Compact manifolds with fixed boundary and large Steklov eigenvalues

0.21 MB·

by  Bruno Colbois

COMPACT MANIFOLDS WITH FIXED BOUNDARY
AND LARGE STEKLOV EIGENVALUES
7
1 BRUNO COLBOIS, AHMAD EL SOUFI, AND ALEXANDRE GIROUARD
0
2
Abstract. Let (M,g) be a compact Riemannian manifold with
n
boundary. Letb>0bethenumberofconnectedcomponentsofits
a
J boundary. For manifolds of dimension 3, we prove that for j =
5 b+1itis possible to obtainanarbitrari≥lylargeSteklov eigenvalue
1 σj(M,eδg) using a conformal perturbation δ C∞(M) which is
∈
supportedinathinneighbourhoodofthe boundary,withδ =0on

] the boundary. For j b, it is also possible to obtain arbitrarily
P
≤
S largeeigenvalues,buttheconformalfactormustspreadthroughout
. the interiorofM. Infact, whenworkingina fixedconformalclass
h
and for δ = 0 on the boundary, it is known that the volume of
t
a (M,eδg) has to tend to infinity in order for some σ to become
j
m
arbitrarily large. This is in stark contrast with the situation for
[ theeigenvaluesoftheLaplaceoperatoronaclosedmanifold,where

a conformal factor that is large enough for the volume to become
1
v unbounded results in the spectrum collapsing to 0. We also prove
5 thatitispossibletoobtainlargeStekloveigenvalueswhilekeeping
2 different boundary components arbitrarily close to each other, by
1
constructing a convenient Riemannian submersion.
4
0
.
1
0
7 1. Introduction
1
:
v The Steklov eigenvalues of a smooth compact connected Riemannian
i manifold (M,g) of dimension n+1 2 with boundary Σ are the real
X
≥
numbersσ forwhichthereexistsanon-zeroharmonicfunctionf : M
r
a R which satisfies ∂ f = σf on the boundary Σ. Here and further ∂ →is
ν ν
the outward normal derivative on Σ. It is well known that the Steklov
eigenvalues form a discrete spectrum 0 = σ < σ σ ,
1 2 3
≤ ≤ ··· ր ∞
where each eigenvalue is repeated according to its multiplicity. The
interplay between the geometry of M and the Steklov spectrum has
recently attracted substantial attention. See [8] and the references
therein for recent development and open problems.
During the firstweek of2017,AGand BC weresupposedto travelto Tours and
work with Ahmad El Soufi to complete this paper. We learned just a few days
before our visit of his untimely death. Ahmad was a colleague and a friend. He
will be dearly missed.
1
2 BRUNO COLBOIS,AHMAD EL SOUFI,AND ALEXANDREGIROUARD
Many developments linking the Steklov eigenvalues of a compact
manifold M with the eigenvalues λ of the Laplace operator on its
j
boundary have appeared. See for instance [13, 3] and more recently
[12], where it was proved that for any Euclidean domain Ω Rn+1
⊂
with smooth boundary, there exists a constant c > 0 such that
Ω
λ σ2 +2c σ , and σ c + c2 +λ .
j ≤ j Ω j j ≤ Ω Ω j
q
TheseresultsindicateastronglinkbetweentheSteklov eigenvaluesofa
manifoldand the geometry of its boundary. In fact, onsmooth surfaces
the spectral asymptotics is completely determined by the geometry of
the boundary [7].
On the other hand, Steklov eigenvalues are not very sensitive to per-
turbation away from the boundary. For instance given any domain
Ω M with compact closure (at a positive distance from the bound-
⊂
ary), it is easy to see that there exists a constant C such that any
Ω
Riemannian metric g which coincides with g on M Ω satisfies
′
\
σ (M,g ) σ (M,g) C for each j N.
j ′ j Ω
| − | ≤ ∈
See for instance Proposition 1.2.
In the present paper, we investigate the following question:
For a given closed Riemannian manifold Σ, how large can σ (M) be
2
among compact Riemannian manifolds M with boundary isometric to
Σ?
