The spectrum of the MartinMoralesNadirashvili minimal surfaces is discrete

DOI 10.1007/s12220-009-9101-z

The Spectrum of the Martin-Morales-Nadirashvili

Minimal Surfaces Is Discrete

G. Pacelli Bessa_·Luquesio P. Jorge_· J. Fabio Montenegro

Received: 3 March 2009 / Published online: 7 August 2009 © Mathematica Josephina, Inc. 2009

Abstract We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean cur-vature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball ofR3is discrete. This gives a positive answer to a question of Yau (Asian J. Math. 4:235–278,2000).

Keywords Proper bounded minimal submanifolds·Discrete spectrum·Essential spectrum

Mathematics Subject Classification (2000) 53C40·53C42·58C40

1 Introduction

An interesting problem in geometry is to understand the relations of the geometry of a Riemannian manifold and its spectrum. Particularly interesting are the geometric restrictions implying that the spectrum is purely continuous or discrete. There are several results along these lines, see [5,6,8,12,21,23] for geometric conditions implying that the spectrum is purely continuous and [2,7,10,14,15] for geometric conditions implying that the spectrum is discrete.

The first and third author were partially supported by CNPq-CAPES, Brazil. G.P. Bessa (

Department of Mathematics, Universidade Federal do Ceara–UFC, Campus do Pici, 60455-760 Fortaleza, CE, Brazil

e-mail:bessa@mat.ufc.br

L.P. Jorge

e-mail:ljorge@mat.ufc.br

Since every complete Riemannianm-manifold can be realized as a complete sub-manifold embedded into a fixed ball of radiusrof ann-dimensional Euclidean space, withndepending only onm, see [20]. It would be important to understand the rela-tions between the spectrum and the extrinsic geometries of bounded embeddings of complete Riemannian manifolds in Euclidean spaces.

A particularly interesting aspect of this problem is the spectrum related part of the so calledCalabi-Yau conjectures on minimal surfaces. S.T. Yau, revisiting Calabi conjectures on minimal surfaces, in his Millennium Lectures [24,25], wrote:It is known[19]that there are complete minimal surfaces properly immersed into a_[open_] ball.. . .Are their spectrum discrete?It is not clear that the Nadirashvili’s complete bounded minimal surface [19] isproperly immersed. However, in [17,18], F. Martin and S. Morales constructed, for every open convex subsetBofR3, a complete proper minimal immersionϕ:D֒_→B, where Dis the disk on R2. The Martin-Morales’ method is a highly non-trivial refinement of Nadirashvili’s method, thus we name, (as we should), these complete minimal surfaces properly immersed into bounded convex subsetsB of R3 as Martin-Morales-Nadirashvili minimal surfaces. In this paper we answer positively Yau’s question in the following corollary of the main theorem.

Theorem 1.1 Letϕ_:M ֒_→B_R3(r)⊂R3be a complete surface,properly immersed

into a ball.If the norm of the mean curvature vectorH ofMsatisfies

sup

M |

H_|<2/r

thenMhas discrete spectrum.

The Theorem1.1is a particular case of our main result, Theorem1.2, that shows that the spectrum of a complete properly immersed submanifoldϕ:M ֒_→BN(r)is

discrete provided the norm of the mean curvature vector H₌Trα is sufficiently small. HereBN(r)⊂N is a normal geodesic ball of radiusrof a Riemannian

mani-foldNandαis the second fundamental form. In the following we denote

Cb(t )=

⎧ ⎨

⎩

√

bcot(√bt ) ifb >0,

1/t ifb₌0, √

−bcoth(√₋bt ) ifb <0.

(1)

Theorem 1.2 Letϕ:M ֒→BN(r)be a completem-submanifold properly immersed

into a geodesic ball,centered atp with radiusr,of a Riemanniann-manifoldN.

Letb₌supK_NradwhereK_Nradare the radial sectional curvatures along the geodes-ics issuing fromp.Assume thatr <min{injN(p), π/2

√

b_},whereπ/2√b_{= +∞}if b_≤0.If the norm of the mean curvature vectorHsatisfies,

sup

M |

H_|< m_·Cb(r),

The idea of the proof of Theorem1.2can be extended to prove that the essential spectrum of submanifolds properly immersed1 in a cylinder BN(r)×R is empty

provided the norm of the mean curvature is small, see Theorem4.1.

