On the fundamental tone of minimal submanifolds with controlled extrinsic curvature

0.15 MB·English

by  Vicent Gimeno

ON THE FUNDAMENTAL TONE OF MINIMAL
SUBMANIFOLDS WITH CONTROLLED EXTRINSIC
CURVATURE
VICENTGIMENO
3
1 Abstract. Theaimofthispaperistoobtainthefundamentaltoneformin-
0
imalsubmanifoldsof theEuclideanorhyperbolicspace under certainrestric-
2 tions on the extrinsic curvature. We show some sufficient conditions on the
y norm of the second fundamental form that allow us to obtain the same up-
a per and lower bound for the fundamental tone of minimal submanifolds in
M a Cartan-Hadamard ambient manifold. As an intrinsic result, we obtain a
sufficientconditiononthevolumegrowthofaCartan-Hadamardmanifoldto
achievethelowestboundforthefundamental tonegivenbyMcKean.
8
2

]
G 1. Introduction
D
Let (M,g) be a Riemannian manifold, the fundamental tone λ (M) of M is
∗
.
h defined by
t
a
m f 2
[ (1.1) λ∗(M)=inf{∫M∫M|∇f2| ; f ∈L21,0(M)\{0}}
3 where L21,0(M) is the completion of C0∞(M) with respect to the norm kϕk2 =
v ϕ2+ ϕ2.
8 MThe funMda|∇men|tal tone is a powerful tool in the challenging area of geometric
4 R R
analysis,being a classicfeature its relationwith the curvature ofthe manifold. We
1
refer to the book of I. Chavel [5] for a discussion of these and related concepts.
1
. For non-positively curved manifolds, lower bounds for the fundamental tone are
1
well-known. Forthesecases,McKeangavethefollowingboundsforthefundamental
0
tone of a Cartan-Hadamardmanifold.
3
1 Theorem 1.1 (See [14]). Let Mn be a complete, simply connected manifold with
:
v sectional curvature bounded above by κ<0. Then
i
X (n 1)2κ
(1.2) λ∗(M) − − .
r ≥ 4
a
Similary,L.P.Cheung andL.F.Leung showedthat minimalsubmanifoldsofthe
hyperbolic space retain this lower bound of the ambient manifold.
Theorem 1.2 (See [8]). Let Mn be a complete minimal submanifold in the hyper-
bolic space Hm(κ) of constant sectional curvature κ<0. Then
(n 1)2κ
(1.3) λ∗(M) − − .
≥ 4
Observe that inequality (1.2) and inequality (1.3) are sharp in the sense that
when M is the hyperbolic space Hn(κ) (considered as a manifold or considered
as a totally geodesic submanifold of Hm(κ)) the fundamental tone λ (M) satisfies
∗
λ∗(M)= −(n−41)2κ (see [5] or [14]).
Moreover, it is known that other minimal submanifolds exist as well as the
hyperbolic space with that lowest fundamental tone. For example, all the classical
1
2 VICENTGIMENO
minimal catenoids in the hyperbolic space H3( 1) given by M. do Carmo and M.
−
Dajczer in [4] achieve equality in (1.3) as shown by A. Candel.
Theorem 1.3 (See [3]). The fundamental tone of the minimal catenoids (given in
[4]) in the hyperbolic space H3( 1) is
−
1
λ (M)= .
∗
4
The purpose of this paper is to provide geometric conditions under which a
submanifold(oramanifold)attainstheequalityintheinequality(1.3)(orinequality
(1.2), respectively). Let us emphasize that the spherical catenoids given in the
previous example, as it was analyzed in [17], have finite L2-norm of the second
fundamental form (denoted by A throughout this paper)
(1.4) A2dµ< .
| | ∞
ZM
This couldsuggestthat the finiteness ofthe L2-normofthe secondfundamental
form for minimal surfaces in the hyperbolic space implies the lowest fundamental
tone. This is in fact true, and is the statement of our first theorem.
