Equidimensional Isometric Extensions

0.25 MB·

by  Micha Wasem

EQUIDIMENSIONAL ISOMETRIC EXTENSIONS
MICHAWASEM
Abstract. Let Σ be a hypersurface in an n-dimensional Riemannian manifold M,
5 n>2. Westudytheisometricextensionproblemforisometricimmersionsf :Σ→Rn,
1 whereRnisequippedwiththeEuclideanstandardmetric. Usingaweakformofconvex
0 integration suggested by Sz´ekelyhidi, we construct “one-sided” isometric Lipschitz-
2 extensions andobtainanaccompanying densityresult.

r
a
M
1. Introduction
4
2 In this article we analyze the problem of extending a given smooth isometric immersion
f :Σ Rq of a hypersurface Σ (Mn,g) into Rq equipped with the standard Euclidean
G] metric→g = , intheequidime⊂nsionalcaseq =n,i.e.wewilllookforamapv :U Rn
0
h· ·i →
satisfying
D
h. (1.1) v∗g0 =g
t v =f,
a |Σ
m
where U M is a neighborhood of a point in Σ. For high codimension, Jacobowitz (see
⊂
[ [Jac74])establishedconditionsonΣandf suchthattheproblem(1.1)admitsananalytic

(q > n(n+ 1)/2) respectively smooth (q > n(n + 3)/2) solution. He also provided a
2
v curvature-obstruction to the existence of C2-solutions to (1.1). In [HW14] it is proven
8 that this obstruction is also an obstruction to C1-solutions and in the present paper we
9 willprovealongthesamelinesthatitisalsoanobstructiontoLipschitz-solutions(Propo-
9
sition 2.2). However, restricting the neighborhood U to one side of Σ only, one can hope
2
to construct one-sided solutions at low regularity, where curvature does not exist. It is
0
. possible to constructone-sided isometric C1-extensions that satisfy a C0-dense paramet-
1
ric h-principle in codimension greater than one i.e. if q >n+1 (see [HW14]).
0
5
1 This type of flexibility is not expected in the equidimensional case since the classical
: Liouville Theorem (see for example [Cia05, p. 30-31]) implies that any two images of
v
isometric C1-immersions into Euclidean space are congruent, so even if g is flat, there
i
X will not be C1-solutions to (1.1) in general. In this case, it seems natural to relax the
r regularity and consider piecewise C1-maps (think of folding a piece of paper). Indeed,
a
Dacorogna, Marcellini and Paolini ([DMP10], [DMP08]) constructed piecewise isometric
C1-immersions from a flat square into R2 mapping the boundary to a single point.
It is remarkable that for general metrics, there is a curvature obstruction even at low
regularity in the equidimensional case. We will show that there are no differentiable
isometric extensions if g is not flat (Proposition 2.1). Hence we will further relax the
regularityandfocus onLipschitz-mapsinstead. Problem(1.1) isreplacedby the problem
of finding a Lipschitz map v :U Rn that satisfies
→
v g =g a.e.
∗ 0
(1.2)
v =f,
Σ
|
Date:March25,2015.
1
where a.e. refers to the measure on M induced by g. This setting provides enough flexi-
bility to circumvent the curvature obstruction and the one given by Liouville’s Theorem.
In [MSˇ98] and [MSˇ03], Mu¨ller and Sˇver´ak constructed solutions to the following related
Dirichlet problem, where Ω is a bounded domain in Rn:
v g =id n-a.e. in Ω
∗ 0
(1.3) L
v =ϕ on ∂Ω,
where n denotes the n-dimensional Lebesgue measure. They require the boundary da-
L
tum ϕ to be a short map, whichis not the case for ourboundary datum f. As mentioned
before, maps satisfying (1.2) may collapse entire submanifolds to single points, hence
this definition does not reflect a truly geometric notion of isometry. A more natural no-
tion of isometry (which is equivalent if v C1) arises if one requires an isometry to be
∈
a map that preserves the length of every rectifiable curve (see [KEL14]). Our results do
notextendtothisstrongersetting. Seealso[Pet10]foranevenstrongernotionofisomery.
The construction of isometric maps in [KEL14] is based on a Baire-Category approach
and produces residuality results. We will use a weak variant of a “Nash-type” Convex
Integration iteration scheme, which only allows us to prove a density result.