A Riemannian manifoldM is said to bestochastically completeif for some (and thus, for any)(x, t )_∈M_×(0,_+∞)it holds that

M

p(x, y, t )dy₌1,

wherep(x, y, t )is the heat kernel of the Laplacian operator. Otherwise, the man-ifold M is said to be stochastically incomplete (for further details about this see [9]). It seems to have a close relationship between discreteness of the spectrum of a complete noncompact Riemannian manifolds and stochastic incompleteness. For instance, it was proved in [1] that submanifolds satisfying the hypotheses of The-orem1.2, (without the properness condition) are stochastically incomplete. Harmer [10], shows that stochastic incompleteness implies discreteness of the spectrum in a certain class of Riemannian manifolds. Based on these evidences, we believe that the following conjecture should be true.

Conjecture 1.3 A complete noncompact Riemannian manifold has discrete spectrum if and only if is stochastically incomplete.

Remark 1.4 I. Salavessa [22], proved discreteness ofX-bounded minimal submani-folds of Riemannian manisubmani-folds carrying strongly convex vector fieldX.

2 Preliminaries

LetM be a complete noncompact Riemannian manifold. The Laplacian △acting onC₀∞(M)has a unique self-adjoint extension to an unbounded operator acting on

L2(M), also denoted by △, whose domain are those functions f _∈L2(M) such that △f _∈L2(M) and whose spectrum (M)_{⊂ [}0,_∞) decomposes as (M)₌ p(M)∪ess(M)wherep(M)is formed by eigenvalues with finite multiplicity

andess(M)is formed by accumulation points of the spectrum and by the

eigenval-ues with infinite multiplicity. It is said thatMhas discrete spectrum ifess(M)= ∅

and thatMhas purely continuous spectrum ifp(M)= ∅.

IfK_⊂M is a compact manifold, of the same dimension as M, with boundary then there is a self-adjoint extension△′of the Laplacian△ofM\K by imposing Dirichlet conditions. TheDecomposition Principle[7] says that△and△′have the same essential spectrumess(M)=ess(M\K). On the other hand, the bottom of

the spectrum ofM\Kis equal to the fundamental tone ofM\K, i.e. inf(M\K)= λ∗(M_\K), where

λ∗(M_\K)₌inf M\K|grad

f_|2

M\Kf2

, f_∈C₀∞(M_\K)_{\ {}0}

.

To give lower estimates forλ∗(M_\K)we need of the following version of Barta’s Theorem.

Theorem 2.1(Barta [3]) Let_⊂M be an open subset of a Riemannian manifold Mand letf _∈C2(),f_| >0.Then

λ∗()_≥inf

(− f/f ). (2)

Proof LetX_{= −}grad logf be aC1vector field in. It was proved in [4] that

λ∗()_≥inf

(divX− |X|

2₎

=inf

−

△f f

.

The second main ingredient of our proof is the Hessian comparison theorem.

Theorem 2.2 LetMmbe a Riemannian manifold andx0, x1∈Mbe such that there

is a minimizing unit speed geodesicγjoiningx0andx1and letρ(x)=dist(x0, x)be

the distance function tox0.Leta_≤Kγ ≤bbe the radial sectional curvatures ofM

alongγ.Ifb >0assumeρ(x1) < π/2

√

b.Then,we haveHessρ(x)(γ′, γ′)₌0and

Ca(ρ(x))X2≥Hessρ(x)(X, X)≥Cb(ρ(x))X2 (3)

whereX_∈TxMis perpendicular toγ′(ρ(x)).

Let ϕ _: M ֒_→W be an isometric immersion of a complete Riemannian m -manifoldMinto a Riemanniann-manifoldWwith second fundamental formα. Con-sider aC2-functiong_:W_→Rand the compositionf ₌g_◦ϕ_:M_→R. Identifying

Xwithdϕ(X)we have atq_∈Mthat the Hessian off is given by

Hessf (q)(X, Y )₌Hessg(ϕ(q))(X, Y )₊ gradg, α(X, Y )ϕ(q). (4)

Taking the trace in (4), with respect to an orthonormal basis{e1, . . . em}forTqM, we

have the Laplacian off,

f (q)₌

m

i=1

Hessg(ϕ(q))(ei, ei)+

gradg,

m

i=1

α(ei, ei)

. (5)

The formulas (4) and (5) are well known in the literature, see [11].

