Mean Curvature of Hypersurfaces in Killing Submersions with Bounded Shadow
aa r X i v : . [ m a t h . DG ] D ec MEAN CURVATURE OF HYPERSURFACES IN KILLINGSUBMERSIONS WITH BOUNDED SHADOW
VICENT GIMENO
Abstract.
Given a complete hypersurface isometrically immersed in an am-bient manifold, in this paper we provide a lower bound for the norm of themean curvature vector field of the immersion assuming that:1) The ambient manifold admits a Killing submersion with unit-lengthKilling vector field.2)The projection of the image of the immersion is bounded in the basemanifold.3)The hypersurface is stochastically complete, or the immersion is proper. Introduction
Lower bounds for the norm of the mean curvature of an isometric immersionof bounded image in the Euclidean space has been largely studied in order tounderstand the Calabi problem. It is well known that there is no gap on the normof the mean curvature of a bounded immersion. More precisely, it is known thata complete isometric immersion in the Euclidean space with bounded image canbe a minimal immersion. For instance, in his celebrated paper [14], Nadirashviliconstructed a complete minimal surface inside a round ball in R . Later on theconstruction of minimal immersions inside of bounded domains of the Euclideanspace was carried out by Martin, Morales, Tokuomaru, Alarc´on, Ferrer and Meeksamong others (see [12, 13, 20, 2, 7]). More recently, Alarc´on and Forstneriˇc haveproved in [1] that every bordered Riemann surface carries a conformal completeminimal immersion into R with bounded image.Despite of this freedom in the norm of the mean curvature we want stress herethat all this previous examples are geodesically complete but stochastically in-complete and are non-properly immersed in the Euclidean space. A Riemmanianmanifold it is said stochastically complete if e t △ t ≥ △ = div ∇ is the Laplacian and { P t = e t △ } t ≥ is the heat semi-group of the Laplacian (seesection 3.4 for a more detailed description of stochastic completeness).Historically, the first attempt to construct a minimal immersion with boundedimage dealt with the construction of an example with bounded projection. In1980, (before the example of Nadirashvili), Jorge and Xavier in [9] exhibit a non-flat complete minimal surface lying between two parallel planes. This example isstochastically complete (see [5]) but is non-properly immersed. Mathematics Subject Classification.
Primary 58C40, 35P15, (53A10).
Key words and phrases.
Killing submersion, stochastic completeness, isometric immersion,mean curvature.Research partially supported by the Universitat Jaume I Research Program Project P1-1B2012-18, and DGI-MINECO grant (FEDER) MTM2013-48371-C2-2-PDGI from Ministerio de Cienciae Inovaci´on (MCINN), Spain.
The relevant point here is that stochastic completeness or the properness of theisometric immersion implies lower bounds for the norm of the mean curvature ofbounded immersions. In fact, in [3], Al´ıas, Bessa and Dajczer proved that givena complete isometric immersion α : Σ n − → R n with Σ stochastically completeand with bounded projection of the image of α in a 2-plane, namely, there exists ageodesic ball B R R of radius R and a projection π : R n → R such that π ( ϕ (Σ)) ⊂ B R R , the norm of the mean curvature ~H of the immersion is bounded bysup M k ~H k ≥ h ( R )where h ( R ) is the norm of the mean curvature of the generalized cylinder π − ( ∂B R 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 immersionof a complete Riemannian manifold Σ of dimension m > l + 1 . Let B NR ( p ) bethe geodesic ball of N n − l centered at p with radius R . Given q ∈ Σ , assume thatthe radial sectional curvature K rad N along the radial geodesics issuing from p = π N ( ϕ ( q )) ∈ N n − is bounded as K rad N ≤ κ in B NR ( p ) . Suppose that ϕ (Σ) ⊂ B NR ( p ) × R l for R < min { inj N ( p ) , π/ √ κ } , where we replace π/ √ κ by + ∞ if κ ≤ . Then, if Σ is stochastically complete or ϕ : Σ → N n − l × R l , the supremum of the norm ofthe mean curvature vector field is bounded by sup Σ k ~H k ≥ m − lm C κ ( R ) where C κ ( R ) = √ κ cot( √ κR ) if κ > , R < π √ κ /R if κ = 0 √− κ coth( √− κR ) if κ < . In this paper we are interested in a similar case but when we have an isometricimmersion ϕ : Σ n → M n +1 and the ambient manifold M admits a Killing submer-sion π : M n +1 → B n (see sections 2 and 3). Our goal is to obtain lower bounds forthe mean curvature of the immersion ϕ when the projection of the image ϕ (Σ) isbounded, i.e. , there exist a geodesic ball B B R in B such that ϕ (Σ) ⊂ π − ( B B R ). In ourmain results, we prove that if we assume that Σ is stochastically complete, or theimmersion is proper, then the norm of the mean curvature is bounded from belowby a function that depends on R , on an upper bound for the sectional curvaturesof B B R , and on the bundle curvature of the Killing submersion π : M n +1 → B . Outline of the paper. In § § § § 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 aRiemmanian 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-lengthKilling vector field ξ ∈ X ( M ). The tangent space T p M at any point p ∈ M can EAN CURVATURE OF HYPERSURFACES 3 be decomposed T p M = V ( p ) ⊕ H ( p ) in its vertical V ( p ) = ker( dπ p ) and horizontalpart H ( p ) = (ker( dπ p )) ⊥ respectively.The Killing vector field ξ induces a smooth (1 , ∇ ξ , see section 3,given by ∇ ξ ( p ) : H ( p ) → H ( p ) , v → ∇ v ξ, where ∇ denotes the Levi-Civita connection in M . With the Hilbert-Schmidt norm k∇ ξ ( p ) k of the linear map ∇ ξ ( p ) we can define the τ -function as τ ( p ) := (cid:18) k∇ ξ ( p ) k (cid:19) 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)sec( v ∧ ξ ) ≤ τ ( p ) . The above inequality is an equality if dim( B ) = 2 and in such a case τ is alsoknown as the bundle curvature of the submersion and τ coincides with the sec-tional curvature of the vertical planes.In this paper we are interested in hypersurfaces ϕ : Σ → M isometrically im-mersed in M , and such that π ◦ ϕ (Σ) is bounded in B . Namely, there exists ageodesic ball B B R ( o ) in B such that ϕ (Σ) ⊂ π − ( B B R ( o )). In this case the restriction τ ◦ ϕ to the hypersurface of the τ -function is absolutely bounded and we denote by τ Σ := sup Σ τ ◦ ϕ. In the statement of theorem 2.1, we compare the norm of the mean curvatureof Σ with the norm of the mean curvature of the inclusion map of the cylinder ∂B M n ( κ ) R × R in M n ( κ ) × R , where M n ( κ ) is the n -dimensional simply -connectedspace form of sectional curvature κ , i.e. , M n ( κ ) := S n ( κ ) if κ > R n if κ = 0 H n ( κ ) if κ < . 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 (Σ) ⊂ π − ( B B R ( o )) for some geodesic ball B B R ( o ) of radius R centered at o ∈ B . Assumethat the sectional curvatures are bounded sec ≤ κ in B B R ( o ) and that R < min (cid:26) inj( o ) , π √ κ (cid:27) where inj( o ) is the injectivity radius of o and we replace π/ √ κ by + ∞ if κ < . Then, if Σ is stochastically complete, the supremum of the norm of the meancurvature vector field of Σ satisfies sup Σ k ~H k ≥ h nκ ( R ) − τ Σ n In the case of dim( B ) = 2 usually is used a signed τ . In such convention our τ is just | τ | . VICENT GIMENO where h nκ ( R ) is the norm of the mean curvature of the generalized cylinder ∂B M n ( κ ) R × R in M n ( κ ) × R . Because lim t → h nκ ( t ) = + ∞ , as an immediate corollary of the main theorem wecan state Corollary 2.2.