TheoremA. LetM2 beaminimalsurfaceimmersedinthehyperbolicspaceHm(κ)
of constant sectional curvature κ < 0. Suppose that M2 has finite total extrinsic
curvature, i.e, A2dµ< . Then, the fundamental tone satisfies
M| | ∞
κ
(1.5) R λ∗(M)= − .
4
Our approachto the problem makes use of our previous results about the influ-
enceoftheextrinsiccurvaturerestrictionsonthevolumegrowthofthesubmanifold
(see[11],[10],[12]or 2),andalsotherelationbetweenthefinitevolumegrowthand
§
the fundamental tone as a new tool (see the Main Theorem).
For higher dimensional minimal submanifolds of the hyperbolic space, under an
appropriate decay of the norm of the second fundamental form on the extrinsic
distance (see 2 for a precise definition) we obtain
§
Theorem B. Let Mn be a minimal submanifold properly immersed in the hyper-
bolic space Hm(κ) of constant sectional curvature κ < 0. Suppose moreover that
n > 2 and the submanifold is of faster than exponential decay of its extrinsic cur-
vature. Namely, there exists a point p M such that
∈
δ(r (x))
p
A ,
x
| | ≤ e2√−κrp(x)
where δ(r) is a function such that δ(r) 0 when r and r is the extrinsic
p
→ → ∞
distance function. Then, the fundamental tone satisfies
(n 1)2κ
(1.6) λ∗(M)= − − .
4
We also prove for minimal submanifolds of the Euclidean space the following
well known result (see for example [2])
Theorem C. Let Mn be a minimal submanifold immersed in the Euclidean space
Rm with finite total scalar curvature, i.e, Andµ< . Then
M| | ∞
(1.7) λ∗(M)=R 0 .
Remark a. Asfarasweknow,thequestion1.5givenin[2]abouttheexistence(or
not)of complete minimalsubmanifolds properlyimmersedin the Rm with positive
fundamental tone is still an open question. In any case, if such kind of immersion
ON THE FUNDAMENTAL TONE OF MINIMAL SUBMANIFOLDS 3
exists,fromourMainTheoremthe immersionshouldbe ofinfinite volumegrowth,
and from Theorem E the submanifold has no an extrinsic doubling property.
Finally,asanintrinsicversionofourMainTheoremonemayobtainthefollowing
theorem in the direction of the McKean’s Theorem 1.1
Theorem D. Let Mn be a complete, simply connected manifold with sectional
curvature bounded from above by κ<0. Suppose moreover that there exists a point
p M such that
∈
Vol(BM(p))
(1.8) sup R < ,
Vol(Bκ) ∞
R>0 R
where BM(p) is the geodesic ball in M centered at p of radius R, and Bκ is the
R R
geodesic ball in Hn(κ) of the same radius R .Then
(n 1)2κ
(1.9) λ∗(M)= − − .
4
Remarkb. Inviewoftheintrinsicversionoftheresultsof[9]forcomplete,simply
connected,2-dimensionalmanifoldM2 withGaussiancurvatureK boundedfrom
M
above by K κ<0, the condition (1.8) can be achieved provided the following
M
≤
(1.10) (κ K )dµ< .
M
− ∞
ZM
In particular (in view of [9]), for every 2-dimensional Cartan-Hadamard manifold
M which is asymptotically locally κ-hyperbolic of order 2 (see [18],[9]) and with
sectional curvatures bounded from above by K κ < 0, the fundamental tone
M
≤
satisfies
κ
λ∗(M)= − .
4
Theproofofthethree previousextrinsictheorems(TheoremA,TheoremBand
TheoremC)usesthefactthatunderthehypothesisofthetheoremsthesubmanifold
has finite volume growth ( lim (R) < , see 2, and 2.1 for precise definitions
R Q ∞ § §
andseealso[19]foranaltern→a∞tivedefinitionintermsofprojectivevolume). Hence,
the theorems will be proved using the following Main Theorem:
Main Theorem. Let Mn be a n dimensional minimal submanifold properly im-
−
mersed in a simply connected Cartan-Hadamard manifold N of sectional curvature
K bounded from above by K κ 0. Suppose that
N N
≤ ≤
(1.11) lim (R)< .
