Annals of Global Analysis and Geometry19: 177–184, 2001.© 2001Kluwer Academic Publishers. Printed in the Netherlands.

177

Curvature Estimates for Stable MinimalHypersurfaces inR4 andR5

QING CHENDepartment of Mathematics, The University of Science and Technology of China, Hefei,Anhui 230026, P. R. China. e-mail: qchen@ustc.edu.cn

(Received: 31 August 1999; accepted: 10 December 1999)

Abstract. We obtain curvature estimates for certain stable minimal hypersurfaces inR4 and R5

without using volume bounds. It follows that ifM is a complete stable minimal hypersurface inR4

or R5, thenM is a hyperplane whenM intersects each extrinsic ball in, at most,N-components.

Mathematics Subject Classifications (2000):53C40, 53C42.

Key words: Bernstein theorem, curvature estimate, stable minimal hypersurface.

1. Introduction

This paper is concerned with curvature estimates for stable minimal hypersurfacesin (n+ 1)-dimensional Euclidean spaceRn+1. A minimal hypersurfaceM in Rn+1

is called stable if the second variation of the volume functional is nonnegativeon any relatively compact open subset ofM. The classical result of this area is theBernstein Theorem [5] which states that a complete minimal graph inR3 is a plane.The Bernstein conjecture, which says that iff : Rn → R is an entire solution ofminimal surface equation

(1+ |∇f |2)n∑i=1

∂2f

∂x2i

−n∑

i,j=1

∂f

∂xi

∂f

∂xj

∂2f

∂xi∂xj= 0, (1)

thenf is a linear function. This was solved affirmatively by Almgren [1] forn =3,4 and by Simons [16] forn ≤ 7 (see also [7, 12]). Counterexamples to theconjecture forn ≥ 8 were found by Bombieriet al. [6].

Whenn = 2, Heinz [13] studied solutions of (1) defined over a disk of radiusR and centered atx0 ∈ R2. He then proved there is an absolute constantC suchthat |A|(x0) ≤ C/R2, whereA is the second fundamental form of the graph of thesolutions of Equation (1). In the case whenf is an entire solution of (1), by lettingR → ∞, Heinz’s result implies Bernstein’s Theorem. In the higher-dimensioncase, a fundamental is the work of Schoenet al. [21], which extended Heinz’sestimate to the case 2≤ n ≤ 5 by using Simons’ identity [16] for the Laplacian ofthe second fundamental form of the minimal submanifolds.

178 QING CHEN

Another extension of the Bernstein Theorem is to study curvature estimatesfor stable minimal hypersurfaces. In the casen = 2, Fischer-Colbrie and Schoen[11] and de Carmo and Peng [8] proved independently, that ifM2 is an immersedand complete stable minimal surface inR3, thenM is a plane. Schoenet al. [21]and Schoen and Simon [20] obtained curvature estimates for stable minimal hy-persurfaces in a ball under the hypothesis that the volume is bounded, forn ≤6.

In this paper, we present a Heinz-type curvature estimate for a class of stableminimal hypersurfaces inR4 and R5. Denote the ball inRn+1 of radius t andcentered atx byBx(t).

DEFINITION 1.1. LetM be ann-dimensional Riemannian manifold andφ:M →Rn+1 be an isometric immersion. We sayφ is anN-connected immersion if thereare at mostN connected components ofφ−1(Bx(t)), for anyx ∈ Rn+1 and anyt > 0. We say that an immersion is finite connected if it isN-connected for somepositive integerN .

Our main result is, which does not require volume bounds in the estimates, asfollows:

THEOREM 1.2.Let M be an immersed stable minimal hypersurface inRn+1,n = 3 or 4. SupposeM is connected, the immersion isN-connected,x0 ∈ M, andthe boundary ofM is outside the ballBx0(R), then there is a constantC dependingonn andN such that|A|(x0) ≤ C/R2. whereA is the second fundamental form ofM.

This theorem readily implies the following Bernstein Theorem.

COROLLARY 1.3.LetM be a properly immersed, connected and complete stableminimal hypersurface inRn+1. If the immersion is finite connected, thenM is ahyperplane providedn = 3 or 4.