Let π : M → B be a Killing submersion with a Killing vector fieldof unit-length, suppose that B has bounded geometry, i.e. , (1) The injectivity radius inj( B ) of B is positive, r inj := inj( B ) > . (2) The sectional curvatures of B are bounded from above by a positive constant, sec( B ) ≤ κ < .Suppose moreover that, τ M := sup M τ < ∞ Then there exists a constant R c ( r inj , κ, τ M ) ∈ R + ∪{ + ∞} depending only on r inj , κ ,and τ M such that if a complete Riemannian manifold admits an isometric minimalimmersion ϕ : Σ → π − ( B B R c ( r inj ,κ,τ M ) ( x )) for some x ∈ B , then Σ is stochasticallyincomplete.Remark . From the main theorem we can give an estimate for R c ( r inj , κ, τ M ) asmin ( r inj , sup
Example 2.4 (Application of the theorem 2.1 to the E ( κ, τ ) spaces) . A simply-connected homogeneous 3-dimensional space with 4-dimensional isometry groupis always a Riemmanian fibration over to a simply-connected 2-dimensional realspace form M ( κ ) and the fibers are integral curves of a unit Killing field, see[6, 19]. Namely, every simply-connected homogeneous 3-dimensional space with4-dimensional isometry group admits a Killing submersion with unit-length Killingvector 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 Riemannianproducts S ( κ ) × R for κ >
0, and H ( κ ) × R for κ <
0. In the case of τ = 0 we haveBerger spheres for κ >
0, the Heisenberg group Nil for κ = 0 and the universalcover ^ P SL ( R ) of P SL ( R ) for κ < E ( κ, τ )inside of π − ( B M ( κ ) R ), with R < π/ √ κ when κ >
0, then by the main theoremthe norm of the mean curvature vector field satisfiessup Σ k ~H k ≥ h κ ( R ) − τ C κ ( R ) − τ ) . In the particular case when κ < Σ k ~H k ≥ h κ ( R ) − τ > (cid:0) √− κ − τ (cid:1) . And therefore if − κ ≥ τ ≥
0, any minimal surface immersed in π − ( B M κ R ) for any R >
EAN CURVATURE OF HYPERSURFACES 5
In the case when the Killing submersion admits a smooth section such that thenormal exponential map is a diffeomorphism, the lower bound for the norm ofthe mean curvature vector field can be improved replacing the hypothesis on thestochastic completeness of Σ in theorem 2.1 by the properness of the immersion asit 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 Killingsubmersion π : M → B with unit-length Killing vector field, suppose moreover that ϕ (Σ) ⊂ π − ( B B R ( o )) 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 R ( o ) and that R < min (cid:26) inj( o ) , π √ κ (cid:27) where inj( o ) is the injectivity radius of o and we replace π/ √ κ by + ∞ if κ < .Suppose moreover, that π admits a smooth section s : B B R ( o ) → M and the normalexponential map exp : s ( B B R ( o )) × R → π − (( 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 vectorfield of Σ satisfies sup Σ k ~H k ≥ h nκ ( R ) where h nκ ( R ) is the norm of the mean curvature of the generalized cylinder ∂B M n ( κ ) R × R in M n ( κ ) × R . In the case of E ( κ, τ ) spaces with κ ≤ E ( κ, τ ) as(see [11]) the space E ( κ, τ ) = n ( x, y, z ) ∈ R : 1 + κ x + y ) > o endowed with the Riemannian metric such that the following three vector fields E = h κ x + y ) i ∂∂x − τ y ∂∂z ,E = h κ x + y ) i ∂∂y + τ x ∂∂z ,E = ∂∂z constitutes an orthonormal basis in each tangent space. Observe that π ( x, y, z ) → ( x, y ) is a Riemannian submersion from E ( κ, τ ) to M ( κ ) whose fibers are theintegral curves of the unit-length Killing vector field E . Moreover, s ( x, y ) → ( x, y,
0) constitutes a smooth global section from M ( κ ) to E ( κ, τ ). The normalexponential map satisfiesexp(( x, y, , t ) = exp ( x,y, ( tE ) = ( x, y, t ) . In this spaces the hypothesis of theorem 2.5 are therefore fulfilled and hence we canstate
Corollary 2.6.
Let ϕ : Σ → E ( κ, τ ) be a proper isometric immersion from thecomplete and non-compact surface to E ( κ, τ ) with κ ≤ . Suppose that π ( ϕ (Σ)) is VICENT GIMENO contained in some ball B M ( κ ) R ( o ) of radius R in M ( κ ) , then the supremum of thenorm of the mean curvature vector field of Σ satisfies sup Σ k ~H k ≥ h κ ( R ) where h κ ( R ) is the norm of the mean curvature of the generalized cylinder ∂B M ( κ ) R × R in M ( κ ) × R .Remark . The E ( κ, τ ) spaces includes for τ = 0, E ( κ < , τ = 0) = H ( κ ) × R and E ( κ = 0 , τ = 0) = R × R = R . In this cases the above corollary is adirect application of [3]. For the case of τ = 0 we have the Heisengerb groupNil = E ( κ = 0 , τ = 0) for κ = 0 and in the case of negative curvature κ < P SL ( R ) , namely ] P SL ( R ) = E ( κ = − , τ = 0). By using the above corollary, any properly immersednon-compact surface ϕ : Σ → Nil with bounded projection π ( ϕ (Σ)) ⊂ B R R ( o ) ⊂ R , has bounded from below the supremum of the norm of the mean curvaturevector field by(1) sup Σ k ~H k ≥ R .
