c 2009 Birkh¨auser Verlag Basel/Switzerland 0003-889X/09/010067-10
published onlineJuly 7, 2009
DOI 10.1007/s00013-009-0012-9 Archiv der Mathematik
The influence of the boundary behavior on isometric
immersions in the hyperbolic space
L. Jorge, H. Mirandola, and F. Vitorio
Abstract. This paper studies how the behavior of a proper isometric immersion into the hyperbolic space is influenced by its behavior at infin-ity. Our first result states that a proper isometric minimal immersion into the hyperbolic space with the asymptotic boundary contained in a sphere reduces codimension. This result is a corollary of a more general one that establishes a sharp lower bound for the sup-norm of the mean curva-ture vector of a Proper isometric immersion into the Hyperbolic space whose Asymptotic boundary is contained in a sphere. We also prove that a properly immersed hypersurfacef : Σn→
Hn+1 with mean curvature
satisfying supp∈ΣH(p) < 1 has no isolated points in its asymptotic
boundary. Our main tool is a Tangency principle for isometric immer-sions of arbitrary codimension.
Mathematics Subject Classification (2000). Primary 53C42; Secondary 53C40.
Keywords. Mean curvature, Hyperbolic space, Proper immersion, Asymptotic boundary, Tangency principle.
1. Introduction. Several works motivated principally by the Alexandrov reflection method showed that a properly embedded hypersurface into the hyperbolic space with constant mean curvature inherits the symmetry of its boundary (see [1,2,8,9]).
Them-dimensional hyperbolic spaceHmcarries a natural compactification:
Hm=Hm∪Sm−1(∞)
whereSm−1(∞) is identified with the asymptotic classes of geodesic rays inHm and carries, in a natural way, the standard conformal structure (isometries of Hmbecome conformal automorphisms ofSm−1(∞)). Theasymptotic boundary
of a subsetB⊂Hmis defined by:
∂∞B=B∩Sm−1(∞)
where B is the closure ofB in Hm. By asphere in Sm−1(∞) we denote the
asymptotic boundary of a complete totally geodesic hypersurface ofHm. If we considerHmin its Poincar´e ball model thenSm−1(∞) coincides with the
stan-dard unit sphere S1m−1 ⊂ Rm and the spheres of Sm−1(∞) are the geodesic
hyperspheres ofSm−1 1 .
Theorem 1.1. Letf : Σ→Hmbe a proper isometric immersion of a connected
Riemannian manifold Σ. Assume that the asymptotic boundary ∂∞f(Σ) is
contained in a sphere S. Let Λm−1 ⊂ Hm be the complete totally geodesic
hypersurface that hasS as its asymptotic boundary. Then, the mean curvature off satisfies:
sup p∈Σ
H(p) ≥tanh(dH(f(p),Λ)), (1)
for all p∈ Σ, where dH is the hyperbolic distance. Furthermore, if for some p∈Σthe equality in(1)is satisfied thenf(Σ)is contained in a totally umbilical hypersurfaceΓm−1⊂Hm with mean curvatureH
Γ= supp∈ΣH(p).
In Example 3 of [10], the second author exhibits examples of properly embedded surfaces Σ2⊂H3whose asymptotic boundary is a sphereS=∂
∞Λ, where Λ is a complete totally geodesic surface ofH3,and the mean curvature
H of Σ2 satisfies:|H(p)|<tanh(d
H(p),Λ),for allp∈Σ.
We recall that an immersionf : Σ→Hmreduces codimension if its image
f(Σ) is contained in a totally geodesic hypersurface ofHm.
Corollary 1.1. Let f : Σ → Hm be a proper minimal isometric immersion.
If the asymptotic boundary ∂∞f(Σ) is contained in a sphere then f reduces
codimension.
Corollary1.1generalizes a result by do Carmo and Lawson [1] that states that a properly immersed minimal hypersurface into Hm whose asymptotic boundary is a sphere is totally geodesic.
