The Journal of Geometric AnalysisVolume , Number ,

Monotonicity inequalities for the r-area and adegeneracy theorem for r-minimal graphs

Cleon S. Barroso∗ Levi L. de Lima† Walcy Santos‡

ABSTRACT. We establish monotonicity inequalities for the r-area of acomplete oriented properly immersed r-minimal hypersurface in Euclideanspace under appropriate quasi-positivity assumptions on certain invariants ofthe immersion. The proofs are based on the corresponding first variationalformula. As an application, we derive a degeneracy theorem for an entirer-minimal graph whose defining function f has first and second derivativesdecaying fast enough at infinity: its Hessian operator D2f has at least n − rnull eigenvalues everywhere.

Key words: r-area, graphs, monotonicityMS Subject Classification: 53A10 - 53C42

1. Introduction

Let M ⊂ Rn+1 be a smooth orientable hypersurface. Recall that onehas a globally defined map N : M → Sn, the unit normal Gauss map,well determined up to sign, which fixes an orientation for M and inducesfor x ∈ M the shape operator Ax : TxM → TxM, given by Ax(v) =−(DvN)(x), where D is the standard covariant derivative in Rn+1. Sinceeach Ax is a symmetric endomorphism of TxM , it has n real eigenvalues,

∗Department of Mathematics, Federal University of Ceara, Fortaleza, Brazil. E-mail:cleonbar@mat.ufc.br.

†Department of Mathematics, Federal University of Ceara, Fortaleza, Brazil, partiallysupported by CNPq and FINEP. E-mail: levi@mat.ufc.br.

‡Institute of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil,partially supported by CNPq and FAPERJ. E-mail: walcy@im.ufrj.br.

c© The Journal of Geometric Analysis

ISSN 1050-6926

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which are the principal curvatures of the immersion. Arranging these innondecreasing order gives a well-defined Rn-valued continuous function κ =(κ1, . . . , κn) and are termed the principal curvatures of the immersion. Foreach r = 0, 1, . . . , n, let Sr = Sr(κ) be the rth elementary symmetric functionof degree r in the entries of κ, also called the r-mean curvature of theimmersion. From these data we can define the Newton transformations onTM,

P−1 = 0,P0 = I,Pr = SrI− Pr−1A, r ≥ 1,

(1.1)

or equivalently,

Pr =r∑

i=0

(−1)iSr−iAi. (1.2)

For the ease of notation we set, for r ≥ 1,

Rr = Pr−1A, (1.3)

and remark thattrRr = rSr. (1.4)

Regarding this as a field of symmetric endomorphisms of TM, we say thatRr is quasi-positive if Rr ≥ 0 everywhere and has a strictly positive eigen-value somewhere.

Remark 1.1 In view of (1.4) we see that the quasi-positivity of Rr impliesSr ≥ 0 and Sr(x) > 0 for some x ∈ M.

We will be interested in hypersurfaces M which are r-minimal in thesense that Sr+1 = 0 identically. In particular, minimal hypersurfaces are0-minimal in our terminology.

It is well-known [R1], [BC] that r-minimal hypersurfaces are criticalpoints for a geometric variational problem, namely, that associated to thefunctional

Ar(M) =∫

MSr dM, (1.5)

under compactly supported variations. Notice that A0(M) =∫M dM is the

standard n-dimensional area of the immersion. Moreover, if M ⊂ Rn+1

3

is a compact and strictly convex immersion then the well-known Cauchy-Crofton formula of Integral Geometry shows that (1.5) agrees with the(properly normalized) measure of the set of affine (r + 1)-planes meetingM (see [BZ] and [LL] for details). Thus, in case Sr ≥ 0 everywhere, it isnatural to call the extended real number in (1.5) the r-area of M and askingwhether this is finite or infinite. A more ambitious goal would be to obtainlower bounds for the growth rate of the r-area of the regions obtained bycutting M with large balls, in terms of the corresponding radius. Theseestimates are well-known in the minimal case (see [BZ] for a derivation)and here we provide an extension of these results under a quasi-positiveassumption on Rr.