The Nash-KuiperTheorem(see [Nas54,Kui55]) usesaniterationscheme thatstartswith
ashort map u:(Mn,g) Rq andaddsCorrugations (q =n+1)orSpirals (q >n+2)at
→
allscalestocorrectthemetricerrorsuccessivelywhilecontrollingtheC1-normoftheper-
turbed maps during the process. This leads to the convergencein C1. The basic building
block is a family of loops with averagezero in a suitable (q n)-dimensional sphere. The
−
method does not apply to the equidimensional case, since here, the sphere is degenerate
and consists of two isolated points. However,the points define an interval I R and one
⊂
may choose a loop in I that is concentrated on the two boundary points (see Lemmata
4.1 and 4.2 for precise statements). Using this loop we obtain a Corrugation function
that leads to weaker estimates than the corresponding ones in codimension one, but still
ensure the required Lipschitz-regularity of a solution to (1.2).
In local coordinates, the setting for the problem (1.2) can be reformulated as follows:
Consider an n-polytope (P,g) in Rn with an appropriate metric g such that the ori-
gin is contained in P˚ and let the isometric immersion f : B Rn be prescribed on
B :=P (Rn 1 0 ). →
−
∩ ×{ }
Definition 1.1
The sets P (Rn 1 R ) and P (Rn 1 R ) are called one-sided neighborhoods of
− >0 − 60
∩ × ∩ ×
B.
LetΩ¯ beaone-sidedneighborhoodofB. AmapubelongstoC (Ω¯,Rn)ifthereisafinite
p∞
simplicialdecompositionof Ω¯ into non-degeneraten-simplices such thatthe restrictionof
uontoeachofthesimplicesissmooth. Similarly,amapubelongstothespaceAff (Ω¯,Rn)
p
ifthereisa finite simplicialdecompositionofΩ¯ intonon-degeneraten-simplicessuchthat
the restriction of u onto each of the simplices is affine.
Definition 1.2
Let Ω¯ be a one-sided neighborhood of B. A map u C (Ω¯,Rn) C0(Ω¯,Rn) is called
∈ p∞ ∩
subsolution adapted to (f,g), if u =f and g u g >0 in the sense of quadratic forms
B ∗ 0
with equality on B only (i.e. g |u g is positiv−e definite on Ω¯ B and zero on B).
∗ 0
− \
We are now ready to state our main result:
2
Theorem 1.3
Let u : Ω¯ Rn be a subsolution adapted to (f,g). Then for every ε > 0, there exists a
Lipschitz m→ap v :Ω¯ →Rn satisfying v|B =f, v∗g0 =g Ln-a.e. and ku−vkC0(Ω¯) <ε.
We obtain a Corollary regarding isometric extensions of the standard inclusion ι : S1 ֒
D¯2 R2 to maps S2 D¯2, where D¯2 denotes the closed two-dimensional unit disk an→d
⊂ →
S1 is the equator of S2. Here µS2 denotes the standard measure on S2.
Corollary 1.4 (Isometric Collapse of S2)
ThereexistinfinitelymanyLipschitz-maps v :S2 D¯2 satisfyingv S1 =ιandv∗g0 =gS2
→ |
µS2-a.e.
This Corollary shows the strong interaction between codimension and regularity: The
standard inclusion S1 ֒ R2 0 ֒ R3 can be extended to an isometric immersion
v C1,α(S2,R3) in a u→nique ×wa{y }(up→to reflection across the R2 0 -plane) provided
∈ ×{ }
α > 2 (see [Bor58a, Bor58b, Bor59a, Bor59b, Bor60, CDS12]), but infinitely many iso-
3
metric extensionsv C1(S2,R3) exist(see [HW14]). We will alsoshowthat no isometric
extension v C1(S∈2,R2) can exist. More precisely, we will show that the maps v from
∈
Corollary1.4 cannot be locally C1 nor locally injective, hence the restrictionto Lipschitz
maps is not excessive.
Acknowledgements. The present article forms part of my doctoral thesis. I would like
to sincerely thank my advisor Norbert Hungerbu¨hler for suggesting the present problem
to me, for his encouragement, patience and constant support. Furthermore I would like
to thank La´szl´o Sz´ekelyhidi for helpful email communication.
2. Obstructions
We will first show that curvature is an obstruction against merely differentiable (not
necessarilyC1)isometricimmersionsintheequidimenisonalcasewhichleadstothechoice
of the Lipschitz-regularity in (1.2).
Theorem 2.1
An n-dimensional Riemannian manifold (Mn,g) can be locally isometrically embedded by
a differentiable map into (Rn,g ) if and only if g is flat and in this case, the map is in
0
fact of class C .
