Embedded positive constant r-mean curvature hypersurfaces in Mm ×

Let M be an m-dimensional Riemannian manifold with sectional curvature bounded from below. We consider hypersurfaces in the (m + 1)-dimensional product manifold M × R with positive constant r-mean curvature. We obtain height estimates of certain compact vertical graphs in M × R with boundary in M × {0}. We apply this to obtain topological obstructions for the existence of some hypersurfaces. We also discuss the rotational symmetry of some embedded complete surfaces in S2 × R of positive constant 2-mean curvature.


INTRODUCTION
If M m+1 is an (m + 1)-dimensional oriented Riemannian manifold and m is a hypersurface in M, the r-mean curvature of , denoted by H r , is the weighted r'th symmetric function of the second fundamental form (see Definition 2.1).The hypersurfaces with constant r-mean curvature include those of constant mean curvature, and of constant Gauss-Kronecker curvature.In some situations, for example, M = R m+1 , S m+1 , H m+1 (−1), a hypersurface of constant r-mean curvature is a critical point for a certain variational problem (See the related works in (Reilly 1973), (Rosenberg 1993), (Barbosa and Colares 1997) and (Elbert 2002)).
Heinz discovered that a compact graph in R m+1 with zero boundary values, of constant mean curvature H = 0, is at most a height 1 H from its boundary.A hemisphere in R m+1 of radius 1 H shows this estimate is optimal.Using Alexandrov reflection techniques, it follows that a 184 XU CHENG and HAROLD ROSENBERG compact embedded hypersurface with constant mean curvature H = 0 and boundary contained in x m+1 = 0, is at most a distance 2 H from the hyperplane x m+1 = 0. Height estimates have been obtained by the second author (Rosenberg 1993) for compact graphs in R m+1 with zero boundary values and with a higher order positive constant r-mean curvature.He also obtained such height estimates in space forms.For example, in R 3 , such a graph of positive constant Gauss curvature can rise at most 1 √ H 2 from the plane containing the boundary.In general, the maximum height is 1 . For r = 1, this result in hyperbolic space H m+1 (−1) was proved by Korevaar, Kusner, Meeks and Solomon (Korevaar et al. 1992).We refer the reader to (Rosenberg 1993) for applications of these height estimates.
In (Hoffman et al. 2005), Hoffman, Lira and Rosenberg obtained height estimates for some vertical graphs in M 2 × R of nonzero constant mean curvature, where M is a Riemannian surface.
In this paper, we will consider orientable hypersurfaces in an (m + 1)-dimensional oriented product manifold M × R with positive constant r-mean curvature.Here M is an m-dimensional Riemannian manifold with sectional curvature bounded below.The typical models are when M = R m , S m , H m (−1).We will obtain height estimates for such compact vertical graphs with boundary in M ×{0}.A significant difference between this case and the previous cases we discussed is the nature of the linearized operator.It is no longer a divergence form operator and we do not have a flux formula.We prove that, Theorem 1.1.(Th.4.1).Let M be an m-dimensional oriented Riemannian manifold.Let be a compact vertical graph in the (m + 1)-dimensional product manifold M × R with positive constant H r , for some 1 ≤ r ≤ m, with boundary in M × {0}.Let h : → R denote the height function of .
We give some applications of Theorem 1.1 using the Alexandrov reflection technique.First we give the extrinsic vertical diameter estimate of compact embedded surfaces in M × R with An Acad Bras Cienc (2005) 77 (2) CONSTANT r-MEAN CURVATURE HYPERSURFACES 185 positive constant r-mean curvature (under the same curvature assumptions as in Theorem 1.1.See Th.4.2);Secondly, we prove that if M is an m-dimensional compact Riemannian manifold and is a noncompact properly embedded hypersurface in M m ×R with positive constant r-mean curvature (under the same curvature assumptions as in Theorem 1.1), then the number of ends of is not one (Th.4.3).Thus we give a topological obstruction for the existence of such a hypersurface.For example, if is a noncompact properly embedded hypersurface in S m × R of positive constant r-mean curvature for 1 ≤ r ≤ m, then the number of ends of is more than one (Cor.4.1).