The proof of the main theorem makes use of theLp-estimate (p ∈ [4,4+√4/n)) ofthe second fundamental form of stable minimal hypersurfaces obtained by Schoenet al. [21, theorem 1]. Indeed thisLp-estimate, together with volume growth con-ditions, can also imply the Bernstein Theorem forn ≤ 5 (see [21] for details).Our first observation is:let M be a properly immersed complete stable minimalhypersurface inRn+1, n = 3 or 4, andA be the second fundamental form of theimmersion. If the immersion is finite connected andsup|A| < ∞, thenM is ahyperplane. This can be done by using theLp-estimate mentioned above and theBesicovitch Covering Lemma. We remark that this observation is crucial in theproof of our main theorem. We then employ the rescaling technique developed by

STABLE MINIMAL HYPERSURFACES INR4 AND R5 179

Anderson [2, 3] together with a smooth compactness theorem to prove our maintheorem.

We would like to point out that many authors have obtained curvature estimatesfor minimal surfaces in 3-manifolds (e.g., [4, 14, 18, 19]), where Schoen’s estimate[18] does not depend on area bonds. In higher dimensions, besides the curvatureestimates of [21] and [20] for stable minimal hypersurfaces, Anderson [3] obtainedcurvature estimates for minimal submanifolds without stability assumptions, byusing theLn-norm of the second fundamental form. When eitherL2-norm orLn-norm of the second fundamental form of a complete stable minimal hypersurfacein Rn+1 (n ≥ 3) is finite, de Carmo and Peng [9] and Shen and Zhu [22] provedthat it is a hyperplane.

2. Preliminary Result

Let M be an immersed minimal hypersurface inRn+1, andA the second funda-mental form ofM. M is said to be stable if the second fundamental form of thevolume functional is nonnegative on any compact domain ofM, which is given ininequality (see, for example, [16])∫

M

|A|2f 2 ≤∫M

|∇f |2,

wheref is anyC1 function with compact support onM, and∇ is the covariantderivative ofM.

By using Simons’ formula for the Laplacian of the second fundamental form ofminimal submanifolds, Schoenet al. (theorem 1 of [21], see also [9]) obtained thefollowing Lp-estimates of the second fundamental form of stable minimal hyper-surfaces in Euclidean space.

THEOREM 2.1.LetM be an immersed minimal stable hypersurface inRn+1. Foreachp ∈ [4,4+√4/n), and for each nonnegative smooth function with compactsupport inM, we have∫

M

|A|pf p ≤ C∫M

|∇f |p,

whereA is the second fundamental form ofM andC is a constant depending onlyonn andp.

LEMMA 2.2. Let φ: M → Rn+1 be a properly immersed complete minimal hy-persurface andA the second fundamental form of the immersion. Suppose thatsup|A| ≤ 1/2, then for anyp ∈ M, the volume of each connected component ofφ−1(Bφ(p)(1/2)) is bounded by a constantC1 depending only onn.

Proof. Let 6 be a component ofφ−1(Bφ(p)(1/2)). Taking q ∈ 6 such thatφ(6) ⊂ Bφ(q)(1). By the hypothesis,6 is a graph over a domain in a unit ball of

180 QING CHEN

Tφ(q)6 with Lipschitz constant less than 1. This implies that the volume of6 isimmediately bounded. 2Recall the following Besicovitch Covering Lemma [17] (see [10] for a proof).

LEMMA 2.3. Suppose thatB is a collection of close balls inRn+1. LetA be theset of centers and suppose the set of all radii of balls is a bounded set. Then thereare pairwise disjoint sub-collectionsB1,B2, . . . ,Bλ ⊂ B (λ = λ(n)) such that⋃λj=1 Bj still coverA.

Now we are ready to show following uniqueness theorem.

THEOREM 2.4.LetM be a properly immersed complete stable minimal hypersur-face inRn+1, n = 3 or 4, andA the second fundamental form ofM. Suppose thatthe immersion isN-connected, andsup|A| <∞, thenM is an immersed union ofhyperplanes.

Proof. It is obvious that theN-connectness of the immersion is invariant undera scaling transformation of the immersion. Using a suitable scaling, we supposethat sup|A| ≤ 1/2. We first show that there is a constantC2 depending only onnandN such that, for allR > 1,

Vol(M ∩ B0(R)) ≤ C2(R + 1)n+1.

Consider the collection of closed balls

B = {Bx(1) : x ∈ B0(R) ∩M}.