∞
Proof. The if part of the statement is a classical Theorem in differential geometry (see
for example Theorem 3.1 in [Tay06]). A local isometric immersion f is locally a distance
preserving homeomorphism by the differentiable Local Inversion Theorem (see [Ray02])
andtheconditionf g =g. SuchamapmustbealocalC -isometricdiffeomorphismby
∗ 0 ∞
Theorem2.1in[Tay06]since g issmoothbyassumption. Hencethe statementis asimple
consequenceofthe classicalfact thatthe Riemanniancurvaturetensoris preservedunder
C -isometries. (cid:3)
∞
Recall that Σ is a hypersurface of an n-dimensional Riemannian manifold (M,g) and
f :Σ Rn is an isometric immersion we seek to extend to a neighborhood U of a point
→
in Σ. We denote by A Γ(S2(T Σ) NΣ) the second fundamental form of Σ in M, and
∗
∈ ⊗
let h(X,Y) := g(ν,A(X,Y)), where ν Γ(NΣ) is the unique (up to sign) unit normal
vector field. Let further A¯ Γ(S2(T Σ∈) f NΣ¯) be the second fundamental form of
∗ ∗
Σ¯ := f(Σ) in Rn with assoc∈iated scalar f⊗undamental form h¯( , ) := A¯( , ),ν¯ , where
ν¯ Γ(f NΣ¯) is again a unit normal vector that is unique up to· s·ign. h · · i
∗
∈
3
Proposition 2.2 (Lipschitz-Obstruction)
Ifthereexistsaunitvectorv T Σsuchthat h(v,v) > h¯(v,v), noisometricLipschitz-
p g
extension u:U Rn can ex∈ist. | | | |
→
Proof. We argue by contradiction. Suppose u exists and let (for ε > 0 small enough)
γ :[0,ε) Σ U be a geodesic with γ(0)=p and γ˙(0)=v such that d (p,γ(t)) can be
M
→ ∩
realized by a minimizing geodesic σ :[0,1] U for all t.
→
Claim: For every δ > 0 there exists a curve c :[0,1] U joining p and γ(t) such that u
→
is differentiable in c(t) 1-a.e. in [0,1] and such that
L
1
c˙(t) dt<d (p,γ(t))+δ.
g M
| |
Z0
Proof of the claim: Let D U be the set where u is differentiable, Z :=[0,1] Bn 1(0)
⊂ × ρ−
and consider the map z : Z U,(s,x) exp (s(s 1)x), where x Bn 1(0)
→ 7→ σ(s) − ∈ ρ− ⊂
σ˙(s) T U. If ρ > 0 is small enough, the restriction of z onto (0,1) Bn 1(0) is a
⊥ ∈ σ(s) × ρ−
diffeomorphism onto its image, hence D :=z 1(D) has full n-measure in Z. We obtain
−
L
1
ZZ|∂sz|gdsdx=ZZ∩De|∂sze|gdsdx=ZBρn−1(0)Z0 χDe|∂sz|gdsdx
hence for n 1-a.e. x it holds that
−
L
1 1
χDe(s,x)|∂sz(s,x)|gds= |∂sz(s,x)|gds.
Z0 Z0
Since
1
lim ∂ z(s,x) ds=d (p,γ(t)),
s g M
x→0Z0 | |
we can choose an appropriate x such that c(t):=z(t,x) has the desired properties. This
finishes the proof of the claim. Let p¯=f(p) and γ¯ =f γ. Observe that
◦
1 d 1
u(σ(1)) u(σ(0)) 6 (u σ)(t) dt= σ˙(t) dt<d (p,γ(t))+δ.
g M
| − | dt ◦ | |
Z0 (cid:12) (cid:12) Z0
(cid:12) (cid:12)
WeconcludethatdRn(p¯,γ¯(t))6(cid:12)(cid:12)dM(p,γ(t)),(cid:12)(cid:12)sinceδ wasarbitrary. Sincewehavekg(p)=
( Mγ˙)(0)=( Σγ˙)(0)+A(v,v), we find (see [HW14, Lemma 2.2])
∇γ˙ ∇γ˙
h(v,v)2
d (p,γ(t))=t | |gt3+O(t4) for t 0.
M
− 24 →
This together with an analogous computation of the geodesic curvature of γ¯ in p¯gives
1
dRn(p¯,γ¯(t))−dM(p,γ(t))= 24 |h(v,v)|2g −|A¯(v,v)|2 t3+O(t4) for t→0
(cid:0) (cid:1)
contradicting dRn(p¯,γ¯(t))6dM(p,γ(t)). (cid:3)
3. Subsolutions
For the construction of subsolutions, we refer to a variant of Proposition 2.4 in [HW14]:
Proposition 3.1
Let f : Σ Rn be an isometric immersion. Suppose there exist unit normal fields
ν Γ(NΣ)→and ν¯ Γ(f NΣ¯) such that h( , ) h¯( , ) is positive definite, then around
∗
∈ ∈ · · − · ·
every p Σ, there exists a subsolution adapted to (f,g).