In this paper, we also consider the properties of symmetry of certain embedded hypersurfaces of positive constant r-mean curvature.Alexandrov (Alexandrov 1962) showed an embedded constant mean curvature hypersurface in the Euclidean space must be a standard round hypersurface.The symmetric properties of hypersurfaces (with or without boundary) in the Euclidean space and the other space forms with constant mean curvature or higher order constant r-mean curvatures have been studied in varied degrees.We discuss the rotational symmetry of a complete embedded surface in S 2 × R of positive constant 2-mean curvature and obtain that, Theorem 1.2.(Th.5.1; Cor.5.1).Let S 2 be the unit sphere in R 3 and D be the open upper hemisphere in S 2 .Let be a complete orientable embedded surface in S 2 ×R with H 2 = constant > 0. If ⊂ D × R, then has the following properties: is topologically a sphere; and (ii) is a compact surface of revolution about o × R, o ∈ D, that is, is foliated by round circles in D × {t}, the centers of which are o × t, where t ∈ (h min , h max ) and h min , h max are the maximum and minimum of the height of respectively.Moreover, there exists a horizontal level D × {t 0 }, t 0 = h min +h max 2 , such that divides into two (upper and lower) symmetric parts; and (iii) Let φ 0 ∈ 0, π 2 denote the polar angle between the north pole of the open upper hemisphere D and o .Then H 2 > − 1 2 log sin φ 0 (if φ 0 = 0, H 2 > 0) and the generating curve of the rotational surface can be denoted by λ(u) = (φ(u), t (u)) where −1 ≤ u ≤ 1, and φ denotes the polar coordinate (about o ) of S 2 , and: where C is a real constant.
We conjecture a compact immersed surface in S 2 × R of positive constant 2-mean curvature is rotationally symmetric about a vertical line.Moreover, such a surface in H 2 × R is rotational when it is topologically a sphere.Cienc (2005) 77 (2) Let M m+1 be an (m + 1)-dimensional oriented Riemannian manifold and let m be an m-dimensional orientable Riemannian manifold.Suppose x : → M is an isometric immersion.We choose a unit normal field N to and define the shape operator A associated with the second fundamental form of , i.e., for any p ∈ ,

An Acad Bras
where ∇ is the Riemannian connection in M.
Let λ 1 , . . ., λ m denote the eigenvalues of A. The r-th symmetric function of λ 1 , . . ., λ r , denoted by S r , is defined as Particularly, H 1 = H is the mean curvature; H m is the Gauss-Kronecker curvature.H 2 is , modulo a constant, the scalar curvature of , when the ambient space M m+1 is Einstein (Elbert 2002).
We also introduce endomorphisms of T ( ), the Newton transformations, defined by It is obvious that T r , r = 0, 1, . . ., m are symmetric linear operators and T r = S r I − AT r−1 .
Let e 1 , . . ., e m be the principal directions corresponding respectively to the principal curvatures λ 1 , . . ., λ m .For i = 1, . . ., m, let A i denote the restriction of the transformation A to the (m − 1)dimensional subspace normal to e i , and let S r (A i ) denote the r-symmetric function associated to A i .
One has the following properties of T r and S r .
Proposition 2.1 can been verified directly.
See the proof of (i) and (ii) of Proposition 2.2 in (Hardy et al. 1989. p.52).The inequality in (iii) can be obtained from (ii).
Given a function f in C 2 ( ) for p ∈ , the linear operator Hessian of f is defined as Hessf (X) = ∇ X (∇f ), X ∈ T p , where ∇ is the induced connection on .
With T r and Hess, we can define a differential operator L r as follows: Given a local coordinate frame ∂ ∂x i of at p, by direct computation, we have locally the expression of L r , where are the connection coefficients of ∇.
From the above local expression, we know that the linear operator L r is elliptic if and only if T r is positive definitive.Clearly, L 0 = tr (Hessf ) = div(∇f ) is elliptic.
In this paper, the ambient space which we study is an (m + 1)-dimensional product manifold M m × R, where M is an m-dimensional Riemannian manifold.will be a hypersurface in M × R with positive constant r-mean curvature, i.e, H r = constant > 0.
We use t to denote the last coordinate in M × R.