By Lemma 2.2, there exist pairwise disjoint sub-collectionsB1,B2, . . . ,Bλ (λ =λ(n)) such that

B0(R) ∩M ⊂λ⋃j=1

⋃B∈Bj

B

.We need to estimate #Bj , the number of elements ofBj , for eachj . Since⋃

B∈Bj

B ⊂ B0(R + 1)

andBj is a pairwise disjoint collection,

ωn+1(R + 1)n+1 = Vol(B0(R + 1)) ≥∑B∈Bj

Vol(B) = ωn+1#Bj ,

whereωn+1 is the volume of the unit ball inRn+1. Hence #Bj ≤ (R + 1)n+1 foreachj . By hypothesis and Lemma 2.3,

Vol(M ∩ Bx(1)) ≤ NC1

STABLE MINIMAL HYPERSURFACES INR4 AND R5 181

holds for everyx ∈ M. Therefore, by settingC2 = C1Nλ, we have

Vol(M ∩ B0(R)) ≤λ∑j=1

∑B∈Bj

Vol(M ∩ B)

≤ C1N

λ∑j=1

#Bj

≤ C2(R + 1)n+1.

Now letf = γ ◦ r, whereγ is aC1 function onR1 with

γ |(−∞,R/2) = 1, γ |[R,+∞) = 0 and |dγ | ≤ 3

R.

Since|∇r| ≤ 1, Theorem 2.1 implies∫M∩B(R/2)

|A|p ≤ 3pC

RpVol(M ∩ B(R)) ≤ 3pCC2

Rp(R + 1)n+1.

Choosingp = 5+ ε ∈ [4,4+√8/n) in the above inequality and lettingR→∞,we must have|A| = 0. This completes the proof. 2Finally, we state the following well-known result and for the proof, refer to [2, 15].

THEOREM 2.5 (Smooth compactness theorem).Let {Mi} be a sequence of con-nected and properly immersed minimal submanifolds in an open subsetU in Rn+p.Suppose that there is a constantC such that the second fundamental formAMi

ofMi satisfies

|AMi|2(x) ≤ C, ∀x ∈ Mi

for eachi, then there is a subsequence{Mi} which converges in theCk-topology,for all k ≥ 2, to a properly immersed minimal submanifoldM∞ in U .

3. Curvature Estimate

In this section we actually prove the following result, which immediately impliesthe main theorem mentioned in the Introduction.

THEOREM 3.1.LetM be an immersed stable minimal hypersurface inRn+1, n =3 or 4. Suppose that the immerston isN-connected,0 ∈ M, and the boundary ofM is outside the ballB0(1), then there is a constantC3 independent ofM such that

supx∈M∩B0(1)

(1− |x|)|A|(x) ≤ C3,

182 QING CHEN

whereA is the second fundamental form ofM.

To prove the main theorem, one can apply Theorem 3.1 to a stable minimal hyper-surfaceσR(M), whereσR: Rn+1 → Rn+1 is the scaling transformation defined byσR(x) = x0+ (x − x0)/R. Then, it is not difficult to see that

|AM |(x0) = 1

R|AσR(M)|(x0) ≤ C3

R,

whereAM andAσR(M) are the second fundamental forms ofM andσR(M), respec-tively.

Proof of Theorem 3.1. If the conclusion were false, there must exists a sequenceof stable minimal hypersurfaces{Mi}, each being as in the theorem, such that

supx∈M∩B(1)

(1− |x|)|Ai |(x)→+∞,

whereAi is the second fundamental form ofMi. Let xi ∈ Mi ∩ B(1) be the pointrealizing the supremum of(1− |x|)|Ai |(x), and denote

ai = |Ai |(xi), bi = (1− |xi |)|Ai|(xi).We haveai ≥ bi →∞ (asi →∞) and|xi | = 1− (bi/ai). Set

M ′i = Mi ∩ B(

1− bi

2ai

),

then

supx∈M ′i|Ai |(x) ≤

{infx∈M ′i

(1− |x|)}−1

bi = 2ai.

Let σi be the dilation ofRn+1 by a factorai and centered atxi, i.e.σi(x) = xi +ai(x − xi) and define the minimal hypersurfacesMi = σi(M

′i ), and denote the

second fundamental form ofMi by Ai . Clearly,

sup|Ai| ≤ 2 and |Ai |(xi) = 1.