For a manifold (M,g) of dimension n+1 3 with boundary Σ, we
≥
will see that it is possible to make σ arbitrarily large by using confor-
2
mal perturbations g = h2g such that h = 1 on Σ. Of course, this imply
′
thatanyeigenvalueσ becomesarbitrarilylargeundersuchaconformal
j
perturbation, but the situation is more interesting than that. Indeed,
let b > 0 be the number of connected components. We will prove
that it is possible to make σ arbitrarily large by using a conformal
b+1
perturbation h2g where h is a smooth function which is different from
1 only in a thin strip located arbitrarily close to the boundary (with
h = 1 identically on Σ). It is also possible to make lower eigenvalues
σ arbitrarily large, but this requires conformal perturbations which
j
penetrates deeply into the manifold M.
One could also ask how small an eigenvalue σ (M) can be. This
j
question is easier, as it is relatively easy to construct small eigenvalues
while keeping the boundary fixed. On surfaces, it is sufficient to create
thin passages (see Figure 3 of [8, Section 4]) while for manifolds of
dimension 3, one can use a conformal perturbation supported inside
≥
the manifold M. See Proposition 2.1.
COMPACT MANIFOLDS WITH LARGE STEKLOV EIGENVALUES 3
Large eigenvalues on surfaces. It was proved in [11] that any compact
surface M with boundary of length L > 0 satisfies
8π
σ (M) (1+genus(M)).
2
≤ L
In [4], a sequence of surfaces (Ml)l N with one boundary component of
∈
fixed length L > 0 was constructed, which satisfies
lim σ (M ) = + .
2 l
l ∞
→∞
These two results give a complete answer to our initial question for
surfaces: it is possible to obtain arbitrarily large σ , but it is necessary
2
to increase the genus of M in order to do so.
Manifolds of higher dimensions. For any compact Riemannian man-
ifold (M,g) of dimension 3 with boundary Σ, we will show that a
≥
conformal perturbation is sufficient to obtain arbitrarilylarge σ . More
2
can be said: let b > 0 be the number of connected components of the
boundary Σ. The next theorem shows that it is possible to make σ
b+1
arbitrarily large using conformal perturbations g which are supported
ǫ
in an arbitrary neighbourhood of the boundary Σ, and which coincide
with g on the boundary. It is also possible to make σ large, but this
2
requires conformal perturbations away from the boundary (See Propo-
sition 3.2). The next theorem is the main result of this paper.
Theorem 1.1. Let (M,g) be a compact connected Riemannian mani-
fold of dimension 3 with b N boundary components Σ , ,Σ .
1 b
≥ ∈ ···
(i) For every neighborhood V of Σ, there exists a one-parameter family
of Riemannian metrics g conformal to g which coincide with g on Σ
ε
and in M V, such that
\
σ (g ) as ε 0.
b+1 ε
→ ∞ →
(ii) There exists a one-parameter family of Riemannian metrics g con-
ε
formal to g which coincide with g on Σ such that
σ (g ) as ε 0.
2 ε
→ ∞ →
The proof of Theorem 1.1 will be presented in Section 3. It is impor-
tant to note that in order to obtain large eigenvalues, it is necessary to
perturb the metric near each points of the boundary.
Proposition 1.2. Let (M,g) be a compact Riemannian manifold with
boundary Σ. Let p Σ and let ǫ > 0. Then any Riemannian metric g
′
∈
on M which coincide with g on B(p,ǫ) M satisfy
⊂
λN(B(p,ǫ),g) σ (M,g ) λD(B(p,ǫ),g),
k ≤ k ′ ≤ k
4 BRUNO COLBOIS,AHMAD EL SOUFI,AND ALEXANDREGIROUARD
where λD(B(p,ǫ),g) and λN(B(p,ǫ),g) are is the k-th eigenvalue of
k k
a mixed Steklov-Dirichlet eigenvalue and Steklov-Neumann eigenvalue
problems respectively.
The proof of this observation is an exercise in the use of min-max
characterisations of eigenvalues. It will be presented in Section 2.
The conformal perturbations which are used in the proof of Theorem
1.1 are such that the volume M tends to infinity as ǫ 0. This
| |gǫ →
is a necessary condition when working in a fixed conformal class [g].
