The Barrier Principle for Minimal Submanifolds of
Arbitrary Codimension
LUQUÉSIO P. JORGE1and FRIEDRICH TOMI2
1Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici,
60455-760 Fortaleza-Ce, Brazil
2Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, 69120 Heidelberg,
Germany
(Received: 19 August 2002; accepted: 6 January 2003)
Communicated by: F. Duzaar (Erlangen)
Abstract. We establish a barrier principle for minimal submanifolds of a Riemannian manifold of arbitrary codimension. We construct examples of barriers for two-dimensional minimal surfaces in
Rn, n
≥ 4, and apply these to deduce existence as well as nonexistence theorems for Plateau’s problem.
Mathematics Subject Classifications (2000):53A10, 49Q05, 35B50.
Key words:minimal surface, barrier principle, Plateau’s problem.
1. Introduction
Let B be a hypersurface in some Riemannian manifold which has nonnegative mean curvature with respect to the normal vector fieldν. It is then a direct conse-quence of Hopf’s maximum principle that whenM ⊂N is a minimal hypersurface which is locally on the side of B to which ν points, and which touches B in some pointp, then M is contained inB in some neighborhood ofp. This is the well-known barrier principle for minimal hypersurfaces. We use Hopf’s maximum principle to prove a corresponding result for minimal submanifolds M ⊂ N of arbitrary dimensionk, 1 ≤k <dimN, replacing the mean convexity ofB by the condition ofk-mean convexity which is the condition that the sum of theksmallest principal curvatures of B be nonnegative. We construct topologically nontrivial domains inRn with 2-mean convex boundaries and use these to deduce existence results to Plateau’s problem for minimal surfaces inRn, n
≥4.
His method is based on the construction of quadratic forms onRnwhich, though indefinite, are subharmonic on anyk-dimensional minimal submanifold. It should be noted that when aC2function ϕ on Rn has the property that its restriction to anyk-dimensional affine subspace is subharmonic, it follows that the (regular) level hypersurfaces ofϕarek-mean convex with respect to−∇ϕ.
2. The Barrier Principle
LetNbe an(n+1)-dimensional Riemannian manifold of classC2. AC2 hypersur-faceB ⊂N is calledk-mean convex in the direction of the normal vector fieldνof
B,1≤k≤n, ifλ1+ · · · +λk ≥0 in all points ofB, whereλ1≤. . .≤λndenote the principle curvatures ofB in the direction ofν. We shall prove the following proposition:
PROPOSITION 2.1. LetMbe ak-dimensional Riemannian manifold,f:M →N
an isometric minimal immersion, B ⊂ N a k-mean convex hypersurface in the
direction of the normalν, p ∈ Mwithf (p)∈B and assume furthermore thatU
andV are connected neighborhoods ofp ∈ Mandf (p) ∈ N, respectively, such
thatV\Bconsists of two components andf (U )is contained in the closure of that
component ofV\Binto whichνpoints. Then it follows thatf (U )⊂B.
The computations in the following lemma are standard. We include them for the convenience of the reader and in order to make the paper self-contained.
LEMMA 2.2. Letf:M → N be as above and letd be a realC2-function onN
such that∇d =1and∇(d◦f )<1. Then we have the inequality
|(d◦f )+trace(A|(f∗T M)T)| ≤ (Hessd)◦f ∇(d ◦f )
2
1− ∇(d ◦f )2,
where, at a pointx ∈ N, Ais the second fundamental form of the hypersurface
d−1(d(x)) in direction of the normal
∇d, and, for a subspace W of TxN, WT
denotes the orthogonal projection ofW onTxd−1(d(x)).
Proof.We compute with a local orthonormal framee1, . . . , ek onM. By means
of the Gauss equation, the minimality off may be written in the form
∇eif∗ei −f∗(∇eiei)=0,
from which we obtain
(d ◦f )=Hess(d◦f )(ei, ei)=Hessd (f∗ei, f∗ei).
For any vectorv∈ TxN we denote byvT its tangential component with respect to the hypersurfaced−1(d(x)), i.e.
Again using∇d =1 we get Hessd (v,∇d) =0 ∀v
and, hence,
Hessd (f∗ei, f∗ei)=Hessd( (f∗ei)T, (f∗ei)T)= −A((f∗ei)T, (f∗ei)T). In order to relate this last expression to a partial trace ofAwe compute
bij := (f∗ei)T, (f∗ej)T =δij−(eid◦f )(ejd◦f ), from which we get the inverse matrix
bij =δij+
(eid◦f )(ejd◦f ) 1− ∇(d◦f )2 , and, hence,
A((f∗ei)T, (f∗ei)T)=bijA((f∗ei)T, (f∗ej)T)
+(1− ∇(d ◦f )2)−1Hessd (f∗ei, f∗ej)(eid◦f )(ejd◦f ) . Since
bijA((f∗ei)T, (f∗ej)T)=trace(A|(f∗T M)T) ,
the statement of the lemma follows.