A beautiful theorem by do Carmo et al. [2] states that a properly embedded hypersurface inHm with constant mean curvatureH ∈[0,1) has no isolated point in its asymptotic boundary. We prove:
Theorem 1.2. Let f : Σ→Hm be a proper immersed hypersurface with mean
curvature satisfyingsupp∈Σ|H(p)|<1. Then the asymptotic boundary∂∞f(Σ)
has no isolated points.
The surface Σ2⊂H3parameterized byφ(u, v) = (u, v, ev) (consideringH3
in the Poincar´e half-space model) is properly embedded with a single point in its asymptotic boundary and mean curvature satisfying|H|= (2 + 3ev)/(2(1 +
e2v)3
2)<1, which shows that the sup-norm in Theorem1.2 is essential. This
example was exhibited by Lluch [7].
Definition 1.1. Let M be an (n+ 1)-dimensional Riemannian manifold and
B⊂M a hypersurface. Givenǫ≥0, we say thatB is (k, ǫ)-mean convex with respect to a normal direction field ν on B, 1≤k ≤ n, if for all points of B
we have:λ1+· · ·+λk ≥kǫ, whereλ1≤ · · · ≤λn are the principal curvatures ofB.
Definition 1.2. Let M be a Riemannian manifold, f : Σ → M an isometric immersion andB ⊂M be a hypersurface. Let p ∈ Σ with f(p) ∈ B and ν
a unit normal vector field onB. We say thatf remains aboveB near p with respect toν(p) if there exist neighborhoodsU ⊂Σ of pandV ⊂M off(p), such thatV−Bhas exactly two connected components andf(U) is contained in the closure of that connected component ofV −B into whichv(p) points.
Proposition 1.1 (Tangency Principle). Let M be a Riemannian manifold and f : Σ→M an isometric immersion. Considerk= dim Σ and let B ⊂M be a(k, ǫ)-mean convex hypersurface with respect to a normal directionν on B. Assume that, for some pointp∈Σ,f(p)∈B andf(Σ) remains aboveB near pwith respect to ν(p). Furthermore, assume that the mean curvature H of f satisfiesH ≤ǫ, nearp. Thenf(U)⊂B, for some neighborhoodU ofpinΣ.
Remark 1.1. Proposition1.1 with f minimal was proved by Jorge and Tomi [6]. The proof of Proposition 1.1 was inspired in the proof of this particular case.
The proof of Theorem1.2uses an important result (see Proposition1.2). To present this result we will to recall the concept of distance between two com-pact sets inSm−1(∞), as defined in [2]. First, two subsetsA
1, A2⊂Sm−1(∞)
are separated by two disjoint spheresS1, S2 if they are contained in distinct
disk-type connected components ofSm−1(∞)−(S1∪S2). Considerd(S1, S2) :=
dH(Λ1,Λ2), where Λi ⊂Hm, i = 1,2, are the totally geodesic hypersurfaces with∂∞Λi=Si. The distanced(A1, A2) fromA1 toA2 is defined by:
d(A1, A2) := ⎧
⎨
⎩
0, if there are no disjoint spheresS1andS2
that separateA1andA2;
sup
d(S1, S2)S1andS2separateA1andA2.
(2)
The distanced(·,·) is conformally invariant, since conformal transformations ofSm−1(∞) are induced by isometries ofHm. Furthermore, form≥2, the dis-tance of a compact set from a point away from this set is infinite. Notice also that if d(A1, A2)<∞ then, by compactness, there exist two disjoint sphere
S1 andS2satisfying:d(A1, A2) =d(S1, S2).
Remark 1.2. Although do Carmo et al. to refer d as a distance, they also observe that the triangle inequality does not hold in general.
In [2], do Carmo, Gomes, and Thorbergsson proved:
Theorem A (Theorem 1 of [2]). LetΣn⊂Hn+1 be a properly embedded
such component. Then there exists a constant dH (depending only H, and
computable) such that
d(A, ∂∞Σ−A)≤dH (3)
and the equality holds if only ifΣn is a rotation hypersurface of spherical type.