R Q ∞
→∞
Then,
(m 1)2κ
(1.12) λ∗(M)= − .
− 4
To obtain the Main Theorem we estimate upper bounds for the fundamental
tone. According to S.T. Yau ([20]) it is important to find upper bounds for the
fundamental tone. In the case of stable minimal submanifolds of the hyperbolic
space,A. Candelgave in 2007the following upper bound for the fundamental tone
Theorem 1.4 (See [3]). Let M be a complete simply connected stable minimal
surface in the 3 dimensional hyperbolic space H3( 1). Then the fundamental tone
− −
of M satisfies
1 3
(1.13) λ∗(M) .
4 ≤ ≤ 4
4 VICENTGIMENO
Usingonlyashypothesisthe finitenessofthe L2 normofthesecondfundamen-
−
tal form, K. Seo has recently generalized the result of Theorem 1.4 without using
the hypothesis about the simply connectedness
Theorem 1.5 (See [17]). Let Mn be a complete stable minimal hypersurface in
Hn+1( 1) with A2dµ< . Then we have
− M| | ∞
R (n 1)2
(1.14) − λ∗(M) n2 .
4 ≤ ≤
AnotherachievementofthispaperisthatusingonlythefinitenessoftheL2 norm
−
ofthe secondfundamentalform(TheoremA),oranappropriatedecayofthenorm
of the second fundamental form, we can also remove the hypothesis about the sta-
bility, and obtain not only an inequality but an equality on the fundamental tone.
Remark c. Note that the finiteness of the L2-norm of the second fundamental
form of a minimal surface in the hyperbolic space does not imply the stability of
the surface. In fact, M. do Carmo and M. Dajczer also proved in [4] that there
exist some unstable spherical catenoids in H3( 1). And observe, moreover, that
−
the codimension of the submanifold plays no role in our theorems, in contrast to
what happens in the theorems from A. Candel and K. Seo where the codimension
must be 1.
The structure of the paper is as follows.
In 2we recallthe definitionofthe extrinsic distance function, the extrinsic ball
§
andthevolumegrowthfunction . ShowingthatunderthehypothesisofTheorems
Q
A, B, and C the submanifold has finite volume growth
(1.15) sup (R)< .
Q ∞
R>0
In 3 we will prove the Main Theorem. Finally as a corollary from the proof
§
of the Main Theorem, supposing that the submanifold has an extrinsic doubling
property we state the Theorem E where is obtained lower and upper bounds for
the fundamental tone of a minimal submanifold properly immersed in a Cartan-
Hadamard ambient manifold.
2. Preliminaries
The proof of Theorems A, B and C is based on upper and lower bounds for
the fundamental tone. The lowerbounds arewellknown(see Theorem1.2), but to
obtainupperboundsweusethesocalledvolumegrowthfunction anditsrelationto
thebehavioroftheextrinsiccurvature. Thevolumegrowthfunctionisthequotient
between the volume of an extrinsic ball of radius R and the geodesic ball of the
same radius R in an appropriate real space form.
Let Kn(κ) denote the n dimensional simply connected real space form of con-
−
stant sectional curvature κ 0, recall that the volume of the geodesic sphere Sκ
and the geodesic ball Bκ of≤radius R in Kn(κ) are (see [13]) R
R
R
Vol(Sκ)=ω S (R)n 1 Vol(Bκ)= Vol(Sκ)ds ,
R n κ − R s
Z0
being ω the volumeofthe geodesicsphereofradius1inRn, whereS isthe usual
n κ
function
t if κ=0,
S (t)=
κ (sinh(√−κt) if κ<0 .
√ κ
−
And the mean curvature pointing inward of the geodesic spheres of radius R is
ON THE FUNDAMENTAL TONE OF MINIMAL SUBMANIFOLDS 5
Ct (t)= S′κ(t) .