In the case of negative curvature if we have a complete and non-compact surfaceΣ properly immersed in ] P SL ( R ) with bounded projection π ( ϕ (Σ)) ⊂ B H ( − R ⊂ H ( − Σ k ~H k ≥
12 cotanh( R ) . Observe that inequalities (1) and (2) are optimal because the right side coincideswith the norm of the mean curvature of the cylinders π − ( ∂B R R ) ⊂ Nil and π − ( ∂B H ( − R ) ⊂ ] P SL ( R ) respectively.3. Preliminaries
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 hasmaximal 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 π − ( x ) is a submanifold of M of dimension dim( M ) − dim( B ) for any x ∈ B . Thesubmanifolds π − ( x ) are called the fibers . A vector field X ∈ X ( M ) is called verticalif it belongs to V , namely, if X ( p ) ∈ V ( p ) for any p ∈ M .If ( M, g ) is moreover a Riemannian manifold, an other distribution called thehorizontal distribution can be constructed as p → H ( p ) = (ker( dπ p )) ⊥ . Likewise, avector 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 T p M = H ( p ) ⊕ V ( p ) . 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 H ( p ) to T π ( p ) B . EAN CURVATURE OF HYPERSURFACES 7
A Riemannian submersion π : M → B is a Killing submersion if the fibers π − ( x ) for any x ∈ B are integral curves of a Killing vector field ξ ∈ X ( M ). Alongthis paper is assumed that the Killing vector field ξ is an unit-length vector field( k ξ k = 1). See [10] for the general discussion of a Killing submersion with a Killingvector 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 aKilling 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 paperin order to characterize a Killing vector field Proposition 3.1 (See [16]) . Let ( M, g ) be a Riemannian manifold. Then, thefollowing 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, h∇ V ξ, W i = −h∇ W ξ, X i for all V, W ∈ X ( M ) . If π : M → B is a Killing submersion, for any p ∈ M , by using the vertical vectorfield ξ , the following linear map ∇ ξ : T p M → T p M, v → ∇ ξ ( v ) = ∇ v ξ can bedefined. Since ξ is a unit-length Killing vector field, by proposition 3.1(3) h∇ v ξ, w i = −h∇ w ξ, v i for any v, w ∈ T p M . This implies that π − ( x ) is geodesic in M because π − ( x ) isthe integral curve of ξ , and by using that k ξ k = 1 and equality (3), we conclude(4) h∇ ξ ξ, v i = −h∇ v ξ, ξ i = − v h ξ, ξ i = 0 , for any v ∈ T p M . Therefore, ∇ ξ ξ = 0, and as we have stated π − ( x ) is a geodesicin M . Moreover, from equation (4) we deduce that ∇ v ξ is perpendicular to ξ , andhence 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 , ∇ ξ and of τ = (cid:16) k∇ ξ k (cid:17) that are relevant for the present paper Proposition 3.2.
Let π : M → B be a Killing submersion with unit-length Killingvector field. Then (1) Given a point p ∈ M and an horizontal vector v ∈ T p M , the sectionalcurvature sec( v ∧ ξ ( p )) of the plane spaned by ξ ( p ) and v is bounded by sec( v ∧ ξ ( p )) ≤ τ ( p ) with equality if dim( B ) = 2 . (2) Given a point p ∈ M and for any horizontal vector v ∈ T p M k∇ v ξ k ≤ τ k v k (3) The function τ : M → R is a basic function, i.e. , it is fiber-independent,namely, if π ( x ) = π ( y ) then τ ( x ) = τ ( y ) .Proof. Given a point p ∈ M and an horizontal vector v ∈ H ( p ) with unit-length, k v k = 1, in order to obtain the sectional curvature sec( v ∧ ξ ) let us consider a vectorfield X ∈ X ( B ) defined in a neighborhood U ∋ π ( p ), such that X ( π ( p )) = dπ ( v ) VICENT GIMENO 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 − ( U ) ∋ p satisfies(5) dπ ( X ) = dπ ( X H ) = Xdπ (( ∇ X X ) H ) = ∇ BX X = 0 ∇ X X = ( ∇ X X ) H + h∇ X X, ξ i ξ = −h X, ∇ X ξ i = 0 . Where here and in what follows the superscript H denotes the horizontal part of avector. Then,sec( v ∧ ξ ) =sec( X ∧ ξ ) = h R ( X, ξ ) X, ξ i = h∇ ξ ∇ X X − ∇ X ∇ ξ X + ∇ [ X,ξ ] X, ξ i = h−∇ X ∇ ξ X + ∇ [ X,ξ ] X, ξ i = h−∇ X ([ ξ, X ] + ∇ X ξ ) + ∇ [ X,ξ ] X, ξ i = h∇ X ([ X, ξ ] − ∇ X ξ ) + ∇ [ X,ξ ] X, ξ i 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 h Z, ξ i = h∇ X ξ − ∇ ξ X, ξ i = h∇ X ξ, ξ i − h∇ ξ X, ξ i = 12 X h ξ, ξ i = 0Therefore,sec( v ∧ ξ ) = h∇ X ( Z − Y ) + ∇ Z X, ξ i = h∇ X Z + ∇ Z X, ξ i − h∇ X Y, ξ i = k Y k = k∇ X ξ k = k∇ v ξ k where we have used that h∇ X Z, ξ i = −h Z, ∇ X ξ i = h X, ∇ Z ξ i = −h∇ Z X, ξ i and h∇ X Y, ξ i = −h Y, ∇ X ξ i = −k Y k . In order to obtain item (1) and (2) of the proposition we only need to relate k∇ ξ ( v ) k with k ξ k . When we focus on p ∈ M and consider an orthonormal basis { E i } ni =1 of H ( p ), for any v ∈ H ( p ), v = P i v i E i ,(6) k∇ ξ ( v ) k = k∇ v ξ k = n X i =1 