LEMMA 2.3. LetAbe a quadratic form on an n-dimensional Euclidean vector
spaceV with the eigenvaluesλ1≤. . .≤λn. Then for anyk-dimensional subspace
W ofV there holds
traceA|W ≥λ1+ · · · +λk.
Proof.We use induction onk+n. The casek+n=2 is trivial. Let us assume
that the statement is true for all A, V ,and W with k+n ≤ m, m ≥ 2, and let
A, V, andW be given withk+n = m+1. Letv1 ∈ V be an eigenvector ofA corresponding to the lowest eigenvalueλ1and let us denote byV1the orthogonal complement ofRv1. IfW is contained inV1then we have by induction hypothesis
traceA|W ≥λ2+ · · · +λk+1≥λ1+ · · · +λk.
IfW is not contained inV1then there is a nonzero vectorw1∈W such that
and we may hence again apply the induction hypothesis toV1, A|V1, and the(k− 1)-dimensional subspaceW1spanned byw2, . . . , wk, thus obtaining
k
j=2
A(wj, wj)=traceA|W1≥λ2+ · · · +λk.
From this we immediately get
traceA|V = |w1|−2A(w1, w1)+ k
j=2
A(wj, wj)≥λ1+ · · · +λk.
We may now easily prove the barrier principle Proposition 2.1. As our functiond
in Lemma 2.2 we choose the (signed) distance functiond(x) =dist(x, B)which is of classC2in a sufficiently small neighborhood ofB. We choose the sign ofdin such a way thatd◦f ≥0. Denoting byλi(x)(i =1. . . n) the principal curvatures ofd−1(d(x))atx, we infer from Lemmas 2.2 and 2.3 the estimate
(d ◦f )+
k
j=1
λj◦f − (Hessd)◦f ∇
(d◦f )2
1− ∇(d◦f )2 ≤0. (2)
Since by hypothesisk
j=1 λj ≥ 0 onB and since the eigenvalues ofAare Lip-schitz continuous functions ofAwe have in some neighborhood ofB the estimate
k
j=1
λj(x)≥ −C1|d(x)|
with a suitable constantC1≥0. Hence (2) implies the differential inequality
(d ◦f )−C1d◦f −C2∇(d◦f )2≤0 (3) with a further constantC2. Hopf’s maximum principle [2] is immediately applica-ble to (3), giving the assertion of Proposition 2.1.
3. Examples and Applications
We wish to construct rotationally symmetric examples of 2-mean convex barriers
B inRn. Consider a splitting Rn
= Rn1+1×Rn2+1 and letSni ⊂ Rni+1 be the
corresponding unit spheres, i = 1,2. Let furthermore I ⊂ Rbe some interval,
S:=I×Sn1×Sn2,and let the immersionf:S−→Rnbe given in the form
f (t, θ, η)=x(t) θ+y(t)η, θ ∈Sn1, η ∈Sn2,
where σ (t) = (x(t), y(t)), t ∈ I, is an immersed curve in R2 with x(t) > 0, y(t) > 0. We observe that, equivalently, f is obtained as the inverse image
)−1(σ )under the submersion)
The Gauss map off is given by
N := y
′
wθ− x′
wη, w =
x′2+y′2.
Define
e1:=
x′
w θ+ y′
wη
and let{e2, . . . , en1+1}and{en1+2, . . . en1+n2+1}be local orthonormal bases inTS
n1
and TSn2, respectively. Then, with respect to the basis {e
j}nj1=+1n2+1, the second fundamental tensorAX = −DXN is given by the diagonal matrix
diag
y′x′′−x′y′′
w3 ,−
y′
xw , . . .− y′
xw, x′
yw, . . . x′
yw
. (4)
As our first example we consider Dierkes’ cones [1]. We taken1 = n−k−1,
n2 = k−1 and as our σ the curve y = cx with a constant c > 0 yet to be determined. According to (4), the principal curvatures are
0, −c
x√1+c2, . . . ,
−c x√1+c2,
1
cx√1+c2, . . . , 1
cx√1+c2
.
Since the smallest curvatures are precisely the firstkentries in this list we get as condition fork-mean convexity−(n−k−1)c+(k−(n−k))c−1≥0 or
c2≤ 2k−n n−k−1.
As Dierkes showed, the quadratic form|z|2
−c2
|w|2 is subharmonic on any k -dimensional minimal submanifold provided thatc2≤2k−n/n−k.
We now specialize to the case thatn1=1 and take asσ the half circle
x =R−rcost, y =rsint, 0< t < π,
where 0< r < R. The principal curvatures are computed as
1
r, −
cost R−rcost,
1
r, . . .