The height function, denoted by h, of in M × R is defined as the restriction of the projection

THE ELLIPTIC PROPERTIES OF THE OPERATOR L r
The following Prop.3.1 is known (cf.Elbert 2002.Lemma 3.10).For completeness, we give its proof.
Proposition 3.1.Let M m+1 be an (m + 1)−dimensional oriented Riemannian manifold and let m be a connected m−dimensional orientable Riemannian manifold.Suppose x : → M is an isometric immersion.If H 2 > 0, then the operator L 1 is elliptic.
Proof.Since L 1 is elliptic if and only if T 1 is positive definite, we will prove that T 1 is positive definite.
By H 2 > 0 and H 2 1 ≥ H 2 , H 1 is nonzero and has the same sign on .We may choose the normal N such that H 1 is positive.So S 1 = mH 1 > 0. By In Prop.3.2, we will prove the ellipticity of L r when the hypersurface with positive r-mean curvature has an elliptic or parabolic point.
Proposition 3.2.Let M m+1 be an (m + 1)−dimensional oriented Riemannian manifold and let m be a connected m−dimensional orientable Riemannian manifold (with or without boundary).
Suppose x : → M is an isometric immersion with H r > 0 for some 1 ≤ r ≤ m.If there exists an interior point p of such that all the principle curvatures at p are nonnegative, then for all 1 ≤ j ≤ r − 1, the operator L j is elliptic, and the j -mean curvature H j is positive.
Proof.Since L j is elliptic if and only if T j is positive definite, it is sufficient to prove that the eigenvalues of T j are positive, that is S j (A i ) > 0 on , for all 1 In the following proof, the ranges of j and i are 1 ≤ j ≤ r − 1, 1 ≤ i ≤ m respectively.Since S r > 0 and the principal curvatures of x at p are nonnegative, we have that at p, at least r principal curvatures are positive.For simplicity, we suppose that λ 1 , λ 2 , . . ., λ r are positive.
Then, by direct verification, at p, S j (A i ) > 0. By continuity, there exists an open intrinsic ball B(p) ⊂ with center p such that the functions S j (A i ) > 0 on B(p).
Define J By continuity, at t 0 , S j (A i ) ≥ 0. We will prove that S j (A i ) > 0 at t 0 and hence t 0 ∈ J .We first show that S r−1 (A i ) > 0 at t 0 .Otherwise, there exists 1 ≤ i ≤ r − 1 such that S r−1 (A i ) = 0 at t 0 .For this i, by S r = λ i S r−1 (A i ) + S r (A i ), we have S r (A i ) = S r > 0 at t 0 .So, at t 0 , S j (A i ) ≥ 0, 1 ≤ j ≤ r − 1 and S r (A i ) > 0. By (ii) in Prop.2.2, at t 0 , the following inequality holds, An Acad Bras Cienc (2005) 77 (2) CONSTANT r-MEAN CURVATURE HYPERSURFACES
So we obtain that at q, S j (A i ) > 0. Hence the L j , for all 1 ≤ j ≤ r − 1 are elliptic.By Prop.2.1 (ii), H j (for all 1 ≤ j ≤ r − 1) are positive.

THE HEIGHT ESTIMATES OF THE VERTICAL GRAPH
In this section, we will consider hypersurfaces in an (m+1)−dimensional product manifold M m ×R of positive constant r-mean curvature.
Lemma 4.1.Let M be an m−dimensional oriented Riemannian manifold and let be an immersed orientable hypersurface in M × R (with or without boundary).Then where 1 ≤ r ≤ m + 1, h denotes the height function of , and n = N, ∂ ∂t .Proof.Fix p ∈ .Let {e i } be a geodesic orthonormal frame of at p.So at p, ∇ e i e j (p) = 0. We can assume {e i } are the principal directions at p, that is, Ae i (p) = λ i e i , where λ i are the eigenvalues (principal curvatures) of A at p (this frame can been obtained by rotating {e i }).
Observe that the vertical translation in M × R is an isometry.Hence ∇h = ∂ ∂t is a Killing vector field, and ∇ e i ∇h = 0, for i = 1, . . ., m.