h∇ v ξ, E i i = n X i =1 n X j =1 ( v j ) h∇ E j ξ, E i i = n X i =1 n X j =1 ( v j ) (cid:18) h∇ E j ξ, E i i + h∇ E i ξ, E j i (cid:19) = n X i =1 n X j =1 ( v j ) h∇ E j ξ, E i i n X i =1 n X j =1 ( v j ) h∇ E i ξ, E j i n X i =1 n X j =1 ( v j ) h∇ E j ξ, E i i n X j =1 n X i =1 ( v i ) h∇ E j ξ, E i i n X i =1 n X j =1 ( v j ) + ( v i ) h∇ E j ξ, E i i = n − X i =1 n X j> (cid:0) ( v j ) + ( v i ) (cid:1) h∇ E j ξ, E i i where we have used that h∇ E i ξ, E j i is symmetric in i, j and h∇ E i ξ, E i i = 0because ξ is a Killing vector field . We now, need to relate k∇ v ξ k with the Hilbert-Schmidt norm k∇ ξ k . Recall that for the linear map ∇ ξ : H ( p ) → H ( p ) the Hilbert-Schmidt norm is given by(7) k∇ ξ k = n X i =1 k∇ ξ ( E i ) k = n X i =1 n X j =1 h∇ E i ξ, E j i =2 n − X i =1 n X j>i h∇ E i ξ, E j i In the particular case when n = 2, by using inequalities (6) and (7), k∇ v ξ k = (cid:0) ( v ) + ( v ) (cid:1) h∇ E ξ, E i = k∇ ξ k k v k = τ k v k when n >
2, taking into account that for any i and j , (cid:0) ( v i ) + ( v j ) (cid:1) ≤ k v k , k∇ v ξ k ≤k v k n − X i =1 n X j> h∇ E j ξ, E i i = τ k v k and item (2) of the proposition follows. By using sec( v ∧ ξ ) = k∇ v ξ k with k v k = 1item (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 thefiber π − ( y ) with γ ξ (0) = p (and ˙ γ ξ (0) = ξ ( p )). It is sufficient to prove that ddt (cid:0) τ ◦ γ ξ ( t ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) t =0 = 0To obtain that let us consider a sufficient small tubular neighborhood of γ (( − ǫ, ǫ ))and the following orthonormal basis { ξ ( γ ( t )) , E , · · · , E n } at γ ( t ), ( n = dim( B ) and { E i } are horizontal vectors). Then(8) ddt (cid:0) τ ◦ γ ξ ( t ) (cid:1) = ξ ( τ ) = ξ n X ij h∇ E i ξ, E j i =2 n X ij h∇ E i ξ, E j i · ξ ( h∇ E i ξ, E j i )But for any i, j (9) ξ ( h∇ E i ξ, E j i ) = h∇ ξ ∇ E i ξ, E j i + h∇ E i ξ, ∇ ξ E j i = h∇ ∇ Ei ξ ξ, E j i + h∇ E i ξ, ∇ E j ξ i = − h∇ E j ξ, ∇ E i ξ i + h∇ E i ξ, ∇ E j ξ i = 0 (cid:3) Hessian and Laplacian in immersions and submersions.
We are inter-ested in the following setting Σ M B ϕ π with ϕ an isometric immersion and π a Killing submersion. Since in this paper wewill assume that Σ is stochastically complete, Σ satisfies a weak maximum principlefor the Laplacian of bounded functions f : Σ → R , see theorem 3.6. Our strategywill be to make use of an specific function f : B → R and to study the Laplacianof the function f ◦ π ◦ ϕ : Σ → R . In this section, we develop in proposition 3.3 therequired relation between △ ( f ◦ π ◦ ϕ ), the mean curvature of the immersion ϕ andthe bundle curvature τ of the submersion π .Let ϕ : Σ → M be an isometric immersion. For any point ϕ ( p ) ∈ M we candecompose the tangent space as T ϕ ( p ) M = dϕ ( T p Σ) ⊕ ( dϕ ( T p Σ)) ⊥ . Let us denoteby ∇ M and ∇ Σ the Levi-Civita connection on M and Σ. For any p ∈ Σ, x, y ∈ T p Σand Y ∈ X (Σ) an extension of y to X (Σ), the second fundamental form II p ( x, y ) isgiven by II p ( x, y ) = ∇ Mdϕ ( x ) Z − dϕ ( ∇ Σ x Y )being Z any extension of dϕ ( Y ) to X ( M ). Since,( ∇ Mdϕ ( x ) Z ) T = dϕ ( ∇ Σ x Y )the second fundamental form II p ( x, y ) ∈ ( dϕ ( T p Σ) ⊥ ) and recall moreover that themean curvature of the immersion ϕ : Σ → M in p is defined by ~H := 1dim(Σ) dim(Σ) X i =1 II p ( E i , E i )for any orthonormal basis { E i } of T p Σ.Let f : M → R be a smooth function, the gradient of f and the gradient of therestricted function f ◦ ϕ : Σ → R satisfy the following relation h∇ f ◦ ϕ, v i Σ = d ( f ◦ ϕ )( v ) = df ( dϕ ( v )) = h∇ f, dϕ ( v ) i M . The Hessian of the restriction f ◦ ϕ is given then by(10) Hess Σ f ◦ ϕ ( x, y ) = h∇ Σ x ∇ f ◦ ϕ, y i Σ = x h∇ f ◦ ϕ, y i Σ − h∇ f ◦ ϕ, ∇ Σ x Y i Σ = x h∇ f ◦ ϕ, y i Σ − h∇ f, dϕ ( ∇ Σ x Y ) i M = dϕ ( x ) h∇ f, dϕ ( y ) i M − h∇ f, ∇ Mdϕ ( x ) dϕ ( Y ) − II p ( x, y ) i M =Hess M f ( dϕ ( x ) , dϕ ( y )) + h∇ f, II p ( x, y ) i M . If Σ is an hypersurface of M ( i.e. , dim(Σ) = n ) there exists (at least locally) a vectorfield ν normal to Σ and such that ( dϕ ( T p Σ)) ⊥ = span { ν } . Given an orthonormalbasis { E i } ni =1 of T p Σ, { dϕ ( E i ) } ni =1 ∪ { ν ϕ ( p ) } is an orthonormal basis of T ϕ ( p ) M andhence △ Σ f ◦ ϕ ( p ) = n X i =1 Hess Σ f ◦ ϕ ( E i , E i ) = n X i =1 Hess M f ( dϕ ( E i ) , dϕ ( E i )) + n h∇ f, ~H i M = △ M f ( ϕ ( p )) − Hess M f ( ν ϕ ( p ) , ν ϕ ( p ) ) + n h∇ f, ~H i M But now we are interested in the particular case when f : M → R is the lift ofa basic function f : B → R , namely f = f ◦ π . EAN CURVATURE OF HYPERSURFACES 11