Theorem 1.1 Let M ⊂ Rn+1 be a complete oriented properly immersedr-minimal hypersurface, 1 ≤ r < n, with Rr quasi-positive. Then, givenp ∈ Rn+1, there exists R0 > 0 such that if R ≥ R0 and MR = M ∩ BR(p)we have

Ar(MR) ≥ Ar(MR0)(

R

R0

)n−rr+1

. (1.6)

In particular, Ar(M) = +∞.

The last assertion in the theorem follows by sending R → +∞ in (1.6)since by Remark 1.1 we can chose R0 so that Ar(MR0) > 0. Anotherimmediate consequence of (1.6) is the following area estimate.

Theorem 1.2 Let M ⊂ Rn+1 be as in Theorem 1.1 and assume further-more that Sr ≤ β for some β > 0. Then there holds

A0(MR) ≥ β−1Ar(MR0)(

R

R0

)n−rr+1

, R ≥ R0. (1.7)

In particular, A0(M) = +∞.

The proof of Theorem 1.1, and in particular of (1.6), is given in Section2 and, similarly to what happens in the minimal case, a crucial ingredientis the first variation formula for the r-area functional (1.5) applied to vari-ations given by homotheties. However, in case r ≥ 1 an extra term appearsinvolving the Newton operator Pr−1 applied to the gradient of the variationalfunction (cf. the last integral in (2.1)). This term may look somewhat in-tractable at first sight but our main contribution here is the elementary ob-servation that it can be expressed in terms of Rr and the variational vector

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field (see Proposition 2.1). This classical argument, which goes through byestablishing a suitable differential inequality for the function R 7→ A0(MR),can be easily transplanted to the case r ≥ 1. We also remark that (1.6)reduces to the standard estimate in the minimal case r = 0 [BZ].

Remark 1.2 As observed in [R2], if r is even, then the tensor Rr is infact an intrinsic object and can be expressed in terms of the Riemanniancurvature tensor RM of M. For example, it follows easily from Gauss equa-tion that R2 = Ric, the Ricci tensor of M, and and that a general Rr isa suitable contraction of the rth curvature tensor of M, as introduced byThorpe [T], much in the same way as Ric is derived from RM. The situa-tion is entirely different in case r is odd, for then it is already clear that Rr

changes sign if we change the orientation of M. This means of course thatin Theorem 1.1 with r odd, after possibly changing the sign of N , it sufficesto assume that Rr does not change sign and is non-degenerate, i.e. has anonzero eigenvalue somewhere.

Consider now a smooth function f : Rn → R whose graph M ⊂ Rn+1

is r-minimal, r ≥ 1, and meets the quasi-positive condition in Theorem 1.1with respect to the unit normal vector

N =(−Df, 1)

W, (1.8)

where Df = (D1f, . . . , Dnf) is the gradient of f and W =√

1 + |Df |2.Then (1.6) applies and gives a very precise lower bound for the growth rateof the r-area of M, namely,

Ar(MR) ≥ CRn−rr+1 , R ≥ R0, (1.9)

where C > 0 depends only on f . Now if we impose extra conditions onf so as to get an upper bound for the growth rate of the r-area involvinga power of R which is strictly lower than the one appearing in (1.9), wewould get a contradiction, thus showing that no such graph exists. Fortu-nately, this argument works and the extra condition involves a control ofthe first and second derivatives of f at infinity. In order to explain this, letus still denote by f the restriction of our original function to the spheresSn−1

R (0) ⊂ Rn, by γ the unit outward vector field to each Sn−1R (0) and by

Dγf the corresponding normal derivative. Moreover, if ∂ is the covariantderivative on Sn−1

R (0), we denote by ∂2f the intrinsic Hessian of f . This

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is a symmetric linear operator and Sr[f ] will represent the rth elementarysymmetric functions of the n− 1 eigenvalues of ∂2f . Similarly, the Hessianoperator of the original function f is denoted by D2f . Now we can state adegeneracy theorem for a certain class of r-minimal graphs.

Theorem 1.3 If f : Rn → R defines an entire r-minimal graph with Rr ≥0 and satisfying the property that, for each 0 ≤ k ≤ r− 1, there are positiveconstants Ck and ηk such that

∣∣∣Sr−k−1[f ] (Dγf)k+1 (x)∣∣∣ ≤ Ck

|x|ηk, (1.10)

for all |x| large enough and

ηk >nr − kr − k − 1

r + 1, (1.11)

then rank D2f ≤ r everywhere on Rn.