∈
4
3.1. Approximation of Subsolutions by Piecewise Affine Maps. We need to in-
troduce the notion of an adapted piecewise affine subsolution and therefore the following
approximation result (see [Sai79] for a proof):
Proposition 3.2
Let u C (Ω¯,Rn) C0(Ω¯,Rn). For every ε > 0 there exists a map v Aff (Ω¯,Rn)
∈ p∞ ∩ ∈ p ∩
C0(Ω¯,Rn) such that ku−vkC0(Ω¯)+k∇u−∇vkL∞(Ω¯) <ε.
Remark 3.3
Note that if an adapted subsolution u is approximated by a piecewise affine map v, we
cannot ensure that u =v since this would require u to be piecewise affine already.
B B B
Tocircumventthispr|oblem,|letΩ¯ := x Ω¯,dist(x,B)6| ℓ anddefineη C (Ω¯,[0,1])
ℓ ℓ ∞
{ ∈ } ∈
to be
1, if x Ω¯
ℓ
η (x):= ∈
ℓ (0, if x∈Ωcℓ/2 := x∈Ω¯,dist(x,B)> 2ℓ
and 0<η <1 elsewhere.
ℓ (cid:8) (cid:9)
Ωc
ℓ
ℓ
Ω¯
ℓ
Definition 3.4
For fixed ℓ > 0, decompose Ω¯ and the closure of its complement in Ω¯ separately into
ℓ/2
non-degenerate simplices, approximate u by v in the sense of Proposition 3.2 and finally
replace v by w := u+η (v u). This map has the property that w u on Ω¯ and
ℓ/2 ℓ/4
− ≡
the restriction of w to Ωc is a piecewise affine map. If such a map has in addition the
ℓ/2
property that g wT w >0 n-a.e., but wT w =g on B, we will callit a piecewise
−∇ ∇ L ∇ ∇
affine subsolution adapted to (f,g).
The estimates
kw−ukC0(Ω¯) 6ku−vkC0(Ω¯)
k∇w−∇ukL∞(Ω¯) 6k∇ηℓ/2kC0(Ω¯)ku−vkC0(Ω¯)+k∇u−∇vkL∞(Ω¯)
ensure that we can approximate an adapted subsolution and its first derivatives by an
adapted piecewise affine subsolution.
4. Convex Integration
Westartwithapiecewiseaffinesubsolutionu:Ω¯ Rnadaptedto(f,g),thatispiecewise
→
affineonΩc anddecomposethemetricdefectintoasumofprimitivemetrics(see[GA13,
ℓ/2
p. 202, Lemma 1] for an explanation) as
m
(g u g ) = a2(x)ν ν ,
− ∗ 0 x k k⊗ k
k=1
X
where the a2 are nonnegative on Ω¯ B, zero on B, belong to C (Ω¯,Rn) and extend
k \ p∞
continuously to B. The sum is locally finite with at most m 6 m terms being nonzero
0
forafixedpointx. Theν Sn 1 arefixedunitvectors. Weintendtocorrectthismetric
k −
∈
defect by successively adding primitive metrics, i.e. metric terms of the form a2ν ν.
⊗
5
Adding a primitive metric is done in a step. A stage consists then of m steps, where the
numberm N isfinite butmaychangefromstageto stage. Fix orthonormalcoordinates
∈
inthe targetsothatthe the metric u g canbe writtenas uT u, where u=(∂ ui) .
∗ 0 j ij
For a specific unit vector ν Sn 1 and a nonnegative fu∇nctio∇n a C (∇Ω¯) we aim at
− ∞
finding v : Ω¯ Rq satisfyin∈g vT v uT u+a2ν ν. Nash∈solved this problem
→ ∇ ∇ ≈ ∇ ∇ ⊗
using the Nash Twist, i.e. an ansatz of the form
1
(4.1) v(x)=u(x)+ N (a(x)λ x,ν )β (x)+N (a(x),λ x,ν )β (x) ,
1 1 2 2
λ h i h i
(cid:20) (cid:21)
where N(s,t) := s( sint,cost) satisfies the circle equation ∂ N2+∂ N2 = s2 and β are
− t 1 t 2 i
mutuallyorthogonalunitnormalfields,requiringthuscodimensionatleasttwo(q >n+2).