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) △ Σ f ◦ ϕ ( p ) = △ B f ( π ◦ ϕ ( p )) − Hess B f ( dπ ( ν ) , dπ ( ν )) + n h∇ f, ~H τ i where (12) ~H τ := ~H + 2 n h ν, ξ i∇ ξ ( ν H ) Proof. If f : M → R is the lift of a basic function f : B → R ,Hess M f ( X, Y ) = h X, ∇ Y ∇ f i M = Y h X, ∇ f i M − h∇ Y X, ∇ f i M Observe that since h∇ f, X i M = df ( X ) = d ( f ◦ π )( X ) = df ( dπ ( X )) = h∇ f , dπ ( X ) i B then ∇ f is an horizontal vector field in X ( M ), π -related with ∇ f ∈ X ( B ). Let usdecompose ν = ν H + ν V in its horizontal and vertical part, thenHess M f ( ν, ν ) =Hess M f ( ν H , ν H ) + 2Hess M f ( ν H , ν V ) + Hess M f ( ν V , ν V )=Hess M f ( ν H , ν H ) + 2 h ν, ξ i Hess M f ( ν H , ξ ) + h ν, ξ i Hess M f ( ξ, ξ )=Hess M f ( ν H , ν H ) − h ν, ξ ih∇ ν H ξ, ∇ f i − h ν, ξ i h∇ ξ ξ, ∇ f i =Hess M f ( ν H , ν H ) − h ν, ξ ih∇ ξ ( ν H ) , ∇ f i = ν H ( h ν H , ∇ f i ) − h∇ ν H ν H , ∇ f i − h ν, ξ ih∇ ξ ( ν H ) , ∇ f i = dπ ( ν )( h dπ ( ν ) , ∇ f i B ) − h∇ B dπ ( ν ) dπ ( ν ) , ∇ f i B − h ν, ξ ih∇ ξ ( ν H ) , ∇ f i =Hess B f ( dπ ( ν ) , dπ ( ν )) − h ν, ξ ih∇ ξ ( ν H ) , ∇ f i where we have used that dπ ( ∇ ν H ν H ) = ∇ B dπ ( ν ) dπ ( ν ) see [15]. Therefore, △ Σ f ◦ ϕ ( p ) = △ M f ( ϕ ( p )) − Hess B f ( dπ ( ν ) , dπ ( ν )) + 2 h ν, ξ ih∇ ξ ( ν H ) , ∇ f i + n h∇ f, ~H i M Moreover given the orthonormal basis { E i } ni =1 ∪ { ξ } (with E i horizontals), △ M f ( ϕ ( p )) = n X i =1 Hess M f ( E i , E i ) + Hess M f ( ξ, ξ )= n X i =1 Hess M f ( E i , E i ) = n X i =1 Hess B f ( dπ ( E i ) , dπ ( E i )) = △ B f ( π ( ϕ ( p )) . Hence, finally △ Σ f ◦ ϕ ( p ) = △ B f ( π ◦ ϕ ( p )) − Hess B f ( dπ ( ν ) , dπ ( ν )) + 2 h ν, ξ ih∇ ξ ( ν H ) , ∇ f i + n h∇ f, ~H i In order to simply the expression we will make us of the τ -mean curvature ~H τ of Σ defined in (12). Then(13) △ Σ f ◦ ϕ ( p ) = △ B f ( π ◦ ϕ ( p )) − Hess B f ( dπ ( ν ) , dπ ( ν )) + n h∇ f, ~H τ i (cid:3) Radial functions on the base manifold.
Suppose that f : B → R is aradial 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 theHessian and Laplacian of f Proposition 3.4.
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 moreoverthat the sectional curvatures of B are bounded from above and below for any planeof the tangent space, k ≤ sec( B ) ≤ κ. Then, for any function F : → R → R with F ′ ≥ , (14) − Hess B f x ( X, X ) ≥ − F ′′ ( t ) h∇ r o , X i − F ′ ( t ) sn ′ k ( t )sn k ( t ) (cid:0) k X k − h X, ∇ r o i (cid:1) △ B F ≥ F ′′ ( t ) + ( n − F ′ ( t ) sn ′ K ( t )sn K ( t ) where here f = F ◦ r o , t = r ( x ) and sn K ( t ) := sin( √ Kt ) √ K if K > t if K = 0 sinh( √− Kt ) √− K if K < Proof.
By using the definition of the Hessian and the chain rule,Hess B f x ( X, X ) = h∇ X ∇ f, X i = h∇ X F ′ ∇ r , X i = F ′′ ( t ) h∇ r , X i + F ′ ( t ) h∇ X ∇ r , X i = F ′′ ( t ) h∇ r , X i + F ′ ( t )Hess B r o ( X, X ) . Therefore, △ B f ( x ) = F ′′ ( t ) + F ′ ( t ) △ B r o ( x ) . But if the sectional curvatures of the base manifold are bounded as k ≤ sec ≤ κ ,see Theorem 27 of [17],sn ′ κ ( r o ( x ))sn κ ( r o ( x )) (cid:0) k X k − h X, ∇ r o i (cid:1) ≤ Hess B r o ( X, X )and Hess B r o ( X, X ) ≤ sn ′ k ( r o ( x ))sn k ( r o ( x )) (cid:0) k X k − h X, ∇ r o i (cid:1) Then, sn ′ κ ( r o ( x ))sn κ ( r o ( x )) ( n − ≤ △ B r ≤ sn ′ k ( r o ( x ))sn k ( r o ( x )) ( n − F ′ > − Hess B f x ( X, X ) ≥ − F ′′ ( t ) h∇ r o , X i − F ′ ( t ) sn ′ k ( t )sn k ( t ) (cid:0) k X k − h X, ∇ r o i (cid:1) △ B F ≥ F ′′ ( t ) + ( n − F ′ ( t ) sn ′ κ ( t )sn κ ( t ) EAN CURVATURE OF HYPERSURFACES 13 (cid:3)
If we have a Killing submersion π : M → B we can lift the radial function f to e f = f ◦ π and using equation (11) of proposition 3.3 we obtain for F ′ > △ Σ f ( z ) ≥ F ′′ ( t ) (cid:0) − h∇ r o , dπ ( ν ) i (cid:1) + ( n − F ′ ( t ) sn ′ κ ( t )sn κ ( t ) − F ′ ( t ) sn ′ k ( t )sn k ( t ) (cid:0) k dπ ( ν ) k − h dπ ( ν ) , ∇ r o i (cid:1) + nF ′ ( t ) h∇ r o , ~H τ i where f = e f ◦ ϕ and t = r ( π ◦ ϕ ( z )). This above inequality can be rewritten inthe 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 k ≤ sec( B ) ≤ κ. Let F : R → R be a smooth and non-decreasing function, let r o : B → R be thedistance function in B to o ∈ B , i.e. , r o ( p ) = dist B ( o, p ) , denote by f = F ◦ r o ◦ π .Then, (17) △ Σ f ( z ) ≥ F ′′ ( t ) + ( n − F ′ ( t ) sn ′ κ ( t )sn κ ( t ) − F ′ ( t ) sn ′ k ( t )sn k ( t ) k dπ ( ν ) k + (cid:18) F ′ ( t ) sn ′ k ( t )sn k ( t ) − F ′′ ( t ) (cid:19) h∇ r o , dπ ( ν ) i + nF ′ ( t ) h∇ r o , ~H τ i where t = r o ( π ( ϕ ( z ))) and ~H τ is given by definition (12). Stochastic Completeness, weak maximum principle and Omori-Yaumaximum principle.