Notice that if r = 0 the above condition on rank D2f implies that f islinear, thus Theorem 1.3 should be thought of a weak form of a Bernsteintype theorem for r-minimal graphs. Also, one would like to stress thateven though the righthand side in (1.10) is a rather complicated expressioninvolving derivatives of f of first and second order, f itself does not appearexplicitly there. Moreover, one always has nr− kr− k − 1 > 0, so that thepoint of (1.11) is a very precise control at infinity of the derivatives of firstand second order of f . In order to get a flavor of this kind of control weexamine here the simplest cases. For r = 1 (1.10) reduces to

|Dγf(x)| ≤ C1

|x|η1, η1 >

n− 1

2, (1.12)

a very neat condition indeed. But notice that the nonnegativity of R1 = Aimplies, by a simple argument using the Hartman-Nirenberg theorem [CY],that M is a cylinder over a plane curve whose curvature is nonnegative, sothat (1.12) can only hold for such a cylinder if the curve in question is a lineand hence f is linear. Thus, if r = 1 the reasoning sketched above actuallyyields a Bernstein type theorem.

For r = 2, (1.10) gives the conditions

|∆fDγf(x)| ≤ C0

|x|η0, η0 >

2n− 1

3, (1.13)

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where ∆ is the intrinsic Laplacian on Sn−1R (0), and

| (Dγf)2 (x)| ≤ C1

|x|η1, η1 >

2(n− 2)

3, (1.14)

and we see here that if η1 is taken large enough, thus yielding a very rapiddecay at infinity for Dγf , then ∆f can even be allowed to grow polynomiallywith R. Obviously, a similar remark holds if r ≥ 3 with ∆f replaced bySm[f ] for 1 ≤ m ≤ r − 2.

Remark 1.3 In [CNS] is proven a Bernstein type theorem for entire graphswith prescribed r-mean curvature satisfying |Df(x)| = o(|x|1/2) at infinity,but as it is already illustrated there with the use of one of the cylindersdescribed above, their argument does not apply to 1-minimal graphs. More-over, consideration of similar cylinders shows in general that a decayingcondition at infinity for Df , similar to (1.10) with k = r − 1, is essentialin order to have a Bernstein theorem for r-minimal graphs, r ≥ 1.

Acknowledgements. The authors would like to thank Prof. Maria LuizaLeite for helpful comments and suggestions.

2. The monotonicity inequality for the r-area and the proof ofTheorem 1.1

Our proof of Theorem 1.1 is based on the first variation formula forthe r-area functional (1.5). Assume that one has, for some interval I ⊂R with 1 ∈ I, a smooth variation t ∈ I 7→ Nt ⊂ Rn+1 of a compactoriented hypersurface N = N1, possibly with nonempty boundary ∂N. Thevariation is described by a family of immersions F : I × N → Rn+1 andE = ∂F/∂t(1, .) is a vector field along F , the associated variational vectorfield. Notice that we neither assume that E is normal to N nor do weprescribe the values of E along ∂N. Under these general conditions one hasthe first variational formula for the r-area functional [R1], [AdCE]:

d

dtAr(Nt)|t=1 = −(r + 1)

∫

NSr+1g dN +

+∫

∂NSr〈E, ν〉 d∂N +

∫

∂N〈Pr−1(grad g), ν〉 d∂N,(2.1)

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where g = 〈E, N〉 is the so-called variational function, ν is the outward unitco-normal vector to ∂N and grad g is the intrinsic gradient of g.

We are going to use this for a rather special choice of variation. Fixp ∈ Rn+1. Given M as in Theorem 1.1, the properness assumption impliesthe existence of R0 > 0 such that for each R ≥ R0, the intersection MR =M ∩ BR(p) has a nonempty boundary. Now consider the one parameterfamily of homotheties F : I ×MR → Rn+1, F (t, x) = p + t(x− p), so thatE(x) = x− p, which implies |E| ≤ R. For each such R, this yields a familyof r-minimal hypersurfaces MR,t = F (t,MR) with Ar(MR,t) = tn−rAr(MR)and (2.1) reduces to

(n− r)Ar(MR) =∫

∂MR

Sr〈E, ν〉 d∂MR +∫

∂MR

〈Pr−1(grad g), ν〉 d∂MR.