Indeed, the following inequality for g [g] was proved in [9]:
′
∈
2
A+Bkn+1
σ (M,g ) Σ 1/n < ,
k ′
| | I(M)(n 1)/n
−
where Ais a constant which depends ong, B depends onthe dimension
and I(M) is the isoperimetric ratio
Σ g′
I(M) = | | .
n/(n+1)
M
| |g′
In each of the constructions, the diameter also becomes unbounded.
In Theorem 3.5, we construct a sequence g of Riemannian metrics on
m
M such that g = g and such that (M,g ) has uniformly bounded
m Σ Σ m
diameter and σ becomes arbitrarily large.
2
(cid:12)
To conclude th(cid:12)is introduction, note that it is difficult to obtain lower
bounds for Steklov eigenvalues. Under relatively strong convexity as-
sumptions this was already investigated by Escobar in [6]. More re-
cently Jammes [10] proposed an interesting inequality in the spirit of
Cheeger:
h(M)j(M)
σ (M) .
2
≥ 4
Here h(M) is the classical Cheeger constant and j(M) was introduced
in [10]. It is challenging to obtain effective lower bounds on σ (M)
2
using this inequality, mainly because it is difficult to estimate h(M)
and j(M). Moreover, it it is interesting to note that the metrics which
we construct in Theorem 1.1 and 3.5 have small Cheeger and Cheeger-
Jammes constants, despite the eigenvalue σ being arbitrarily large.
2
1.1. Plan of the paper. In the next section, we present the vari-
ational characterization of the Steklov and mixed Steklov-Dirichlet
eigenvalue problems and deduce some simple consequences. In Section
3 we prove the main results of the paper by first working in cylinders
COMPACT MANIFOLDS WITH LARGE STEKLOV EIGENVALUES 5
and then using quasi-isometric control of eigenvalues to obtain Theo-
rem 1.1. We also prove Theorem 3.5 which provides an example where
two boundary components are arbitrarily close to each other.
2. Variational characterisation and quasi-isometric
control of eigenvalues
Let M be a smooth compact Riemannian manifold with boundary
Σ. Let (M) be the set of k-dimensional linear subspaces of C (M).
k ∞
H
It is well known that the Steklov eigenvalue σ is given by
k
df 2dv
σ (M,g) = min max M | |g g, (1)
k E∈Hk06=f∈E ´ Σ|f|2dvg
´
where dv is the volume form. It the following we will use conformal
g
metrics of the form g = h2g, where h is a smooth function on M such
′
that h = 1 identically on the boundary Σ. In the min-max characteri-
zation of σ (M,h2g), the denominator is the same as above, while the
k
numerator is
ˆ |df|2g′dvg′ = ˆ |df|2ghn−1dvg.
M M
In the following, we will often write df for df .
g
| | | |
We seize this opportunity to prove one of the simple statement from
the introduction.
Proposition 2.1. Let M be a compact smooth Riemannian manifold
of dimension n+ 1 at least 3, with boundary Σ. For each p Σ and
∈
each ǫ > 0, there exists a sequence of conformal deformations h2 g
m
such that h > 0 is a smooth function which is identically equal to 1
m
on Σ and on the complement of the ball B(p,ǫ) M, and such that
lim σ (M,h2 g) = 0 for each k N. ⊂
m→∞ k m ∈
Proof. Given ǫ > 0, let p Σ and consider a smooth function f
∈ ∈
C (M) which is supported in B(p,ǫ) M and which does not vanish
∞
⊂
at p. Let h be a sequence of positive smooth functions on M such
m
that h = 1 on Σ and lim h = 0 uniformly on compact subsets
m m m
→∞
of B(p,ǫ) interior(M). It follows that the conformal deformations
∩
g˜ = h2 g satisfy
m m
df 2dv df 2hn 1dv
lim M | |g˜ g˜m = M | |g m− g = 0.
m ´ f2dv ´ f2dv
→∞ Σ g˜m Σ g
´ ´
Using k functions f (j = 1, ,k) with disjoint support in B(p,ǫ)
j
···
instead of a single function f, the result now follows from the min-max
characterization of σ . (cid:3)
k
6 BRUNO COLBOIS,AHMAD EL SOUFI,AND ALEXANDREGIROUARD
We will also use the following mixed Steklov-Dirichlet problem on a
domain Ω M:
⊂
∆f = 0 in Ω, f = 0 on ∂Ω Σ, ∂ f = λf on ∂Ω Σ.