Using the techniques developed in [2] to prove Theorem A, we prove the following result:
Proposition 1.2. Let Σ be a connected n-dimensional Riemannian manifold andf : Σn →Hn+1 a proper isometric immersion. Assume that the
asymp-totic boundary ∂∞f(Σ) has at least two connected components and let A be
any such component. Assume further that the mean curvature of f satisfies:
supp∈Σ|H(p)| < 1. Then there exists a constant d (depending only supp∈Σ
|H(p)|, and computable) such that
d(A, ∂∞f(Σ)−A)≤d
and the equality holds if only if f(Σ) is a rotation hypersurface of spherical type.
Remark 1.3. Theorem1.1(and, consequently, Corollary1.1), assuming thatf
has codimension one, remain true considering, in the place of the mean curva-ture, any normalized symmetric function of the principal curvatures (namelyr -mean curvatures), in particular, escalar and Gauss-Kronecker curvatures. The proof of Theorem1.1, in this more general context, differ simply of the use of the following tangency principle which follows as a consequence of Theorem 1.1 of [3].
Proposition 1.3. Let M1 andM2 be hypersurfaces of a Riemannian manifold
N and a pointp∈M1∩M2 satisfying TpM1=TpM2. Let η be a unit normal
vector field of M2. Assume that M2 is umbilical and nontotally geodesic and
M1 remains aboveM2with respect toν(p). Furthermore, assume that, nearp,
the mean curvatureH ofM2 with respect toν satisfies:
H ≥ min
1≤r≤n|H 1 r|,
whereH1
r,r= 1, . . . , n, are the r-mean curvatures of M1. Then, near p,M1
coincides withM2.
2. Proof of Proposition 1.1. To prove Proposition 1.1 we will need of the following lemmas.
Lemma 2.1. Letf : Σ→M be as in Proposition1.1. Letd∈C2(M)satisfying
that∇d= 1and∇(d◦f)<1. Then we have the inequality
∆(d◦f) + Trace
A|(f∗TΣ)
T
≤kH+(Hessd)◦f
∇(d◦f)2
1− ∇(d◦f)2, (4)
where, at a point x∈M,A is the second fundamental form of the hypersur-face Bx = d−1(d(x)) in the direction of the normal ∇d and, for a subspace
Proof. Denote∇and∇the Riemannian connection of Σ andM, respectively. Fix{e1, . . . , ek} an orthonormal frame on Σ. By the Gauss Formula,
∇f∗eif∗ei=f∗(∇eiei) +α(ei, ei), (5) it follows that
∆(d◦f) = k
i=1
Hess (d◦f)(ei, ei) =kH,∇d+ k
i=1
Hessd(f∗ei, f∗ei). (6)
For any vector v ∈ TxM, we denote by vT its component tangent to the hypersurface B = d−1(d(x)). Using that ∇d(x) = 1 we have that vT =
v− v,∇d(x) ∇d(x) and Hessd(v,∇d) = 0, for allv∈TxM. Thus we have
Hessd(f∗ei, f∗ej) = Hessd(f∗ei)T,(f∗ej)T=−A(f∗ei)T,(f∗ej)T. (7) In order to relate the Eq. (7) to a partial trace of Awe compute
bij:=(f∗ei)T,(f∗ej)T=δij−ei(d◦f)ej(d◦f), from which we get the inverse matrix
bij =δij+
ei(d◦f)ej(d◦f) 1− ∇(d◦f)2 .
Using (7) we have
A
(f∗ei)T,(f∗ei)T
= k
j=1
bijA
(f∗ei)T,(f∗ej)T
+ k
j=1
Hessd(f∗ei, f∗ej)
ei(d◦f)ej(d◦f) 1− ∇(d◦f)2 .
Since
Trace
A|(f∗TxΣ)T
= k
i,j=1
bijA
(f∗ei)T,(f∗ej)T
,
using (6) and (7) we have ∆(d◦f) + Trace
A|(f∗TxΣ)T
=kH,∇d+ k
i,j=1
Hessd(f∗ei, f∗ej)
ei(d◦f)ej(d◦f) 1− ∇(d◦f)2
≤kH+(Hessd)◦f ∇(d◦f)
2
1− ∇(d◦f)2,
which proves Lemma2.1.