κ
S (t)
κ
On the other hand, given a submanifold M immersed in a simply connected
Cartan-HadamardmanifoldN ofsectionalcurvatureK boundedfromaaboveby
N
K κ 0, the extrinsic ball MR centered at p M of radius R is the sublevel
N ≤ ≤ p ∈
set of the extrinsic distance function r , where the extrinsic distance function is
p
Definition 2.1 (Extrinsic distance function). Let ϕ : M N be an immersion
→
from the manifold M to the simply connected Cartan-Hadamard manifold of sec-
tional curvature K bounded from above by K κ 0. Given a point p M,
N N
the extrinsic distance function r :M R+ is defi≤ned≤by ∈
p
→
r (x)=distN(ϕ(p),ϕ(x)) ,
p
where distN denotes the usual geodesic distance function in N.
Therefore the extrinsic ball MR centered at p M of radius R is
p ∈
MR := x M : r (x)<R .
p { ∈ p }
With the extrinsic ball and the geodesic ball we can define the volume growth
function
Definition 2.2 (Volume growth function). Let ϕ : Mn N be an immersion
→
from the n dimensional manifold M to the simply connected Cartan-Hadamard
−
manifold of sectional curvature K bounded from above by K κ 0. The
N N
volume growth function :R+ R+ is given by ≤ ≤
Q →
Vol(MR)
p
(R):= .
Q Vol(Bκ)
R
Where Vol(Bκ) is the geodesic ball of radius R in Kn(κ).
R
2.1. Volume growth of minimal submanifolds of controlled extrinsic cur-
vature. FirstlytoprovetheTheoremsA,BandCwehavetostudythebehaviorof
the volume comparison function for minimal submanifolds of a Cartan-Hadamard
manifold. Let us recall the following monotonicity formula
Proposition 2.3 (Monotonicity formula, see [15]). Let ϕ : Mn N be an im-
→
mersion from the n dimensional manifold M to the simply connected Cartan-
−
Hadamard manifold of sectional curvatureK bounded from above by K κ 0.
N N
≤ ≤
Then, the volume growth function is a non-decreasing function.
Secondly to prove Theorems A, B, and C we need to check that an appropriate
controloftheirsecondfundamentalformallowustoapplytheMainTheorem( 3).
§
In order to apply the Main Theorem the submanifold should have finite volume
growth
(2.1) lim (R)< .
R Q ∞
→∞
ButwecanapplytheMainTheoremunderthe assumptionsoftheTheoremsA,
B and C because of the following three theorems that we recall here:
Theorem 2.4 (See [1],[6] and [11]). Let Mn be a minimal submanifold immersed
in the Euclidean space Rm. If Mn has finite total scalar curvature
Andµ< .
| | ∞
ZM
Then
sup (R)< .
Q ∞
R>0
6 VICENTGIMENO
Theorem2.5(See[7],[11]and[16]). LetM2 beaminimalsurfaceimmersedinthe
hyperbolic space Hm(κ) of constant sectional curvature κ < 0 or in the Euclidean
space Rm. If M has finite total extrinsic curvature, namely A2dµ < , then
M| | ∞
M has finite topological type, and
R
1
sup (R) A2dµ+χ(M) ,
Q ≤ 4 | |
R>0 ZM
being χ(M) the Euler characteristic of M.
And
Theorem 2.6 (see [12]). Let Mn be a minimal n dimensional submanifold prop-
erly immersed in the hyperbolic space Hm(κ) of con−stant sectional curvature κ<0.
If n > 2 and the submanifold is of faster than exponential decay of its extrinsic
curvature, namely, there exists a point p M such that
∈
δ(r (x))
p
A ,
x
| | ≤ e2√−κrp(x)
where δ(r) is a function such that δ(r) 0 when r . Then the submanifold
→ → ∞
has finite topological type, and
sup (R) (M) ,
Q ≤E
R>0
being (M) the (finite) number of ends of M.