ν
\ ∩
Ithas discrete spectrum 0 < λD λD . Thek-theigenvalue
1 ≤ 2 ≤ ··· ր ∞
is given by
f 2dv
λD = min max Ω|∇ | g, (2)
k E∈Hk,006=f∈E ´∂Ω Σf2dvg
´ ∩
where = E : f = 0 on ∂Ω Σ 0 f E . For more
k,0 k
H { ∈ H \ ×{ } ∀ ∈ }
informations on mixed Steklov problems see for instance [2] and [1].
We are now ready for the proof of Proposition 1.2.
Proof of Proposition 1.2. Let (φ ) be a sequence of eigenfunctions cor-
k
responding to λD(B(p,ǫ)), which are extended by 0 elsewhere in M.
k
Using the subspace E = span(φ , ,φ ) in the min-max characteri-
k 1 k
zation of σ (M,g ) completes the p·r·o·of. (cid:3)
k ′
The following proposition is borrowed from [4]. It is classical and
follows directly from the min-max characterization of the eigenvalues.
We believe that this principle was used for the first time in [5], in the
context of the Laplace-Beltrami operator acting on differential forms.
Proposition 2.2. Let M be a compact manifold of dimension n, with
smooth boundary Σ and let g ,g be two Riemannian metrics on M
1 2
which are quasi-isometric with ratio A 1, which means that for each
≥
x M and 0 = v T M we have
x
∈ 6 ∈
1 g (x)(v,v)
1
A.
A ≤ g (x)(v,v) ≤
2
Then the Steklov eigenvalues with respect to g and g satisfies the
1 2
following inequality:
1 σ (M,g )
k 1 A2n+1.
A2n+1 ≤ σ (M,g ) ≤
k 2
Note also that if the metrics g and g are quasi-isometric with ratio
1 2
A 1,thengivenasmoothfunctionhonM, thetwoconformalmetrics
≥
h2g and h2g are also quasi-isometric with the same ratio A. This will
1 2
be useful in the proof of Theorem 1.1 when going from cylindrical
boundaries to arbitrary manifolds.
COMPACT MANIFOLDS WITH LARGE STEKLOV EIGENVALUES 7
3. Large Steklov eigenvalues on manifolds with fixed
boundary
Let M be a compact manifold of dimension (n + 1) with b 1
≥
boundary components: Σ = Σ ... Σ . We will prove Theorem 1.1
1 b
∪ ∪
by first working under the extra hypothesis that the boundary Σ of M
has a neighbourhood which is isometric to the product Σ [0,L) for
×
some L > 0. This is not a strong hypothesis since it is always satisfied
up to a quasi-isometry (See the proof of Theorem 1.1 below).
In the present context, we note g the restriction of the Riemann-
0
ian metric g to the boundary Σ, and d the corresponding exterior
0
derivative on C (Σ). The spectrum of the Laplace operator on Σ is
∞
denoted
0 = λ = = λ < λ + .
1 b b+1
··· ≤ ··· → ∞
Theorem1.1willfollowfromProposition3.1andProposition3.3below.
Proposition 3.1. Let (M,g) be a Riemannian manifold of dimension
n+1 3, with boundary Σ and assume that there exists a neighborhood
≥
V of Σ which is isometric to the product Σ [0,L) for some L > 0.
×
For every ε > 0 sufficiently small, there exists a Riemannian metric
g = h2g conformal to g which coincides with g in the complement of
ε ε
Σ (ε,4ε) and such that
×
A
σ (g ) ,
b+1 ε
≥ ε
where A = 1 min λ (Σ), 1 > 0.
4 { b+1 4}
Proof. For every positive ε < min L, 2 , define a Riemannian metric
{4 L}
g = h2g on M where h 1 is a smooth function which is identically
ε ε ε ≥
equal to 1 in the complement of Σ (ε,4ε) and, for (x,t) Σ [2ε,3ε],
× ∈ ×
h (x,t) = ε 2.