Let Σ be a complete and non compact Riemannian man-ifold. The heat kernel of Σ is a function p t ( x, y ) on (0 , ∞ ) × Σ × Σ which is theminimal positive fundamental solution to the heat equation ∂v∂t = △ v. In other words, the Cauchy problem ∂v∂t = △ vv | t =0 = v ( x )has a solution v ( x, t ) = Z Σ p t ( x, y ) v ( y ) dV ( y )provided that v is a bounded continuous positive function. The manifold Σ is saidto be stochastically complete, see [8], if Z Σ p t ( x, y ) dV ( y ) = 1for any x ∈ Σ and any t >
0. The main property of stochastic completeness whichis used in this paper is that if a Riemannian manifold is stochastic complete a weak maximum principle is satisfied for bounded functions in C . More precisely, if Σ isstochastically 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 (Σ) with sup Σ u < ∞ there exists a sequence { x k } , k = 1 , , . . . , such that, for every k , u ( x k ) ≥ sup Σ u − /k and △ u ( x k ) ≤ /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 ( M ) which isbounded sup M u = u ∗ < ∞ , there exists a sequence { x i } i ∈ N ⊂ M such that u ( x i ) > u ∗ − i , k∇ u ( x i ) k < i , △ u ( x i ) < i . In this paper we will use the following sufficient condition for the Omori-Yau max-imum principle
Theorem 3.7 (See [4]) . Let Σ be a connected non-compact Riemannian manifold.Suppose that Σ admits a C function f : Σ → R satisfying (1) f ( x ) → ∞ when x → ∞ (2) k∇ f k ≤ G ( f ) outside a compact subset of Σ . (3) △ f ≤ G ( f ) outside a compact subset of Σ .with G ∈ C ( R + ) , positive near infinity and such that G / ∈ L (+ ∞ ) and G ′ ( t ) ≥ − A (log( t ) + 1) for t large enough and A ≥ . Then, the Omori-Yau maximum principle for theLaplacian 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 − ( B B R ( p )) for some geodesic ball B B R ( o ) of radius R centered at o ∈ B . Assumethat the sectional curvatures are bounded sec ≤ κ in B B R ( o ) and that R < min (cid:26) inj( o ) , π √ κ (cid:27) where inj( o ) is the injectivity radius of o and we replace π/ √ κ by + ∞ if κ < . Then, if Σ is stochastically complete, the supremum of the norm of the meancurvature vector field of Σ satisfies sup Σ k ~H k ≥ h nκ ( R ) − τ Σ n where h nκ ( R ) is the norm of the mean curvature of the generalized cylinder ∂B M n ( κ ) R × R in M n ( κ ) × R .Proof. Since π ( ϕ (Σ)) is bounded and contained in the geodesic ball B B R ( o ) for some o ∈ B R ∗ := sup π ( ϕ (Σ)) r o ≤ R < ∞ , with r o ( · ) = dist B ( o, · ) . EAN CURVATURE OF HYPERSURFACES 15
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 −∞ < k ≤ inf p ∈ B B R ( o ) sec(Π p ) ≤ sec(Π p ) ≤ κ. In order to simplify the argument of the proof let us choose k <
0, and let us definethe function F k : R → R , t → F k ( t ) = Z t sn k ( s ) ds Now we are going to compute the Laplacian of f = F k ◦ r o ◦ π ◦ ϕ . By usinginequality (17) of corollary 3.5,(18) △ Σ f ( z ) ≥ sn ′ k ( t ( z )) + ( n − k ( t ( z )) sn ′ κ ( t ( z ))sn κ ( t ( z )) − sn ′ k ( t ( z )) k dπ ( ν ) k − n sn k ( t ( z )) k ~H τ ( z ) k Since k ≤
0, then sn ′ k ≥ △ Σ f ( z ) ≥ ( n − k ( t ( z )) sn ′ κ ( t ( z ))sn κ ( t ( z )) − n sn k ( t ( z )) k ~H τ ( z ) k Now, we are going to apply theorem 3.6 to f : Σ → R because since sup Σ r ◦ π = R ∗ , and F k is an increasing function, sup Σ f = Z R ∗ sn k ( s ) ds < ∞ . Then, thereexists a sequence { x i } , such that f ( x i ) ≥ sup Σ f − i , and △ f ( x i ) ≤ i Therefore t ( x i ) → R ∗ when i → ∞ , and by inequality (19),(20) 1 i ≥ ( n − k ( t ( x i )) sn ′ κ ( t ( x i ))sn κ ( t ( x i )) − n sn k ( t ( x i )) k ~H τ ( x i ) k Then,(21) k ~H τ ( x i ) k ≥ ( n − n sn ′ κ ( t ( x i ))sn κ ( t ( x i )) − n sn k ( t ( x i )) i But k ~H τ k = k ~H + n h ν, ξ i∇ ξ ( ν H ) k ≤ k ~H k + n kh ν, ξ ikk∇ ξ ( ν H ) k . Hence, denoting θ = arccos( h ν, ξ i ) and applying proposition 3.2,(22) k ~H τ k ≤k ~H k + 2 n | cos ( θ ) | k ν H k τ = k ~H k + τn | sin(2 θ ) |≤k ~H k + τ Σ n Therefore,(23) sup Σ k ~H k ≥ ( n − n sn ′ κ ( t ( x i ))sn κ ( t ( x i )) − n sn k ( t ( x i )) i − τ Σ n Letting now i tend to infinity,(24) sup Σ k ~H k ≥ ( n − n sn ′ κ ( R ∗ )sn κ ( R ∗ ) − τ Σ n · Since R ∗ ≤ R < π/ √ κ , sn ′ κ sn κ is a decreasing function and therefore(25) sup Σ k ~H k ≥ ( n − n sn ′ κ ( R )sn κ ( R ) − τ Σ n · Finally the theorem follows by taking into account that ( n − n sn ′ κ ( R )sn κ ( R ) is the norm ofthe mean curvature of the generalized cylinder ∂B M n ( κ ) R × R in M n ( κ ) × R . (cid:3) Remark . In the proof of theorem 2.1 we have used in inequality (23) that τ ≤ τ Σ and | sin(2 θ ) | ≤
1. We could use instead the following factors to improve the result τ ∗ := lim ρ → R ∗ sup (cid:8) τ ( x ) : x ∈ Σ \ π − ( B ρ ( p )) (cid:9) α := lim ρ → R ∗ sup (cid:8) | sin(2 θ ( x )) | : x ∈ Σ \ π − ( B ρ ( p )) (cid:9) and the inequality would be(26) sup Σ k ~H k ≥ ( n − n sn ′ κ ( R )sn κ ( R ) − τ ∗ αn · Proof of theorem 2.5
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 thenormal exponential mapexp : s ( B B R ( o )) × R → π − (( B B R ( o ))) , ( p, z ) → exp( p, z ) = exp p ( zξ )is a diffeomorphism.Given the section s : B B R ( o ) → M we can trivialize π − ( B B R ( o )) ≈ s ( B B R ( o )) × R using the following map T : π − ( B B R ( o )) ≈ s ( B B R ( o )) × R given by q ∈ π − ( B B R ( o )) → T ( q ) = ( p ( q ) , z ( q ))where p and z are the following two functions(27) p : π − ( B B R ( o )) → s ( B B R ( o )) , p ( q ) := s ( π ( q )) z : π − ( 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 thehypothesis 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 − : s ( B B R ( o )) × R → π − ( B B R ( o )) , ( p, z ) → T − ( p, z ) = exp p ( zξ ) . We will need furthermore the expression of the gradient of the z function which isgiven in the following lemma. Lemma 5.1.