The Lemma below was proved in [6].
Lemma 2.2. Let A be a quadratic form on an n-dimensional vector space V with the eigenvaluesλ1≤ · · · ≤λn. Then
Trace(A|W)≥λ1+· · ·+λk,
Finally, consider the signed distance function
d:x∈V →d(x) =
⎧
⎨
⎩
dist(x, B), ifxbelongs to the connected component ofV −B for whichν points;
−dist(x, B), otherwise
Thend∈C2(V) (consideringV sufficiently small) and satisfies∇d= 1.
Fur-thermore, ∇(d◦f)<1 (considering U sufficiently small), since ν =∇d and
f∗(TpM) is a subspace of Tf(p)B. Since kj=1λj ≥kǫ on B (by hypothesis) and each eigenvalueλj ofAis a Lipschitz continuous function, in a sufficiently small neighborhood ofB, it follows the following estimate:
k
j=1
λj(x)≥kǫ−C1|d(x)|,
with a suitable constantC1 ≥0. Using that H ≤ ǫand (4) we obtain the
differential inequality
∆(d◦f)−C1(d◦f)−C2∇(d◦f)2≤0, (8)
with a further constantC2. By hypothesis it follows thatd◦f ≥0, inV, and
d◦f(p) = 0. Therefore, Hopf’s maximum principle (see, for instance, [5]) is immediately applicable to (8), which proves Proposition1.1.
3. Proofs of Proposition1.2and Theorem1.2. First we will prove Proposition 1.2. The arguments to follow are inspired in the proof of Theorem A in [2]. Assume that
d(A, ∂∞f(Σ)−A)> d, (9)
where ddepends only supp∈Σ|H(p)| and it will be given soon. Then we will
derive a contradiction. In fact, setB=∂∞f(Σ)−Aand choose totally geodesic hypersurfaces ΛA, ΛB of Hn+1 with hyperbolic distancedH(ΛA,ΛB) > d, so thatA andB are contained in distinct disk-type connected components,DA andDB, of Sn(∞)−(∂∞ΛA∪∂∞ΛB), whereA⊂DA and∂∞DA=∂∞ΛA.
Consider the Poincar´e half-space model for the hyperbolic space
Hn+1=
(x1, . . . , xn+1)∈Rn+1|xn+1>0
such that ∂∞f(Σ) ⊂ {xn+1 = 0} and DA is a disk centered at the origin 0∈Rn+1. Let γbe the geodesic ofHn+1, represented inRn+1, as the half-line
emanating from the origin, andpA,pB the intersection ofγwith ΛA and ΛB, respectively. Considerpthe middle point of the segmentpApB and letg be a geodesic ofHn+1, orthogonal toγat p.
Consider the one-parameter family{Mλ}λ>0of rotational properly
embed-ded hypersurfaces of spherical type, with constant mean curvature
Hλ= sup p∈Σ
H(p)<1
1. The generating curve of each hypersurfaceMλ(in the vertical hyperplane containingγandg) is symmetric relative togand intersectgat a distance
λ=dH(Mλ, γ) ofp;
2. the mean curvature vector of each hypersurfaceMλ points to the con-nected component ofHn+p−M
λ containing the rotation axisγ;
3. the asymptotic boundary ∂∞Mλ consists of two disjoint n-spheres S1λ,
Sλ
2. Furthermore, the function d(λ) = d(S1λ, Sλ2), λ ∈ (0,∞), satisfies:
limλ→0d(λ) = 0, increases initially, reaches a maximum dmax, and de-creases asymptotically to zero asλ→ ∞, and its maximum value dmax depends only supp∈Σ|H(p)|and can be given in terms of an integral; thus,
it can be explicitly computed to any degree of accuracy (see Proposition 5.6 of [4]).
Consider d := dmax in the inequality (9). Notice that ∂∞Mλ does not intersect eitherA orB, since
d(Sλ
1, S2λ)≤d < dH(pA, pB).