ε −
Let φk k N be an orthonormal basis of eigenfunctions of the Laplacian
{ } ∈
on Σ, with ∆φ = λ φ . Denote by Σ ,...,Σ the connected compo-
k k k 1 b
nents of Σ. One has λ = = λ = 0 and, for every j b, one
1 b
··· ≤
1
chooses φ = Σ on Σ and φ = 0 elsewhere.
j j −2 j j
| |
Let f be a smooth function on M with fdv = 0 and f2dv =
Σ g0 Σ g0
1. The restriction of f to Σ [0,L) is dev´eloped in Fourier´series:
×
f(x,t) = a (t)φ (x)
j j
j 1
X≥
with
1
aj(0) = |Σj|−2 ˆ fdvg0 , for j = 1,...,b
Σj
8 BRUNO COLBOIS,AHMAD EL SOUFI,AND ALEXANDREGIROUARD
and, since fdv = 0 and f2dv = 1
Σ g0 Σ g0
´ ´
1 1
a1(0) Σ1 2 + ab(0) Σb 2 = 0 (3)
| | ··· | |
a (0)2 = 1. (4)
j
j 1
X≥
Observethat b a (0)2 isthesquareoftheL2-normoftheorthogonal
j=1 j
projection of f on ker(∆) = span φ ,...,φ , in L2(Σ,g ). From
PΣ { 1 b} 0
(cid:12)
df(cid:12)(x,t) = a (t)φ (x)dt+a (t)d φ (x)
′j j j 0 j
j 1
X≥ (cid:0) (cid:1)
and d φ 2dv = λ , it follows that the Dirichlet energy of f on
Σ| 0 j| g0 j
(M,h´2g) is
ǫ
R (f) := df 2hn 1dv df 2hn 1dv
ε ˆ | | ε− g ≥ ˆ | | ε− g
M Σ (0,L)
×
L
a (t)2 +λ a (t)2 hn 1(t)dt. (5)
≥ ˆ ′j j j ε−
0
j 1
X≥ (cid:0) (cid:1)
Atthispoint, observethateitherthefunctiona decreasesquicklywhen
j
movingawayfromtheboundary(whichcostsenergyfromthefirstterm
in(5))oritremainsbigenough, andthesecondtermcontributesalarge
amount to the energy R (f). This is now explained more precisely.
ǫ
Fix an integer j 1. If a (t) 1 a (0) for all t [2ε,3ε), then
≥ | j | ≥ 2| j | ∈
L 3ε
λ a (t)2hn 1(t)dt λ a (t)2ε 2(n 1)dt
ˆ j j ε− ≥ j ˆ j − −
0 2ε
λ λ
jε 2n+3a (0)2 ja (0)2. (6)
− j j
≥ 4 ≥ 4ε
Otherwise, there exists t [2ε,3ε) with a (t ) 1 a (0) , which
0 ∈ | j 0 | ≤ 2| j |
implies a (0) a (t ) a (0) a (t ) 1 a (0) and using the
| j − j 0 | ≥ | j | − | j 0 | ≥ 2| j |
Cauchy-Schwarz inequality this leads to
L 2ε t0
a (t)2hn 1(t)dt a (t)2dt+ a (t)2ε 2(n 1)dt (7)
ˆ ′j ε− ≥ ˆ ′j ˆ ′j − −
0 0 2ε
1 2ε 2 ε 2(n 1) t0 2
− −
a (t)dt + a (t)dt
≥ 2ε ˆ ′j t 2ε ˆ ′j
(cid:18) 0 (cid:19) 0 − (cid:18) 2ε (cid:19)
COMPACT MANIFOLDS WITH LARGE STEKLOV EIGENVALUES 9
where ε−2(n−1) ε−2(n−1) = 1 1 . Hence (since x2 +y2 1(x+y)2)
t0 2ε ≥ ε ε ≥ 2ε ≥ 2
−
L 1 2ε 2 t0 2
a (t)2hn 1(t)dt a (t)dt + a (t)dt
ˆ ′j ε− ≥ 2ε ˆ ′j ˆ ′j
0 "(cid:18) 0 (cid:19) (cid:18) 2ε (cid:19) #
1 2ε t0 2
a (t)dt+ a (t)dt
≥ 4ε ˆ ′j ˆ ′j
(cid:18) 0 2ε (cid:19)
1 1
= (a (0) a (t ))2 a (0)2. (8)
j j 0 j
4ε − ≥ 16ε
For j b+1 one has λ λ and, combining (6) and (8) leads to
j b+1
≥ ≥
L λ 1 1 A
a (t)2 +λ a (t)2 hn 1(t)dt min b+1, a (0)2 = a (0)2.