1
r
,
and we see that the corresponding surface)−1(σ )will be 2-mean convex provided that r ≤ (1/2)R. We observe, furthermore, that )−1(σ ) is the boundary of the toroidal body
T :=
(z, w)∈R2×Rn−2|(|z| −R)2+ |w|2≤r2
, (5)
which is homeomorphic toS1
×Bn−1, whereBn−1denotes the(n
examples of such domains may be obtained by taking as our generating curve σ
one of the catenaries
y =ϕλ(x)=λ ℓn(x/λ+
x2/λ2
−1), x ≥λ >0. (6) The principal curvatures in the order in which they appear in (4) are
λ1= −
ϕ′′
w3, λ2= −
ϕ′
xw, λj =
1
ϕw (j ≥3).
Sinceϕ ≥ 0, ϕ′ ≥0, andϕ′′≤ 0 we just have to verify the relationsλ1+λ2 ≥0 andλ2+λ3 ≥0 in order to guarantee the 2-mean convexity of our corresponding rotational hypersurface. Since in the casen2=0 (i.e.n=3) this surface is nothing but an ordinary catenoid, we haveλ1+λ2=0. The remaining conditionλ2+λ3≥0 is equivalent to
−ϕ(x)+ x
ϕ′(x) ≥0 (7)
which is easily verified by observing that the derivative of the left-hand side of (7) is nonnegative and the value at x = λ is zero. Again we conclude that the unbounded sets
Tλ := {(z, w)∈R2×Rn−2| |w| ≤ϕλ(|z|)}, (8) whereϕλ is given by (6), are domains with 2-mean convex boundary and funda-mental group isomorphic toZ.
Proceeding as in[4]for the casen =3 we may join several copies of 2-mean convex tori in Rn by suitable 2-mean convex necks (which are easily obtained
by the above construction with n1 = 0) to obtain domains in Rn with 2-mean convex boundaries and fundamental groups isomorphic to the free group of any finite number of generators. The concept of incompressible surfaces[3] may then be applied in such domains in the same way as was done in[4]. When applying the methods of[4] the inequality (2) for isometric minimal immersions has to be replaced by its variant for conformal minimal immersions. As a simple example for the kind of results which can be obtained in this way we formulate
THEOREM 3.1. Let T be a domain in Rn with 2-mean convex boundary and
fundamental group isomorphic toZ, e.g. the domain (5) withr ≤ 1/2R or one
of the domainsTλ in (8). Let Ŵ1 and Ŵ2 be a pair of disjoint rectifiable Jordan
curves in T which are freely homotopic in T and which represent a nontrivial
element ofπ1(T ). ThenŴ1and Ŵ2span a minimal (branched) annulus in T. IfŴ
is a rectifiable Jordan curve inT whose class in π1(T )is an even multiple of a
generator ofπ1(T ), thenŴspans a minimal Möbius strip inT.
tangent to the family of catenariesy = ϕλ(x), i.e. c0 := ϕ(t0)/t0where t0 is the solution of the equation
tϕ1′(t)=ϕ1(t).
The liney=c0x touches the catenaryy =ϕλ(x)in the pointλ(t0, c0t0).
THEOREM 3.2. LetMbe a compact connected minimal surface contained in the tubeT := {(z, w)∈R2
×Rn−2
| |w| ≤d2}and such that∂Mis contained in the
union of the hyperplanes x1 = d1and x1 = −d1, d1 > 0, and intersects each of
these hyperplanes. Then it follows that
d2 ≥c0d1.
Proof.Suppose thatd2< c0d1. Fort∈Rconsider the 2-mean convex domains
At := {(z, w)| |w| ≤ϕλ(|z−(0, t)|)} withλ=d1/t0. Since
ϕλ(|(±d1, x2)|)≥ϕλ(d1)=ϕ(λt0)=λc0t0=c0d1> d2
by assumption, we see that ∂At never intersects ∂M. For large values of |t| the compact surfaceM will be contained in the interior ofAt. Varyingt we therefore find a position in whichMtouches∂At from its 2-mean convex side provided that
Mconnects the two hyperplanes x1 = ±d1. By Proposition 2.1 the surface ought then to be contained in some∂Atwhich is however impossible since∂At∩∂M= ∅
for alltas we have seen.
References
1. Dierkes, U.: Maximum principles and nonexistence results for minimal submanifolds, Manu-scripta Math.69(1990), 203–218.
2. Gilbarg, D. and Trudinger, N. S.:Elliptic Partial Differential Equations of Second Order, second edition, Springer, New York, 1983.
3. Schoen, R. and Yau, S. T.: Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature,Ann. of Math.110(1979), 127– 142.