3 Proof of Theorem1.2

LetK1⊂K2⊂ · · ·be an exhaustion sequence ofMby compact sets. The well known

Decomposition Principle[7] states thatMandM_\Ki have the same essential

spec-trum,ess(M)=ess(M\Ki). Therefore, the Theorem1.2is proved if we show that

By hypothesis we have a completem-submanifoldϕ:M ֒_→BN(r)properly

im-mersed into a ballBN(r)=BN(p, r)with center atpand radiusrin a Riemannian

n-manifold N with radial sectional curvatures K_Nrad along the radial geodesics is-suing from p bounded as a ₌infK_Nrad_≤K_Nrad_≤b₌supK_Nrad in BN(r), where

r <min{inj_N(p), π/2√b_}. Here we replaceπ/2√bby+∞ifb_≤0.

Define a functionv:BN(p, r)→Rby v(y)=φa(ρ(y)), whereφa: [0, r] →R

given by

φa(t )=

⎧ ⎪ ⎨

⎪ ⎩

cos(√at )₋cos(√ar) ifa >0, t < π/2√a,

r2₋t2 ifa₌0,

cosh(√₋ar)₋cosh(√₋at ) ifa <0.

(6)

Observe thatφ (t ) >0 in [0, r),φa(r)=0, φa′(t ) <0 and φ′′a(t )−Ca(t )φa′(t )=0

in[0, r_]. This functionφa we learned from Markvorsen [16]. Let f:M→R

de-fined by f ₌v_◦ϕ and consider an exhaustion sequence of M by compact sets

Ki =ϕ−1(BN(p, ri)), where ri < r, ri → r. By Barta’s Theorem we have that

λ∗(M_\Ki)≥infM\Ki( −△f

f ).

Now by (5) we have

△f (x)₌

m

i=1

HessNv(ϕ(x))(ei, ei)+

gradv,

m

i=1

α(ei, ei)

.

=

m

i=1

HessNv(ϕ(y))(ei, ei)+ gradv, H.

The metric ofN inside the normal geodesic ball BN(p, r)can be written in

po-lar coordinates asds2₌dt2_{+ |}A(t, ξ )_|2dξ2,A(t, ξ )satisfies the Jacobi equation

A′′₊RA₌0 with initial conditionsA(0, ξ )₌0,A′(0, ξ )₌I. We have at the point

ϕ(x)an orthonormal basis{∂/∂t, ∂/∂ξ1, . . . ∂/∂ξn−1}forTϕ(x)N. We may choose an

orthonormal basis forTx(M\Ki)ase1= e1, ∂/∂t ·∂/∂t+e⊥1, wheree⊥1 ⊥∂/∂t and{e2, . . . , em} ⊂ {∂/∂ξ1, . . . ∂/∂ξn−1}. Computing HessNv(ϕ(x))(ei, ei)we have

HessNv(ϕ(x))(e1, e1)

=φ_a′′(t )₋φ′_a(t )_·HessNρ(e1⊥/|e⊥1|, e⊥1/|e⊥1|)

e1,gradρ2

+φ_a′(t )_·HessNρ(e⊥1/|e⊥1|), e⊥1/|e⊥1|))

=φ_a′(t )_·Ca(t )−Hessρ(e⊥1/|e⊥1|, e⊥1/|e1⊥|)

e1,gradρ2

+φ_a′(t )_·HessNρ(e⊥1, e1⊥) (7) and fori_≥2

wheret₌ρ(ϕ(x)). Now,

−△f _{= −}φ′_a(t )_·Ca(t )−Hessρ(e1⊥/|e⊥1|), e1⊥/|e1⊥|))

e1,gradρ2

−φ_a′(t )_·

HessNρ(e1⊥/|e⊥1|), e⊥1/|e1⊥|))+

m

i=2

HessNρ(ei, ei)

−φ_a′(t )gradρ, H

≥ −φ′_a(t )· [m·Cb(t )−sup|H|].

We used thatCa(t )≥Hessρ(e1⊥/|e1⊥|), e⊥1/|e⊥1|))≥Cb(t ), Hessρ(ei, ei)≥Cb(t )

by the Hessian Comparison Theorem and that−φ_a′(t ) >0. Hence

λ∗(M_\Ki)≥ inf

M\Ki −

△f f

≥ inf

M\Ki −φ

′

a(t )

φa(t )[

m_·Cb(t )−sup|H|]

≥ −φ ′

a(ri)

φa(ri)[

m_·Cb(r)−sup|H|]. (9)

Thusλ∗(M_\Ki)→ ∞asri→r. This proves Theorem1.2.

