2 (Σ) with sup Σ u < ∞ there exists a sequence {x k }, k = 1, 2, . .., such that, for every k, u(x k ) ≥ sup Σ u − 1/k and △u(x k ) ≤ 1/k.
On the other hand (see [4]), a Riemannian manifold (M, g) satisfies the Omori-Yau maximum principle for the Laplacian if for any function u ∈ C 2 (M ) which is bounded sup M u = u * < ∞, there exists a sequence In this paper we will use the following sufficient condition for the Omori-Yau maximum principle Theorem 3.7 (See [4]).Let Σ be a connected non-compact Riemannian manifold.Suppose that Σ admits a for t large enough and A ≥ 0.Then, the Omori-Yau maximum principle for the Laplacian holds on Σ.

Proof of theorem 2.1
The statement of the theorem 2.1 is as follows Theorem.Let Σ be a complete and non-compact Riemannian manifold.Let ϕ : Σ → M be an isometric immersion.Suppose that M admits a Killing submersion π : M → B with unit-length Killing vector field, suppose moreover that ϕ(Σ) ⊂ π −1 (B where h n κ (R) is the norm of the mean curvature of the generalized cylinder ∂B Moreover, for any 2-plane Π p ⊂ T p B the sectional curvatures sec(Π p ) of any p ∈ B B R (o) will be bounded as In order to simplify the argument of the proof let us choose k < 0, and let us define the function Now, we are going to apply theorem 3.6 to Therefore t(x i ) → R * when i → ∞, and by inequality ( 19), Hence, denoting θ = arccos( ν, ξ ) and applying proposition 3.2, Letting now i tend to infinity, Finally the theorem follows by taking into account that (n−1) is the norm of the mean curvature of the generalized cylinder ∂B and the inequality would be In the statement of theorem 2.5 it is assumed that the Killing submersion π : M → B admits a smooth section s : B B R (o) → M and is assumed as well that the normal exponential map exp : × R given by q ∈ π −1 (B B R (o)) → T (q) = (p(q), z(q)) where p and z are the following two functions (27) p : π −1 (B B R (o)) → s(B B R (o)), p(q) := s(π(q)) z : π −1 (B B R (o)) → R, q = exp p(q) (z(q)ξ) Observe that we can define the z function by an implicit equation because the hypothesis on the injectivity of normal exponential map.Observe moreover that z(q) ≥ dist M (q, s(B B R (o)) because the curve t → γ(t) = exp p(q) (tξ) is a geodesic joining γ(0) = p(q) ∈ s(B B R (o)) with q = γ(z(q)).We can can moreover reverse the T map T −1 : s(B B R (o)) × R → π −1 (B B R (o)), (p, z) → T −1 (p, z) = exp p (zξ).We will need furthermore the expression of the gradient of the z function which is given in the following lemma.Now we can assume that sup Σ H < ∞, (otherwise, there is nothing to be proved), and hence by using proposition 5.2, Σ satisfies the Omori-Yau maximum principle for the Laplacian, and f is bounded in Σ, there exists a sequence {x i } Then

2 12 1 ) 2 )
that are relevant for the present paper Proposition 3.2.Let π : M → B be a Killing submersion with unit-length Killing vector field.Then (Given a point p ∈ M and an horizontal vector v ∈ T p M , the sectional curvature sec(v ∧ ξ(p)) of the plane spaned by ξ(p) and v is bounded bysec(v ∧ ξ(p)) ≤ τ 2 (p)with equality if dim(B) = 2. (Given a point p ∈ M and for any horizontal vector

Proposition 3 . 3 .
Let Σ be an hypersurface immersed in M by ϕ : Σ → M , let M admit a Killing submersion π : M → B with unit-length Killing vector field ξ ∈ X(M ).Let f : B → R be a smooth function on the base manifold.Denote by f = f • π the lift of f .Then,(11)

3. 3 .Proposition 3 . 4 .
Radial functions on the base manifold.Suppose that f : B → R is a radial function with respect to the point o ∈ B, in the sense that f(x) = f (y) if r o (x) = dist B (o, x) = r o (y),then there exists a function F : R → R such that f (x) = F • r o (x) for any x ∈ B. Now, in the following proposition we will obtain bounds on the Hessian and Laplacian of f Let B a Riemannian manifold, let o ∈ B, and denote by r o : B → R the distance function in B to o, i.e., r o (p) = dist B (o, p).Assume moreover that the sectional curvatures of B are bounded from above and below for any plane of the tangent space, k ≤ sec(B) ≤ κ.Then, for any function going to compute the Laplacian of f = F k • r o • π • ϕ.By using inequality (17) of corollary 3.5,

Remark 4 . 1 .
In the proof of theorem 2.1 we have used in inequality (23) that τ ≤ τ Σ and | sin(2θ)| ≤ 1.We could use instead the following factors to improve the result τ zξ) is a diffeomorphism.Given the section s :B B R (o) → M we can trivialize π −1 (B B R (o)) ≈ s(B B R (o)) × R using the following map T : π −1 (B B R (o)) ≈ s(B B R (o))

Lemma 5 . 1 .
Let π : M → B a Killing submersion with unit-length Killing vector field ξ.Suppose that the Killing submersion π : M → B(o) admits a smooth section s : B B R (o) → M and that the normal exponential map exp :s(B B R (o)) × R → π −1 (B B R (o)), (p, z) → exp(p, z) = exp p (zξ)where inj(o) is the injectivity radius of o and we replace π/2 √ κ by +∞ if κ < 0. Suppose moreover, that π admits a smooth section s : B B R (o) → M and the normal exponential mapexp : s(B B R (o)) × R → π −1 ((B B R (o))), (p, z) → exp(p, z) := exp p (zξ) is a diffeomorphism.Then, the supremum of the norm of the mean curvature vector field of Σ satisfies supΣ H ≥ h n κ (R)where h n κ (R) is the norm of the mean curvature of the generalized cylinder ∂BM n (κ) R × R in M n (κ) × R.Proof.Likewise to the proof of theorem 2.1, we are using the test functionf = F k • r o • π • ϕ with F k : R → R, t → F k (t) = t 0 sn k (s)ds Let us setting, R * := sup Σ r o (ϕ(Σ)) < ∞, s)ds < ∞.By using inequality (17) of corollary 3.5(28) △ Σ f (z) ≥sn ′ k (t) 1 − dπ(ν) ∇r o , dπ(ν)2 B R (p)) for some geodesic ball B B R (o) of radius R centered at o ∈ B.Assume that the sectional curvatures are bounded sec ≤ κ in B B Then, if Σ is stochastically complete, the supremum of the norm of the mean curvature vector field of Σ satisfies sup tend to ∞, taking into account that t(x i ) → R * when i → ∞, , 1 i ≥ (n − 1)sn k (t(x i )) sn ′ κ (t(x i )) sn κ (t(x i )) − sn k (t(x i )) sup Σ H − 2 ni letting i