(2.2)

By Remark 1.1, Sr ≥ 0, and the first integral, say I1, in the righthandside of (2.2) can clearly be estimated from above by R

∫∂MR

Sr d∂MR. Now,if we set for brevity h(R) = Ar(MR), so that by the co-area formula,

h(R) =∫ R

0

(∫

∂Ms

Sr|grad ρ|−1 d∂Ms

)ds,

where ρ(x) = |x − p|, and use that |grad ρ| ≤ |Dρ| = 1, we get, for almostevery R,

h′(R) ≥∫

∂MR

Sr d∂MR, (2.3)

so thatI1 ≤ Rh′(R). (2.4)

In order to handle the other integral, say I2, in (2.2) one has to computegrad g.

Proposition 2.1 One has

grad g = −A(ET ), (2.5)

where ET is the tangential projection of E onto MR. In particular,

I2 = −∫

∂MR

〈RrET , ν〉 d∂MR, (2.6)

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This is a standard computation that we include here for the sake ofcompleteness. Fix x ∈ MR and consider a local orthonormal frame ein

i=1

tangent to MR. Without loss of generality, we may take p = 0 and since thevariation is by homotheties, up to parallel displacement we have E = F ,where F (·) = F (1, ·) is the immersion defining MR. We then compute:

ei(g) = ei 〈F,N〉= 〈Dei

F, N〉+ 〈F,DeiN〉

= 〈ei, N〉 − 〈E, A(ei)〉= −〈E, A(ei)〉 ,

so that around x we have ei(g) = −〈E, A(ei)〉 = −〈A(ET ), ei〉. It followsthat the identity

grad g =∑

i

ei(g)ei = −A(ET )

holds at x and hence everywhere along MR, as desired.

Under the quasi-positive assumption in Theorem 1.1, a simple differ-ential inequality for h in the range R ≥ R0 can be easily derived. In effect,by examining the computation leading to (2.5), we see that by performinga tangential rotation if necessary we may assume that the adapted framesatisfies A(ei) = κiei at x. It follows from (1.2) that the ei’s are eigenvectorsof Pr−1 as well: Pr−1ei = µiei. In this notation, (2.6) can be rewritten as

I2 = −∫

∂MR

(∑

i

κiµi〈ET , ei〉〈ei, ν〉)

d∂MR,

and the assumption Rr ≥ 0 means that κiµi ≥ 0 for any i. From this weestimate

I2 ≤ R∫

∂MR

(∑

i

κiµi

)d∂Mr

= R∫

∂MR

trRr d∂Mr

= rR∫

∂MR

Sr d∂Mr

≤ rRh′(R),

where we have used (1.4) and (2.3). This yields the promised differentialinequality for h:

(n− r)h(R) ≤ (r + 1)Rh′(R), (2.7)

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After integrating (2.7) in the range R ≥ R0 we obtain the monotonicityinequality (1.6), thus completing the proof of Theorem 1.1.

We observe that there exists another situation where an estimate sim-ilar to (1.6) can be established. More precisely, assume the existence ofα > 0 such that

Sr ≥ α (2.8)

everywhere along M. From (2.2), (2.6) and (1.2) we get

(n− r)Ar(MR) =∫

∂MR

〈(SrI−Rr)ET , ν〉 d∂MR

=∫

∂MR

〈PrET , ν〉 d∂MR

≤ R∫

∂MR

|Pr| d∂MR.

This suggests looking at the quotient ϕ(κ) = S−1r |Pr|. Recall that we think

of κ as an Rn-valued function on M. Now, (2.8) certainly implies theexistence of ρ > 0 such that Im κ, the image of κ, does not overlap theball of radius ρ centered in the origin of Rn. In particular, ϕ is continuousand uniformly bounded in bounded regions of Im κ. On the other hand,as |κ| → +∞, (1.2) makes it clear that ϕ(κ) ≈ |κ|−r|κ|r ≈ const., so thatwe can find K > 0 (depending only on α, r and n) such that |Pr| ≤ KSr

everywhere on M. It follows that

(n− r)h(R) ≤ KR∫

∂MR

Sr d∂MR ≤ KRh′(R), (2.9)

where we have used (2.3) once again. The discussion above can be summa-rized as follows.