E
3. Proof of the Main Theorem: volume growth behavior and
fundamental tone
The wayto proofthe equality(1.12) inthe MainTheoremis to obtainthe same
upper and lowerbound for the fundamentaltone. Hence, firstofall, we needlower
bounds for the fundamental tone. But the lower bounds are well known, and it is
straight forward in a similar way to Theorem 1.2 that
(n 1)2κ
λ∗(M) − − .
≥ 4
TheabovelowerboundcanalsobeprovedusingtheCheegerisoperimetricconstant
of the submanifold. Taking into account that the Cheeger constant h(M) of a
minimal submanifold Mn properly immersed in a Cartan-Hadamard manifold N
of sectional curvatures K bounded from above by K κ 0 is (see [10])
N N
≤ ≤
(3.1) h(M) (n 1)√ κ ,
≥ − −
therefore,
h(M)2 (n 1)2κ
(3.2) λ∗(M) − − .
≥ 4 ≥ 4
To obtain the upper bounds for the fundamental tone, we use the Rayleigh
quotient definition (1.1) with an appropriate testing function. The first step to
obtain the testing function is to define the real function φ:R R, given by
→
f(t) if t [R,R] ,
(3.3) φ(t)= ∈ 2
(0 otherwise
where the function f :R R is
→
sin(2π(s−R2))
(3.4) f(s)= R .
Vol(Sκ)1/2
s
ON THE FUNDAMENTAL TONE OF MINIMAL SUBMANIFOLDS 7
Now,thesecondsteptotaketoconstructthetestingfunctionΦ,istotransplant
φ to M using the extrinsic distance function by the following definition:
Φ:M R; Φ(x)=φ(r (x)) .
p
→
By the Rayleigh quotient definition and the coarea formula
Φ, Φ dµ (φ)2 r , r dµ (φ)2dµ
λ (M) Mh∇ ∇ i = M ′ h∇ p ∇ pi M ′
∗
≤ Φ2dµ φ2dµ ≤ φ2dµ
R M R M R M
RR (φ′)2 ds R(Rφ(s))2 1 dRs
= 0 ∂Mps |∇r| = R2 ′ ∂Mps |∇r|
(3.5) R RhR φ2 ids R Rφ2(s) hR 1 ids
0 ∂Mps |∇r| R2 ∂Mps |∇r|
RRR(hφR′(s))2 Voil(Mps) ′dRs hR i
= 2 .
R Rφ2(s) (cid:0)Vol(Ms) (cid:1)′ds
R p
2
R (cid:0) (cid:1)
Using the following two lemmas
Lemma 3.1.
(3.6) Q(s)Vol(Ssκ)≤ Vol(Mps) ′ =(lnQ(s))′Vol(Bsκ)Q(s)+Q(s)Vol(Ssκ) .
Proof. Fromthedefiniti(cid:0)onof an(cid:1)dtakingintoaccountthat isanon-decreasing
Q Q
function (by proposition 2.3)
VolMs ′ Vol(Sκ)
(3.7) (ln (s))′ = p s 0 .
Q VolMs − Vol(Bκ) ≥
(cid:0) p(cid:1) s
So, (cid:0) (cid:1)
(3.8) Q(s)Vol(Ssκ)≤ Vol(Mps) ′ =(lnQ(s))′Vol(Bsκ)Q(s)+Q(s)Vol(Ssκ) .
(cid:0) (cid:1) ✷ (cid:3)
Lemma 3.2. There exists an upper bound function Λ:R+ R+ such that:
→
R(φ)2Vol(Sκ)ds
(3.9) 0 ′ s Λ(R) ,
R 0Rφ2Vol(Ssκ)ds ≤
and R
(m 1)κ
(3.10) lim Λ(R)= − −
R 4
→∞
Proof.
(3.11) 0R(φ′(s))2Vol(Ssκ)ds = RR2(f′(s))2Vol(Ssκ)ds .