ˆ ′j j j ε− ≥ { 4 16}ε j ε j
0
(cid:0) (cid:1)
Therefore, thanks to (5) and (4),
A A
R (f) a (0)2 = 1 a (0)2 . (9)
ε j j
≥ ε ε −
!
j b+1 j b
X≥ X≤
Now, every normalized function f which is orthogonal in L2(Σ) to
ker(∆) = span φ ,...,φ ,satisfies a (0)2 = 0and,thenR (f)
{ 1 b} j b j ε ≥
A. Using the max-min principle we ded≤uce that σ (M,g ) A. This
ε P b+1 ε ≥ ε
completes the proof of (i).
(cid:3)
InProposition3.3below, wewillprovethatitisalsopossibletomake
σ arbitrarily large using a conformal perturbation h2g. This is more
2
difficult than for σ . One of the difficulties comes from the fact that
b+1
theconformalperturbationwill needtobesupportedeverywhere inside
the manifold M. This follows from the following easy proposition.
Proposition 3.2. For every Riemannian metric g which coincides
′
with g on Σ and in the complement of Σ [0, L), one has
× 2
2
σ (g ) .
b ′
≤ L
Proof of Proposition 3.2. For every j b, let ψ be the function on M
j
≤
such that ψ is constant equal to zero in the complement of Σ [0,L)
j j
×
and, for (x,t) Σ [0,L),
j
∈ ×
Σ 1 in Σ [0, L],
ψ (x,t) = | j|−2 j × 2
j 2(1 t) Σ 1 in Σ [L,L].
(cid:26) − L | j|−2 × 2
For any Riemannian metric g which coincides with g on Σ and on the
′
complement of Σ×[0, L2], one has M |dψj|2g′dvg′ = L2 and Σψj2dvg′ =
´ ´
10 BRUNO COLBOIS,AHMAD EL SOUFI,AND ALEXANDREGIROUARD
1. Moreover, ψ , ,ψ are mutually orthogonal on the boundary.
1 b
Therefore, using th·e··min-max principle we deduce that σ (g ) 2. (cid:3)
b ′ ≤ L
Proposition 3.3. Let (M,g) be a Riemannian manifold of dimension
n+1 3 with boundary Σ and assume that there exists a neighbourhood
≥
of Σ which is isometric to the product Σ [0,L) for some L > 0.
×
For every ε > 0 sufficiently small, there exists a Riemannian metric
g = h2g conformal to g which coincides with g in the neighbourhood
ε ε
Σ [0,ε) of Σ and such that
×
C
σ (g )
2 ε
≥ ε
where C is an explicit constant which only depends on g.
The following Poincar´e type result will be useful.
Lemma 3.4. Let (M,g) be a compact manifold and denote by µ the
firstpositiveeigenvalueofthe Laplacianof(M,g) withNeumann bound-
ary condition if ∂M is nonempty. Let V and V be two disjoint mea-
1 2
surable subsets of M of positive volume. Every function f C (M)
∞
∈
satisfies
2
µ
df 2dv min( V , V ) f dv f dv .
ˆ | | g ≥ 2 | 1|g | 2|g g − g
M (cid:18) V1 V2 (cid:19)
where f dv := 1 f dv .
M g M g M g
ffl | | ´
Proof of Lemma 3.4. Denote by m = f dv the mean value of f on
M g
M. The functionf m is orthogonaltoffl constant functions onM which
−
implies
df 2dv µ (f m)2dv µ (f m)2dv +µ (f m)2dv .
ˆ | | g ≥ ˆ − g ≥ ˆ − g ˆ − g
M M V1 V2
Using the Cauchy-Schwarz inequality, we get for j = 1,2,
2 2
1
(f m)2dv (f m)dv = V f dv m
ˆ − g ≥ V ˆ − g | j| g −
Vj | j| Vj ! Vj !
and then (since x2 +y2 1(x y)2)
≥ 2 −
2
1
(f m)2dv + (f m)2dv min( V , V ) f dv f dv
ˆ − g ˆ − g ≥ 2 | 1| | 2| g − g
V1 V2 (cid:18) V1 V2 (cid:19)
(cid:3)
which ends the proof.