Using thatdH(Mλ, γ) = λand Mλ∩Σ =∅, for λsufficiently large, we have that there exists λ0 such that Mλ0 touches f(Σ) for the first time (by the
properness off), say at a pointf(q).
Consider Mλ oriented by its mean curvature vector and consider a local orientation atpsuch that the unit normal vectorN(p) off atpcoincides with the unit normal vectorNλ0(f(p)) ofMλ0 atf(p). Then f(Σ) lies aboveMλ0,
nearf(p), with respect to N(p). Since the mean curvature ofxsatisfies:
|H| ≤Hλ0= sup
p∈Σ
|H|
by the tangency principle, we have thatf(Σ) coincides with Mλ0, nearf(p).
Then, by the connectedness of Σ, it follows thatf(Σ)⊂Mλ, which contradicts
∂∞f(Σ)∩∂∞Mλ0 =∅. Therefored(A, B)≤d, and it proves the first part.
Now, suppose that the equality (9) holds. Choose ΛA and ΛB, as above, such that dH(ΛA,ΛB) = d. Proceeding as in the first part of the proof, we obtain that f(Σ) = Mλ0 (by the connectedness of Σ and the completeness
off). This proves the second part, and completes the proof of Proposition1.2. The proof of Theorem1.2depends of the following lemma.
Lemma 3.1. Let f : Σ→Hm be a proper isometric immersion of a connected
Riemannian manifoldΣintoHm. Assume that∂∞f(Σ)has a single point then supp∈Σ|H(p)| ≥1.
Proof. ConsiderHmin the Poincar´e half-space model,
Hm={(x1, . . . , xm)∈Rm|xm>0},
such that∂∞f(Σ) ={0}. GivenR >0, let ΓR⊂Hm be the totally umbilical hypersurface with mean curvature
H0>sup p∈Σ
|H(p)|, (10)
whose asymptotic boundary of its convex part is the ballBR(0)⊂ {xn+1= 0}
sufficiently large, so that f(Σ) is contained in the convex part of ΓR. Notice that the convex part of ΓR tends to ∅ as R → 0. Then, by the properness of f, there existsR0 >0 such that ΓR0 touchesf(Σ) for the first time, say
at a pointf(q). Since f(Σ) is contained in the convex part of ΓR0 it follows
from (10) and Proposition1.1 that there exists a neighborhood U of q in Σ satisfying thatf(U)⊂ΓR0. By the connectedness of Σ it follows thatf(Σ) is
contained in ΓR0, which contradicts∂∞f(Σ) ={0}.
Now we are able to prove Theorem 1.2. Since the distance in Sm−1(∞),
m≥2, of a compact set to a point away from this set is infinite it follows from Proposition1.2that if∂∞f(Σ) has an isolated point then∂∞f(Σ) reduces to a single point. The final of the proof follows from Lemma3.1.
4. Proof of Theorem1.1. We can assume that supp∈ΣH(p)<1. Consider
Hm in the Poincar´e ball model,
Hm=(x1, . . . , xm)∈Rm|x21+· · ·+x2m<1,
such that ∂∞f(Σ) is contained in the asymptotic boundary of the following totally geodesic hypersurface
Λ ={(x1, . . . , xm)∈Hm|xm= 0};
consider PN = (0, . . . ,0,1) and PS = (0, . . . ,0,−1) and let γ : R → Hm be the normalized geodesic such that γ(0) = 0, limt→−∞γ(t) = PS and limt→∞γ(t) =PN.
GivenR≥0, let Λ+Rand Λ−Rbe the complete totally geodesic hypersurfaces ofHmthat intersectγorthogonally at the pointsγ(R) andγ(−R), respectively. LetBR be the connected component of Hm−(Λ+R∪Λ
−
R) that contains γ(0). Since∂∞f(Σ)⊂∂∞Λ and f is proper, the family:
f(Σ)∩(Hm−BR)
R>0
is a decreasing chain of compact sets converging to∅, asR→ ∞. Thus, there existsR0>0, sufficiently large, such thatf(Σ)⊂BR0. Choosed0>0,
suffi-ciently large, so that
tanh(d0)>sup p∈Σ
H(p) (11)
and consider, for eachR≥0, the pointsqR+=γ(R+d0) andqR−=γ(−R−d0).