We have where the T denotes the tangent part of the vector of T (M × R) to .Then Let A t (p) be the shape operator of (t) at p and for 0 ≤ r ≤ m, let S r (t)(p) be the r-th symmetric function of the eigenvalues of A t (p).We are interested in the first variation of S r .To calculate this one may differentiate the equation rS r = tr(T r−1 A).We refer the reader to (Rosenberg 1993) for details of this calculation in space forms.In general ambient spaces, Elbert (Elbert 2002.Proposition 3.2) proved the following result.
Set E s (p) = ∂φ ∂s (p, s) and f s =< E s , N s >, where N s is the unit normal to φ s (D).We have where R N is defined as R N (X) = R(N, X)N, R the curvature operator of M, and E T s denotes the tangent part of E s .
Lemma 4.2.With M and as in Lemma 4.1, assume H r of is constant, for some r, 1 ≤ r ≤ m+1.Then, on , (4.5) Proof.We may choose the variation φ s : Under this variation, S r (s) is constant, i.e., ∂ ∂s S r (s) = 0. Also by the hypothesis S r = constant on , we have E T s (S r ) = 0, for s = 0. Hence, when s = 0, we have (4.6) Thus (4.6) holds on .
We will express tr(T 1 R N ) using the sectional curvature of M in order to prove Theorem 4.1.Given p = (x, t) ∈ M × R. For X ∈ T p M × R, let X h denote the horizontal component of X.If {e i } denotes a geodesic frame of at p and N denotes the unit normal to at p, we have, by direct calculation, 191 where K is the sectional curvature of M.
If e i are also the principal directions of A. We have T r e i = S r (A i )e i .Then Theorem 4.1.Let M be an m-dimensional oriented Riemannian manifold.Let be a compact vertical graph in the product manifold M × R with boundary in M × {0} and with positive constant H r , for some 1 ≤ r ≤ m.Let h : → R denote the height function of .
Proof.At a highest point, all the principal curvatures have the same sign.Since we assume that H r > 0, we know that at this point, all the principal curvatures are nonnegative and the unit normal N to is downward pointing.Since is a vertical graph, we may choose the smooth unit normal field N to to be downward pointing (i.e.,n = N, ∂ ∂t ≤ 0 on ).Hence we can apply Prop.3.1 and Prop.3.2 to obtain that, L r−1 is elliptic, and H i (1 ≤ i ≤ r − 1) are positive.
Define ϕ = ch + n on , where c is a positive constant to be determined.On ∂ , ϕ = n ≤ 0. Since L r−1 is an elliptic operator, we have, by the maximum principle, that if L r−1 ϕ ≥ 0, then ϕ ≤ 0 on .Then h ≤ − n c ≤ 1 c .Now we will choose c such that L r−1 ϕ ≥ 0. By Lemma 4.1 and 4.2, So we may choose c such that (i) Take a = 0 and choose c ≤ H By (iii) When K ≥ a(a < 0) and r = 1, we have a better estimate.Note Similar to the above, we may choose c Remark 4.1.The height estimate in Theorem 4.1 is sharp.Consider a hemisphere of the unitary round sphere S m in R m+1 .It is a vertical graph on R m = R m × {0} of H r = 1 with boundary S m−1 × {0} and has the maximum height 1.
Theorem 4.2.Let M be an m-dimensional Riemannian manifold and let be a compact orientable embedded hypersurface in M × R with H r = constant > 0, for some 1 ≤ r ≤ m.Then is symmetric about some horizontal surface M × {t 0 }, t 0 ∈ R.Moreover, Proof.We will prove that has a horizontal surface of symmetry M(t 0 ) = M × {t 0 }, for some t 0 ∈ R. Note that vertical translation is an isometry, as well as reflection through each horizontal M(t).Hence we can use the Alexandrov reflection technique.One comes down from above with the horizontal surfaces M(t).For t slightly smaller than the highest value of the height of , the part of above M(t) is a vertical graph of bounded gradient over a domain in M(t).The symmetry through M(t) of this part of , is below M(t), contained in the domain bounded by , and meets only along the boundary.One continues to do these reflections through the M(t) translated downwards, until a first accident occurs.This plane is a plane of symmetry of , and the part of above this plane is a graph with zero boundary values.This proves the Theorem.Since is not invariant by symmetry in any M(t), there is no first point of contact of the symmetry of the part of above M(t), with the part of below M(t).But by the Alexandrov reflection technique, the part of above each M(t) is always a vertical graph.This contradicts the height estimate for such vertical graphs.