4 Cylindrically Bounded Submanifolds

Letϕ:Mm_֒→B

N(r)×Rℓ⊂Nn−ℓ×Rℓ,m≥ℓ+1, be an isometric immersion

of a complete Riemannianm-manifoldMm into theBN(r)×Rℓ, whereBN(r) is

a geodesic ball in a Riemannian (n₋ℓ)-manifoldNn−ℓ, centered at a pointpwith radiusr. Let b₌supK_NradwhereK_Nradare the radial sectional curvatures along the geodesics issuing fromp. Assume thatr <min{inj_N(p), π/2√b}, whereπ/2√b= +∞ifb_≤0.

Theorem 4.1 Suppose thatϕ:Mm֒_→BN(r)×Rℓas above satisfies the following.

1. For everys < r,the setϕ−1(BN(s)×Rℓ)is compact inM.

2. supM|H|< (m−ℓ)Cb(r)where|H|(x)is norm of the mean curvature vector of

ϕ(M)atϕ(x).

ThenMhas discrete spectrum.

Proof Observe that the condition 1. is a stronger property than being a proper immer-sion except whenℓ₌0. As before, letri→rbe a sequence of positive real numbers

ri< r and the compacts setsKi =ϕ−1(BN(ri)×Rℓ). We need only to show that

ρ(x)₌distN(p, x),φa given in (6) and a=infKNrad. Letf =v◦ϕ:M→R. We

have by (5)

△f (x)₌

m

i=1

HessN×Rℓv(ϕ(x))(ei, ei)+ gradv, H

=

m

i=1

HessN(φa◦ρ)(ϕ(x))(ei, ei)+ grad(φa◦ρ), H. (10)

Atϕ(x)₌(y1, y2), consider the orthonormal basis

{

Polar basis

∂/∂t, ∂/∂ξ1, . . . ∂/∂ξn−ℓ−1,

Cartesian basis

∂/∂s1, . . . ∂/∂sℓ}

for

T(y1,y2)N

n−ℓ

×Rℓ=Ty1N

n−ℓ

⊕Ty2R

ℓ_.

Choose an orthonormal basis{e1, e2, . . . , em}as follows

ei=ai

∂ ∂t +

n−ℓ−1

j=1

bij

∂ ∂ξj +

ℓ

j=1

cij

∂ ∂sj

.

Using thatφ′_a(t ) <0,φ′′_a=Ca(t )φa′(t )and Hessρ(y1)(∂/∂ξj, ∂/∂ξj)≥Cb(t )for

allj₌1, . . . , n₋ℓ₋1, we have that

Hessφa◦ρ(y1)(ei, ei)

=φ_a′′(t )a_i2₊φ_a′(t )

n−ℓ−1

j=2

b_ij2Hessρ(y1)(∂/∂ξj, ∂/∂ξj)

≤φ_a′′(t )a_i2₊φ′_a(t )

n−ℓ−1

j=2

b2_ijCb(t )

=Ca(t )φa′(t )a

2

i +φa′(t )

1−a2_{i −}

ℓ

k=1

c2_ik

Cb(t )

=φ_a′(t )a_i2(Ca(t )−Cb(t ))+φ′a(t )

1−

ℓ

k=1

c_ik2

Cb(t )

≤φ_a′(t )

1−

ℓ

k=1

c2_ik

sinceCa(t )≥Cb(t )and wheret=ρ(y1). Therefore

−

n

i=1

Hessφa◦ρ(y1)(ei, ei)≥ −φ′a(t )

m₋

m

i=1

ℓ

k=1

c2_ik

Cb(t )

≥ −φ′_a(t )(m−ℓ)Cb(t )

From this we have that

−△f

f (x)≥ − φ′_a(t ) φa(t )

(m₋ℓ)Cb(t )−sup

M |

H_| (11)

so that

inf

M\Ki −

△f f

≥ −φ ′

a(ri)

φa(ri)

(m₋ℓ)Cb(r)−sup

M |

H_|.