For each 0 ≤ R ≤ R0, let Γ+R = Γ+R(d0) and Γ−R = Γ−R(d0) be the
com-plete totally umbilical hypersurfaces of Hm equidistant to Λ+ R and Λ
−
R, that intersect γ orthogonally atq+R andq−
R, respectively. Since Γ + 0 and Γ
−
0 satisfy
∂∞Γ+0 =∂∞Γ−0 =∂∞Λ we have that Γ+0 and Γ
−
0 are hypersurfaces equidistant
to Λ.
Claim 4.1. f(Σ) lies betweenΓ+0 andΓ−
0.
In fact, by contradiction, assume that Claim4.1 is false. Since the region between Γ+0 and Γ
−
0 is given by the intersection of the convex parts of Γ + 0 and
part is analogue). Since f(Σ) ⊂ BR0 we have that f(Σ) is contained in the
connected component of Hn+p−(Γ+ R0 ∪Γ
−
R0) that contains γ(0). Sincef is
proper there exists 0< R1< R0such thatf(Σ) touches Γ+R1 for the first time,
say at a pointf(q). Notice thatf(Σ) is contained in the convex part of Γ+R1.
Since Γ+R1 and Λ
+
R1 are equidistant each other and intersectγ orthogonally at
the pointsqR+1 =γ(R1+d0) andγ(R1), respectively, the mean curvatureHR1
of Γ+R1 is given by:
HR1 = tanh(d0)>sup
p∈Σ
H(p). (12)
It follows from Proposition 1.1that there exists a neighborhood U of q in Σ such thatf(U)⊂Γ+R1. By the connectedness of Σ it follows thatf(Σ)⊂Γ
+ R1,
which contradicts the fact ∂∞f(Σ) ⊂ ∂∞Λ. Then, f(Σ) is contained in the convex part of Γ+0 and Γ
−
0 (by an analogue argument), which proves the claim.
By the arbitrarily ofd0, it follows thatf(Σ) is contained between two
hyper-surfaces Γ+and Γ−equidistant to Λ with mean curvatureH= sup
p∈ΣH(p).
Therefore,
sup p∈Σ
H(p)=H= tanh dH
Γ±,Λ
≥tanh
sup p∈Σ
dH(f(Σ),Λ)
. (13)
This prove the first part of Theorem1.1.
To prove the second part of the theorem we assume that, for somep∈Σ, we have supp∈ΣH(p)= tanh (d(f(p),Λ)). It follows from (13) that iff is
min-imal thenf(Σ)⊂Λ. Now, assume thatf is not minimal. It follows from (13) that d(f(p),Λ) = supp∈Σd(f(p),Λ) = dH(Γ±,Λ). Then f(Σ) touches either
Γ+ or Γ− and it is contained in the convex parts of Γ+ and Γ−. By Proposi-tion1.1and the connectedness of Σ if follows that eitherf(Σ) is contained in Γ+ or Γ−.
Acknowledgement.The authors thank Professor Manfredo do Carmo for useful discussions during the preparation of this paper. The second author thanks Marcos Petrucio Cavalcante for providing the meeting between himself and the third author.
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L. Jorge
Instituto de Matem´atica, Universidade Federal do Cear´a, Fortaleza, CE, CEP 60455-760, Brazil e-mail:luquesio.jorge@ufc.br
H. Mirandola
Instituto Nacional de Matem´atica Pura e Aplicada, Rio de Janeiro, RJ, CEP 22460-320, Brazil
e-mail:heudson@impa.br
F. Vitorio
Instituto de Matem´atica,
Universidade Federal das Alagoas, Macei´o, AL, CEP 57072-970, Brazil e-mail:feliciano.vitorio@gmail.com