Corollary 4.1.Let be a noncompact properly embedded hypersurface in S m × R with H r = constant > 0, for some 1 ≤ r ≤ m.Then the number of ends of is not one.

ROTATIONAL SYMMETRY OF SOME SURFACES IN S 2 × R OF CONSTANT 2-MEAN CURVATURE
If M is the unit sphere S 2 in R 3 , an embedded surface in S 2 × R of positive constant 2-mean curvature has another symmetry under some restriction.If we demand the vertical projection of such a surface on S 2 × {0} is contained in an open hemisphere, then it is also rotational about a vertical line parallel to the vertical R-axis.
We now give some definitions which are modifications of the related concepts in R 3 (cf.Hopf 1983. p.147-148).
Let D be the open hemisphere in S 2 and ∂D be the boundary of D. Let {p, −p} denote a pair of antipodal points on ∂D, and γ denote a semi-circle on D joining p and −p.We call the surface P = γ × R in S 2 × R a vertical geodesic strip.
Definition 5.1.For a point q ∈ D × R, a point q ∈ D × R is called a point of symmetry of q about a vertical geodesic strip P = γ × R if, for a geodesic l passing through q and perpendicular to P , we have q ∈ l, lies on the opposite part of l divided by P , and dist(q, P ) = dist(q , P ).This means q is the image of q by the isometry of S × R which is reflection of each S(t) through γ × {t}.
A vertical geodesic strip P = γ × R in S 2 × R is called a vertical strip of symmetry for a set W ⊂ S 2 × R if, for every point q ∈ W , its point of symmetry q is also in W .
We will obtain the following: Theorem 5.1.Let S 2 be the unit sphere in R 3 and D be an open hemisphere in S 2 .Let be a complete oriented embedded surface in S 2 × R with H 2 = constant > 0. If ⊂ D × R, then has the following properties: is topologically a sphere; and (ii) is a surface of revolution about a vertical line parallel to the vertical R-axis, that is, is foliated by round circles in D × {t}, the centers of which are the same point on D modulo t, where t ∈ (h min , h max ) ⊂ R, and h min , h max are the maximum and minimum of vertical height of respectively; and Hess(h)(e i ), e i (p) = − ∇h, N ∇ e i N, e i (p)= n λ i e i , e i (p) = nλ i Hence L r−1 (h)(p) = i e i , T r−1 Hessh(e i ) (p) = i n(p)λ i T r−1 (e i ) (4.2) = n(p)tr(AT r−1 ) = n(p)rS r .An Acad Bras Cienc(2005) 77 (2) 190 XU CHENG and HAROLD ROSENBERG Since rS r n is independent of the choice of the frame, we obtain that, on L r−1 (h)(p) = rS r n. (4.3 ) In order to prove the following Lemma 4.2, we first recall some known results.Let ⊂ M m+1 be an oriented hypersurface.Let D be a compact domain of .Consider a variation of D, denoted by φ. φ : (− , ) × D → M × R, > 0, such that for each s ∈ (− , ) , the map φ s : {s} × D → M × R, φ s (p) = φ(s, p) is an immersion, and φ 0 = D.
Remark 4.3.Consider the case m = 2 and r = 2 in Theorem 4.1 or Theorem 4.2.From the Gauss equation, we obtain that the intrinsic sectional curvature of has a positive lower bound H 2 .So Bonnet-Myers theorem yields the intrinsic diameter estimate of and hence the vertical height or the extrinsic vertical diameter estimate.That is, (a) when is a vertical graph with zero boundary values, a geodesic on from the highest point to ∂ (and hence the vertical height) is at most π Suppose on the contrary, that has exactly one end E. Since is properly embedded, E must go up or down, but not both.Assume E goes down.Then has a highest point so we can do Alexandrov reflection coming down from above with horizontal surfaces M(t).
Theorem 4.3.Let M be an m-dimensional compact Riemannian manifold and let be a noncompact properly embedded hypersurface in M × R with H r = constant > 0, for some 1 ≤ r ≤ m.If the sectional curvature of M and the r-mean curvature of satisfy the conditions in (i), (ii) or (iii) of Theorem 4.1, then the number of ends of is not one.Proof.