Therefore infM\Ki(− △f

f )→ +∞asri→rproving Theorem4.1.

5 Final Remarks

The proof of Theorem1.2can be easily adapted to prove that the spectrum of a com-plete properly immersed submanifoldϕ:M ֒_→B_⊂N is discrete provided the norm of the mean curvature vectorH is sufficiently small, whereB is a smooth convex bounded subset ofN.

The distance to the boundary∂Bis smooth in aǫ-tubular neighborhoodTǫ(∂B)\

∂Bof the boundary for smallǫ >0. Letǫi→0 a sequence of positive numbers and

letKi=ϕ−1(B\Tǫi(∂B)). We have to show thatλ∗(Ki)→ ∞asi→ ∞.

We can define a positive smooth functiong:Tǫ(∂B)\∂B→Rusing the distance

to the boundary and then definef ₌g_◦ϕand apply a Hessian Comparison Theorem of Kasue [13] together with Barta’s Theorem as in the proof of Theorem1.2. We leave the details to the interested reader.

References

1. Alias, L., Bessa, G.P., Dajczer, M.: On the mean curvature of cylindrically bounded submanifolds. Math. Ann.345, 367–376 (2009).arxiv:0812.0623

2. Baider, A.: Noncompact Riemannian manifolds with discrete spectra. J. Differ. Geom.14, 41–57 (1979)

3. Barta, J.: Sur la vibration fundamentale d’une membrane. C. R. Acad. Sci.204, 472–473 (1937) 4. Bessa, G.P., Montenegro, J.F.: An extension of Barta’s Theorem and geometric applications. Ann.

Global Anal. Geom.31, 345–362 (2007)

5. Donnelly, H.: Negative curvature and embedded eigenvalues. Math. Z.203, 301–308 (1990) 6. Donnelly, H., Garofalo, N.: Riemannian manifolds whose Laplacian have purely continuous

spec-trum. Math. Ann.293, 143–161 (1992)

8. Escobar, E.: On the spectrum of the Laplacian on complete Riemannian manifolds. Comm. Partial Differ. Equ.11, 63–85 (1985)

9. Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc.36(2), 135–249 (1999)

10. Harmer, M.: Discreteness of the spectrum of the Laplacian and Stochastic incompleteness. J. Geom. Anal.19, 358–372 (2009)

11. Jorge, L., Koutrofiotis, D.: An estimate for the curvature of bounded submanifolds. Am. J. Math.103, 711–725 (1980)

12. Karp, L.: Noncompact manifolds with purely continuous spectrum. Mich. Math. J.31, 339–347 (1984)

13. Kasue, A.: A Laplacian comparison theorem and function theoretic properties of a complete Rie-mannian manifold. Jpn. J. Math.8, 309–341 (1981)

14. Kleine, R.: Discreteness conditions for the Laplacian on complete noncompact Riemannian mani-folds. Math. Z.198, 127–141 (1988)

15. Kleine, R.: Warped products with discrete spectra. Results Math.15, 81–103 (1989)

16. Markvorsen, S.: On the mean exit time from a minimal submanifold. J. Differ. Geom.29, 1–8 (1989) 17. Martín, F., Morales, S.: Complete proper minimal surfaces in convex bodies ofR3. Duke Math. J.

128, 559–593 (2005)

18. Martín, F., Morales, S.: Complete proper minimal surfaces in convex bodies ofR3. II. The bahavior of the limit set. Comment. Math. Helv.81, 699–725 (2006)

19. Nadirashvili, N.: Hadamard’s and Calabi-Yau’s conjectures on negatively curved and minimal sur-faces. Invent. Math.126, 457–465 (1996)

20. Nash, J.: The imbedding problem for Riemannian manifolds. Ann. Math. (2)63, 20–63 (1956) 21. Rellich, F.: Über das asymptotische Verhalten der Lösungen von△u+λu=0 in unendlichen

Gebi-eten. Jahresber. Dtsch. Math.-Ver.53, 57–65 (1943)

22. Salavessa, I.: On the spectrum ofX¯-bounded minimal submanifolds.arXiv:0901.1246v1

23. Tayoshi, T.: On the spectrum of the Laplace-Beltrami operator on noncompact surface. Proc. Jpn. Acad.47, 579–585 (1971)

24. Yau, S.T.: Review of geometry and analysis. Asian J. Math.4, 235–278 (2000)