Let π : M → B a Killing submersion with unit-length Killing vectorfield 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 − ( B B R ( o )) , ( p, z ) → exp( p, z ) = exp p ( zξ ) EAN CURVATURE OF HYPERSURFACES 17 is a diffeomorphism. Then the gradient of the map z : π − ( B B R ( o )) → R defined inequation (27) is given by ∇ z = ξ. Proof.
For any q ∈ π − ( B B R ( o )), T q M = dT − ( T p ( q ) s ( B B R ( o ))) + dT − ( T z ( q ) R )Observe moreover that for any v ∈ T p ( q ) s ( B B R ( o )), dz ( dT − ( v )) = ddt z (exp γ ( t ) ( z ( q ) ξ )) | t =0 = ddt z ( q ) = 0for any curve γ : ( − ǫ, ǫ ) → s ( B B R ( o )) with γ (0) = p and ˙ γ (0) = v . Then, dz ( dT − ( v )) = h∇ z, dT − ( v ) i = 0 . This implies that ∇ z ( q ) ∈ dT − ( T z ( q ) R ) . Then ∇ z ( q ) = α ( q ) ξ ( q )but taking now the curve γ ξ ( t ) = exp p ( q ) (( z ( q ) + t ) ξ ) which is the integral curve of ξ with γ ξ ( o ) = q and ˙ γ ξ (0) = ξ ( q ), ddt z ( γ ξ ( t )) | t =0 = ddt t | t =0 = 1 = dz ( ξ ) = h∇ z ( q ) , ξ ( q ) i we conclude that ∇ z ( q ) = ξ ( q ) . (cid:3) Proposition 5.2.
Let Σ be a complete and non-compact Riemannian manifold.Let ϕ : Σ → M be a proper isometric immersion. Suppose that M admits a Killingsubmersion π : M → B with unit-length Killing vector field, suppose moreover that ϕ (Σ) ⊂ π − ( B B R ( o )) 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 R ( o ) and that R < min (cid:26) inj( o ) , π √ κ (cid:27) where inj( o ) is the injectivity radius of o and we replace π/ √ κ by + ∞ if κ < .Suppose moreover, that π admits a smooth section s : B B R ( o ) → M and the normalexponential map exp : s ( B B R ( o )) × R → π − (( B B R ( o ))) , ( p, z ) → exp( p, z ) := exp p ( zξ ) is a diffeomorphism. If moreover, sup Σ k ~H ( p ) k < ∞ Then, the Omori-Yau principle for the Laplacian holds in Σ .Proof. In order to prove that the Omori-Yau principle for the Laplacian holds in Σwe are using theorem 3.7 with the function f = z ◦ ϕ . Hence, we only need to provethat the function f = z ◦ ϕ when ϕ is proper, and the hypothesis of the propositionare fulfilled, satisfies(1) f ( x ) → ∞ when x → ∞ .(2) |∇ f | ≤ G ( f ).(3) △ Σ f ≤ G ( f ) outside of a compact set. with G ( t ) = √ t + 1.Observe that since the immersion is proper, and Σ is a complete and non-compactmanifold, if dist Σ ( x, x ) → ∞ then dist M ( ϕ ( x ) , ϕ ( x )) → ∞ . But for any p ∈ s ( B B R ( o )) dist M ( ϕ ( x ) , ϕ ( x )) ≤ dist M ( ϕ ( x ) , p ) + dist M ( p, ϕ ( x ))Therefore, since B B R ( o ) is compact there exists p ∗ ∈ s ( B B R ( o )) such that the distanceis minimized at p ∗ , i.e. , dist M ( ϕ ( x ) , p ∗ ) = dist M ( ϕ ( x ) , s ( B B R ( o ))) and an other p ∗ ∈ s ( B B R ( o )) such that dist M ( p, ϕ ( x )) ≤ dist M ( p ∗ , ϕ ( x )) for any p ∈ s ( B B R ).Hence, dist M ( ϕ ( x ) , ϕ ( x )) ≤ dist M ( ϕ ( x ) , s ( B B R )) + dist M ( p ∗ , ϕ ( x )) ≤ z ◦ ϕ ( x ) + dist M ( p ∗ , ϕ ( x )) ≤ f ( x ) + dist M ( p ∗ , ϕ ( x ))then, f ( x ) → ∞ when x → ∞ as it was stated. On the other hand, h∇ f, v i = h∇ z, dϕ ( v ) i then, k∇ f k ≤ h∇ f, ∇ f i = h∇ z, dϕ ( ∇ f ) i = k∇ T z k ≤ k∇ z k ≤ k ξ k = 1 ≤ G ( f ( z ))Now we are going to compute △ f . By equation (10), for any orthonormal basis { E i } of T p ΣHess Σ f ( E i , E i ) =Hess M z ( dϕ ( E i ) , dϕ ( E i )) + h∇ z, II p ( E i , E i ) i = h∇ dϕ ( E i ) ∇ z, dϕ ( E i ) i + h∇ z, II p ( E i , E i ) i = h∇ dϕ ( E i ) ξ, dϕ ( E i ) i + h ξ, II p ( E i , E i ) i = h ξ, II p ( E i , E i ) i Then, △ Σ f = n X i =1 Hess Σ f ( E i , E i ) = n h ξ, ~H ( p ) i And hence, △ Σ f ≤ n sup Σ k ~H ( p ) k Then if sup Σ k ~H ( p ) k < ∞ , since f ( x ) → ∞ when x → ∞ , △ Σ f ≤ G ( f )outside of a compact set, and the proposition follows. (cid:3) The statement of theorem 2.5 and its proof is as follows
Theorem.