Furthermore,

distRn+1(xi, ∂Mi) = ai distRn+1(xi, ∂M′i ) ≥ ai

bi

2ai→∞ (*)

asi →∞.By using the method of taking the diagonal, Theorem 2.5 then implies that there

is a subsequence of{Mi} which converges inCk-topology (k ≥ 2), on every rela-tively compact open subset ofRn+1, to a smoothly immersed minimal hypersurfaceM∞. Denote the second fundamental form ofM∞ byA∞ and one has

STABLE MINIMAL HYPERSURFACES INR4 AND R5 183

(1) by (∗),M∞ is a properly immersed complete hypersurface;(2) byCk (k ≥ 2) convergence,M∞ is a stable minimal hypersurface;(3) sup|A∞| ≤ 2 and|A∞|(x∞) = 1, wherex∞ = lim xi .

Further, eachMi is N-connected immersed. Therefore, by the convergence,M∞is alsoN-connected immersed. By Theorem 2.5,M∞ must be a totally geodesichyperplane. This contradicts (3). We complete the proof of Theorem 3.1. 2Acknowledgements

The author would like to thank Professor C. K. Peng for many beneficial sugges-tions, and thank the referee for the useful comments.

This work was partially supported by NNSF, P. R. China.

References

1. Almgren, Jr., F. J.: Some interior regularity theorems for minimal surfaces and extension ofBernstein’s theorem,Ann. of Math. (2)84 (1966), 277–292.

2. Anderson, M. T.: The compactification of a minimal submanifold in Euclidean space by theGauss map, I.H.E.S. Preprint, 1984.

3. Anderson, M. T.: Local estimates for minimal submanifolds in dimensions greater than two,Proc. Sympos. Pure Math.44 (1986), 131–137.

4. Anderson, M. T.: Curvature estimate for minimal surfaces in 3-manifolds,Ann. Sci. EcoleNorm. Sup.18 (1985), 85–105.

5. Bernstein, S.: Sur un théorème de géométrie et ses applications aux équations aux dérivéespartielles du type elliptique,Comm. Soc. Math. Karkov (2)15 (1915), 38–45.

6. Bombieri, E., deGiorge, E. and Giusti, E.: Minimal cone and Bernstein problem,Invent. Math.7 (1969), 243–268.

7. deGiorge, E.: Una estensione del teorema di Bernstein,Ann. Scuola Norm. Sup. Pisa, III19(1965), 79–85.

8. de Carmo, M. and Peng, C. K.: Stable complete minimal surfaces inR3 are planes,Bull. Amer.Math. Soc.1 (1979), 903–905.

9. de Carmo, M. and Peng, C. K.: Stable complete minimal hypersurfaces,Beijing Sympos.Differential Equations Differential Geom.3 (1980), 1349–1358.

10. Federer, H.:Geometric Measure Theory, Grundlehren Math. Wiss. 153, Springer-Verlag, NewYork, 1969.

11. Fischer-Colbrie, D. and Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature,Comm. Pure Appl. Math.33 (1980), 121–132.

12. Fleming, W. H.: On the oriented plateau problem,Rend. Circ. Mat. Palermo2 (1962), 1–22.13. Heinz, E.: Über Lösungen der Minimalflächengleichung,Nachr. Wiss. Göttingen Math. Phys.

Kl II (1952), 51–56.14. Osserman, R.: On the Gauss curvature of minimal surfaces,Trans. Amer. Math. Soc.96 (1960),

115–128.15. Pitts, J.:Existence and Regularity of Minimal Surfaces on Riemannian Manifolds, Math.

Notes 27, Princeton Univ. Press, 1981.16. Simons, J.: Minimal varieties in Riemannian manifolds,Ann. of Math.66 (1968), 62–105.17. Simon, L.:Lectures on Geometric Measure Theory, C.M.A. Australian National University,

Vol. 3, 1983.

184 QING CHEN

18. Schoen, R.: Estimates for stable minimal surfaces in three dimensional manifolds, in:Ann. ofMath. Stud.103, Princeton Univ. Press, 1983.

19. Schoen, R. and Simon, L.: Regularity of simply connected surfaces with quasicomformal Gaussmap, in:Ann. of Math. Stud.103, Princeton University Press, 1983.

20. Schoen, R. and Simon, L.: Regularity of stable minimal hypersurfaces,Comm. Pure Appl.Math.34 (1981), 741–797.

21. Schoen, R., Simon, L. and Yau, S.-T.: Curvature estimate for minimal hypersurfaces,ActaMath.134(1975), 276–288.

22. Shen, Y. B. and Zhu, X. H.: On stable complete minimal hypersurfaces inRn+1, Amer. J. Math.120(1998), 103–116.