R 0Rφ(s)2Vol(Ssκ)ds R RR2 f(s)2Vol(Ssκ)ds
R R
But
2 −m2−1Ctκ(s)sin(2π(sR−R2))+ 2Rπ cos(2π(sR−R2)) 2
(f (s)) =
′ (cid:16) Vol(Sκ) (cid:17)
(3.12) s
(m−41)2Ctκ(s)2sin(2π(sR−R2))2+ 4Rπ22 + 2(mR−1)πCtκ(s) .
≤ Vol(Sκ)
s
8 VICENTGIMENO
And, since Ctκ(t) is a non-increasing function and RR2 sin(2π(sR−R2))2ds= R4
0R(φ′(s))2Vol(Ssκ)ds (m−1)2CtR (R/2)2+ 8π2
(3.13) R 0Rφ(s)2Vol(Ssκ)ds ≤ 4 κ R2
4π(m 1)
R + − Ct (R/2) .
κ
R
Then, letting
(m 1)2 8π2 4π(m 1)
(3.14) Λ(R):= − Ct (R/2)2+ + − Ct (R/2) ,
4 κ R2 R κ
and taking into account that
(3.15) lim Ct (R/2)=√ κ ,
κ
R −
→∞
the lemma is proven. ✷ (cid:3)
Denotingnow,F(R):= (m−41)2Ctκ(R/2)2+ 4Rπ22 + 2(mR−1)πCtκ(R/2) andδ(R):=
RR2 (lnQ(s))′ds, applyin(cid:16)g the lemma 3.1 and the lemma 3.2 to inequa(cid:17)lity (3.5) we
Rget since Vol(Bsκ) is a non-decreasing function
Vol(Sκ)
s
λ∗(M)≤Q((RR))RRR2(φ′(s))2(lnQ(s))R′Vφo2l((sB)sκV)odls(S+κ)RdRR2s(φ′(s))2Vol(Ssκ)ds
Q 2 R2 s
(R) 4 R R
≤Q(R) R R (φ′(s))2(lnQ(s))′Vol(Bsκ)ds+Λ(R)!
Q 2 Z2
(R) Vol(Bκ) 4
Q R F(R)δ(R)+Λ(R)
≤ (R) Vol(Sκ) R
Q 2 (cid:20) R (cid:21)
Letting R tend to infinity and taking into account that
(m 1)2κ
lim F(R)= − ,
R − 4
→∞
lim δ(R)=0 ,
R
→∞
Vol(Bκ) 4 4 if κ=0,
lim R = m 1
(3.16) R→∞ Vol(SRκ) R (0 i−f b<0.
(R)
lim Q =1 ,
R (R)
→∞Q 2
(m 1)2κ
lim Λ(R)= − .
R − 4
→∞
we conclude the proof of the theorem.
Remark d. Observe that the finiteness of the volume growth function
(3.17) lim (R)< ,
R Q ∞
→∞
is only used in the proof of the Main Theorem to achieve
(R)
(3.18) lim Q =1 .
R (R)
→∞Q 2
Therefore,wecouldusetheslightlyweakerassumptiononthevolumegrowthgiven
by the limit (3.18) instead the assumption on the finite volume growth of the
ON THE FUNDAMENTAL TONE OF MINIMAL SUBMANIFOLDS 9
submanifold (limit (3.17)), or use an extrinsic doubling property to obtain two
sides estimates for the fundamental tone
TheoremE. LetMn bean dimensionalminimalsubmanifold properly immersed
−
in a simply connected Cartan-Hadamard manifold N of sectional curvature K
N
bounded from above by K κ 0. Suppose that the immersion has an extrinsic
N
≤ ≤
doubling property, namely
(R)
(3.19) Q <C .
(R)
Q 2
Then,
(m 1)2κ C(m 1)2κ
(3.20) − λ∗(M) − .
− 4 ≤ ≤− 4
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10 VICENTGIMENO
DepartmentofMathematics,UniversitatJaumeI,Castello´ de la Plana,Spain
E-mail address: [email protected]