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 sub-mersion π : M → B with unit-length Killing vector field, suppose moreover that ϕ (Σ) ⊂ π − ( B 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 R ( o ) and that R < min (cid:26) inj( o ) , π √ κ (cid:27) EAN CURVATURE OF HYPERSURFACES 19 where inj( o ) is the injectivity radius of o and we replace π/ √ κ by + ∞ if κ < .Suppose moreover, that π admits a smooth section s : B B R ( o ) → M and the normalexponential map exp : s ( B B R ( o )) × R → π − (( 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 vectorfield of Σ satisfies sup Σ k ~H k ≥ h nκ ( R ) where h nκ ( R ) is the norm of the mean curvature of the generalized cylinder ∂B M n ( κ ) R × R in M n ( κ ) × R .Proof. Likewise to the proof of theorem 2.1, we are using the test function f = F k ◦ r o ◦ π ◦ ϕ with F k : R → R , t → F k ( t ) = Z t sn k ( s ) ds Let us setting, R ∗ := sup Σ r o ( ϕ (Σ)) < ∞ , then, sup Σ f = Z R ∗ sn k ( s ) ds < ∞ . By using inequality (17) of corollary 3.5(28) △ Σ f ( z ) ≥ sn ′ k ( t ) (cid:0) − k dπ ( ν ) k (cid:1) + ( n − k ( t ) sn ′ κ ( t )sn κ ( t )+ n h∇ f, ~H τ i Since sn ′ k ( t ) ≥ △ Σ f ( z ) ≥ ( n − k ( t ) sn ′ κ ( t )sn κ ( t ) + n h∇ f, ~H τ i =( n − k ( t ) sn ′ κ ( t )sn κ ( t ) + h∇ f, ~H i + h∇ f, n h ν, ξ i∇ ξ ( ν H ) i≥ ( n − k ( t ) sn ′ κ ( t )sn κ ( t ) + sn k ( t ) h∇ r o , ~H i − n k∇ f k≥ ( n − k ( t ) sn ′ κ ( t )sn κ ( t ) − sn k ( t ) sup Σ k ~H k − n k∇ f k Now we can assume that sup Σ k ~H k < ∞ , (otherwise, there is nothing to be proved),and hence by using proposition 5.2, Σ satisfies the Omori-Yau maximum principlefor the Laplacian, and f is bounded in Σ, there exists a sequence { x i } f ( x i ) ≥ sup Σ f − i , k∇ f ( x i ) k < i , △ Σ f ( x i ) < i . Then, 1 i ≥ ( n − k ( t ( x i )) sn ′ κ ( t ( x i ))sn κ ( t ( x i )) − sn k ( t ( x i )) sup Σ k ~H k − ni letting i tend to ∞ , taking into account that t ( x i ) → R ∗ when i → ∞ ,sup Σ k ~H k ≥ ( n −
1) sn ′ κ ( R ∗ )sn κ (( R ∗ ) ≥ h κ ( R ) , and the theorem is proved. (cid:3) Acknowledgments.
I am grateful to professor Pacelli Bessa for his valuable helpand for his useful comments and suggestions during the preparation of the presentpaper.
References [1] A. Alarc´on and F. Forstneriˇc. The Calabi-Yau problem, null curves, and Bryant surfaces.
Math. Ann. , 363(3-4):913–951, 2015.[2] A. Alarcn, L. Ferrer, and F. Martn. Density theorems for complete minimal surfaces in 3.
Geometric and Functional Analysis , 18(1):1–49, 2008.[3] L. J. Al´ıas, G. P. Bessa, and M. Dajczer. The mean curvature of cylindrically boundedsubmanifolds.
Math. Ann. , 345(2):367–376, 2009.[4] L.J. Alas, P. Mastrolia, and M. Rigoli. Maximum principles and geometric applications.
Springer Monographs in Mathematics , pages 1–570, 2016.[5] A. Atsuji. Remarks on harmonic maps into a cone from a stochastically complete manifold.
Proceedings of the Japan Academy Series A: Mathematical Sciences , 75(7):105–108, 1999.[6] B. Daniel. Isometric immersions into 3-dimensional homogeneous manifolds.
Comment. Math.Helv. , 82(1):87–131, 2007.[7] L. Ferrer, F. Martn, and W.H. Meeks. Existence of proper minimal surfaces of arbitrarytopological type.
Advances in Mathematics , 231(1):378–413, 2012.[8] A. Grigor ′ yan. Analytic and geometric background of recurrence and non-explosion of theBrownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. (N.S.) , 36(2):135–249,1999.[9] L. P. de M. Jorge and F. Xavier. A complete minimal surface in R between two parallelplanes. Ann. of Math. (2) , 112(1):203–206, 1980.[10] A. M. Lerma and J. M. Manzano. Compact stable surfaces with constant mean curvature inKilling submersions.
Ann. Mat. Pura Appl. (4) , 196(4):1345–1364, 2017.[11] J.M. Manzano and B. Nelli. Height and area estimates for constant mean curvature graphsin E ( κ, τ )-spaces. Journal of Geometric Analysis , 27(4):3441–3473, 2017.[12] F. Mart´ın and S. Morales. Complete proper minimal surfaces in convex bodies of R . DukeMath. J. , 128(3):559–593, 2005.[13] F. Mart´ın and S. Morales. Complete proper minimal surfaces in convex bodies of R . II. Thebehavior of the limit set. Comment. Math. Helv. , 81(3):699–725, 2006.[14] N. Nadirashvili. Hadamard’s and Calabi-Yau’s conjectures on negatively curved and minimalsurfaces.
Invent. Math. , 126(3):457–465, 1996.[15] B. O’Neill. The fundamental equations of a submersion.
Michigan Math. J. , 13:459–469, 1966.[16] B. O’Neill.
Semi-Riemannian geometry , volume 103 of
Pure and Applied Mathematics . Aca-demic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983. With applicationsto relativity.[17] P. Petersen.
Riemannian geometry , volume 171 of
Graduate Texts in Mathematics . Springer-Verlag, New York, 1998.[18] S. Pigola, M. Rigoli, and A. G. Setti. A remark on the maximum principle and stochasticcompleteness.
Proc. Amer. Math. Soc. , 131(4):1283–1288, 2003.[19] P. Scott. The geometries of 3-manifolds.
Bull. London Math. Soc. , 15(5):401–487, 1983.[20] M. Tokuomaru. Complete minimal cylinders properly immersed in the unit ball.
KyushuJournal of Mathematics , 61(2):373–394, 2007.
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