Theorem 2.1 Let M ⊂ Rn+1 be a complete oriented properly immersed r-minimal hypersurface, 1 ≤ r < n, with Sr ≥ α for some α > 0. Then, givenp ∈ Rn+1, there exists R0 > 0 such that if R ≥ R0 and MR = M ∩ BR(p)we have

Ar(MR) ≥ Ar(MR0)(

R

R0

)n−rK

, (2.10)

for some K > 0. In particular, Ar(M) = +∞.

Similar to (1.7) we have the area estimate

A0(MR) ≥ β−1A0(MR0)(

R

R0

)n−rK

, (2.11)

in case Sr is pinched between the positive constants α < β.

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3. A degeneracy theorem for r-minimal graphs

In this section we will give the proof of a degeneracy theorem for certainentire r-minimal graphs, so let f be as in Theorem 1.3. Here, it is convenientto set p = 0 in Theorem 1.1 so that (1.9) should be interpreted in this way.With the choice of orientation for the graph of f given by (1.8), one findsin Theorem 3.6 (page 379) of [R2] the identity

Si[f ] = W iSi + W i+2〈Riξ, ξ〉, 1 ≤ i < n, (3.1)

where, as before, W =√

1 + |Df |2. As usual, here Si[f ] is the ith elementary

symmetric function in the eigenvalues of D2f , the Hessian of f , ξ is theorthogonal projection of the constant vector (0, . . . , 0, 1) onto M, the graphof f , and Si and Ri are the standard invariants of M. From this and thenonnegativity of Rr one easily obtains

WSr ≤ Sr[f ], (3.2)

that is, the r-mean curvature of M is controlled from above by the corre-sponding elementary symmetric function in the eigenvalues of D2f . If, asexplained above, MR = M ∩ BR(0) and UR is its projection over Rn, oneclearly has UR ⊂ BR(0), the ball of Rn with radius R centered at the origin,so we estimate using (3.2):

Ar(MR) =∫

MR

Sr dM

=∫

UR

SrW dx

≤∫

BR(0)Sr[f ] dx,

where of course dx is the standard volume element of Rn.

We now appeal to another remarkable integral identity in [R2] [R3],a generalization of which appears in a paper by Trudinger [T]. A versionof this identity (see equation (4.7) in [T] and set g = Df and Ω = BR(0)there) reads as

∫

BR(0)Sr[f ] dx =

∫

Sn−1R (0)

Hr[f ] dσn−1(R). (3.3)

Here, Sn−1R (0) = ∂BR(0), dσn−1(R) is the corresponding area element and

Hr[f ] =r−1∑

k=0

1

k + 1Sr−1,k[W , ∂2f ] (Dγf)k+1 , (3.4)

11

where W = Dγ is minus the shape operator of Sn−1r (0) ⊂ Rn with respect

to γ and

Sr−1,k[W , ∂2f ] =1

(r − 1)!

∑δ

i1...ir−1

j1...jr−1Wi1j1 . . .Wikjk

∂ik+1jk+1f . . . ∂ir−1jr−1f.

(3.5)In the expression above, ∂ijfn−1

i,j=1 are the entries of the Hessian matrix ∂2fwith respect to some local orthonormal frame tangent to Sn−1

R (0) and thesum is taken over all choices of integers 1 ≤ i1, . . . , ir−1, j1, . . . , jr−1 ≤ n− 1with δ

i1...ir−1

j1...jr−1being equal to 1 (respectively −1) if i1, . . . , ir−1 are distinct

and (j1, . . . , jr−1) is an even (respectively odd) permutation of (i1, . . . , ir−1),and 0 otherwise.

Now we can slightly simplify (3.5) by using the standard fact thatWij = δij/R. In effect,

Sr−1,k[W , ∂2f ] =1

(r − 1)!Rk

(r − 1

k

) ∑δ

ik+1...ir−1

jk+1...jr−1∂ik+1jk+1

f . . . ∂ir−1jr−1f

=1

k!RkSr−k−1[f ],

where we have used the fact that Sr−k−1[f ] coincides with the sum of theprincipal minors of order r − k − 1 of ∂2f and hence is given by

Sr−k−1[f ] =1

(r − k − 1)!

∑δ

ik+1...ir−1

jk+1...jr−1∂ik+1jk+1

f . . . ∂ir−1jr−1f.

After replacing the above expression for Sr−1,k[W , ∂2f ] in (3.4) we get

Hr[f ] =r−1∑

k=0

1

(k + 1)!RkSr−k−1[∂

2f ] (Dγf)k+1 ,

so that using (3.3) and returning to our estimate of Ar(MR) for R ≥max1, R0 we find that

Ar(MR) ≤r−1∑

k=0

1

(k + 1)!Rk

∫

Sn−1R (0)

|Sr−k−1[f ] (Dγf)k+1 | dσn−1(R)

≤r−1∑

k=0

C ′kR

−k−ηk+n−1, (3.6)

where (1.10) has been used in the last inequality. But (1.11) means preciselythat, for each k,

−k − ηk + n− 1 <n− r

r + 1,

12

and we see that (1.9) and (3.6) are mutually contradictory if we furtherassume that Rr is quasi-positive. Thus no such graph exists and we are leftwith the case in which Rr = 0 identically. From this and (1.4) we deducethat Sr = 0 and then Pr = SrI −Rr = 0. It follows that Rr+1 = PrA = 0and again Sr+1 = 0, so that Pr+1 = Sr+1I − Rr+1 = 0. Starting fromthese identities, induction easily shows that, for i = r, . . . , n, Si = 0 andPi = Ri = 0 identically, so that appealing to (3.1) we also have Si[f ] = 0for i ≥ r. This clearly implies rank D2f ≤ r, thus completing the proof ofTheorem 1.3.

Remark 3.1 Note incidentally that the above established identities Si = 0,i ≥ r, imply that νA, the nullity of A, is at least n− r everywhere on M. Inparticular, if ν0 is the minimal nullity and U ⊂ M is a connected componentof the nonempty open set where νA = ν0, then a standard argument (see[D]) shows that U is foliated by ν0-planes, ν0 ≥ n − r. Moreover, thegrowth conditions (1.10) easily imply that these planes are parallel in theobvious sense so that U is actually an extrinsic product M′ × Rν0, whereM′ ⊂ Rn−ν0+1. We firmly believe that this cylindrical structure can beextended to the whole of M, but we have not been able to completely clarifythis point.

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[BC] Barbosa, J. L., Colares, A. G., Stability of hypersurfaces with constant r-meancurvature, Ann. Global Anal. Geom., 15 (1997), no. 3, 277-297.

[BZ] Burago, Yu. D., Zalgaller, V. A., Geometric inequalities, Springer-Verlag,Berlin, 1988.

[CNS] Caffarelli, L., Nirenberg, L., Spruck, J., On a form of Bernstein’s theo-rem. Analyse mathmatique et applications, 55-66, Gauthier-Villars, Montrouge,1988.

[CY] Cheng, S. Y., Yau, S. T., Hypersurfaces with constant scalar curvature, Math.Ann. 225 (1977), no. 3, 195-204.

[D] Dajczer, M., Submanifolds and Isometric Immersions, Mathematics LectureSeries 13, Publish or Perish, Inc. Houston 1990.

[LL] Lazaro, I., de Lima, L., A Cauchy-Crofton formula and monotonicity inequal-ities for the Barbosa-Colares functionals, Asian J. Math., 7 (2003) 081-090.

[R1] Reilly, R., Variational properties of functions of the mean curvatures for hy-persurfaces in space forms, J. Diff. Geo., 8 (1973), 465-477.

[R2] Reilly, R., On the Hessian of a function and the curvatures of its graph, Michi-gan Math. J., 20 (1973), 373-383.

[R3] Reilly, R., Applications of the Hessian operator in a Riemannian manifold,Indiana Univ. Math. J. 26 (1977). 459-472.

[T] Thorpe, J. A., Sectional curvatures and characteristic classes, Ann. of Math.(2) 80 1964 429-443.

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[Tr] Trudinger, N. S., Isoperimetric inequalities for quermassintegrals, Ann. Inst.H. Poincar Anal. Non Linaire, 11 (1994), no. 4, 411-425.