The improvement to codimension one was first achieved by Kuiper [Kui55] with the use
of a different ansatz (Strain). We will present the Corrugation introduced by Conti, de
Lellis and Sz´ekelyhidi [CDS12] since it will be illustrative for the codimension zero case.
Define
ξ := u uT u −1 ν and ζ :=⋆(∂1u ∂2u ... ∂nu),
∇ · ∇ ∇ · ∧ ∧ ∧
where ⋆ denotes the Hodge star with respect to the usual metric and orientationin Rn+1
(cid:0) (cid:1)
and let e e
ξ ζ
ξ := , ζ := .
ξ 2 ζ ξ
|e| | |e| |
The Corrugation then has the form
e e e
1
(4.2) v(x)=u(x)+ Γ (a(x)ξ(x),λ x,ν )ξ(x)+Γ (a(x)ξ(x),λ x,ν )ζ(x) ,
1 2
λ | | h i | | h i
(cid:20) (cid:21)
whereΓ C∞(R S1,R2),(s,t) Γe(s,t)isafamilyofloopssatisefyingthecircleequation
∈ × 7→
(∂ Γ +1)2+∂ Γ2 =1+s2. Observe that both, N and Γ satisfy
t 1 t 2
(4.3) ∂ Ndt=0 and ∂ Γdt=0.
t t
IS1 IS1
The key point is that the first derivatives of the main building block satisfy an average
condition like (4.3) (periodicity of N and Γ in the second variable) and an appropriate
circle equation. In both cases, the crucial ingredients to control the C1-norm during the
iteration are the following C1-estimates: The maps N and Γ satisfy (see [HW14] for a
reference on the second estimate)
∂ N(s,t) = s
t
(4.4) | | | |
∂ Γ(s,t) 6√2s
t
| | | |
forall(t,s) R S1. Inthe equidimensionalcase,wherethere isno normalvectoratall,
∈ ×
let ξ := u T ν and ξ :=ξ ξ 2. A similar ansatz like (4.1) or (4.2) is
− −
∇ · | |
1
(4.5) v(x)=u(x)+ L(a(x)ξ(x),λ x,ν )ξ,
e ee
λ | | h i
where L:R S1 R2,(s,t) L(s,t) is a smoothfamily of loopsstill to be constructed.
e
× → 7→
Direct computations show that v = u + ∂ Lξ ν + O(λ 1) and therefore by the
t −
∇ ∇ ⊗
definition of ξ:
1
vT v = uT u+ (2∂ L+∂ L2)ν ν+O(λ 1).
t t −
∇ ∇ ∇ ∇ ξ 2 ⊗
| |
In order to obtain vT v = uT u+a2ν ν+O(λ 1), ∂ L needs to satisfy the circle
− t
∇ ∇ ∇ ∇ e ⊗
equation (1+∂ L)2 = 1+ ξ 2a2 =: 1+s2. Since the circle here is zero-dimensional, it
t
| |
consistsoftwoisolatedpointsandthereisnosmoothmap∂ Ltothatcirclebeingzeroin
t
average. We will circumventethis problem by replacing the equality in the circle equation
by a pointwise and an average inequality (see Lemma 4.1 and Figure 4). Note that the
6
definition of ξ in (4.5) requires u to be immersive. If u does not have full rank, we
∇
will choose ξ to be a unit vector field in ker uT which will lead to the circle equation
∇
∂ L2 =a2 (see Lemma 4.2).
t (cid:0) (cid:1)
Lemma 4.1 (Regular Corrugation)
For everyε>0and c>0thereexistsamapL C ([0,c] S1),(s,t) L(s,t)satisfying
∞
∈ × 7→
the following conditions:
(4.6) (1+∂ L)2 61+s2
t
1
(4.7) s2 ∂ L2 dt<ε
t
2π IS1 −
(cid:0) 1 (cid:1)
(4.8) ∂ Ldt=0
t
2π IS1
Proof. Consider the 2π-periodic extension of the function pˆ : [0,2π] [ 1,1], where
s
→ −
s [0,c]:
∈
1, x π 1+ 1 ,π 3 1
pˆ (t):= − ∈ 2 √1+s2 2 − √1+s2
s
( 1 else.h (cid:16) (cid:17) (cid:16) (cid:17)i
For 0 < ε < 1 fixed, choose 0 < δ < επ and let ϕ denote the usual symmetric
2 2(1+c2) δ
standardmollifier. The functionp:R S1 R,(s,t) ϕ pˆ (t)is smoothandadirect
δ s
× → 7→ ∗
computation using Fubini’s Theorem implies
1 1
p(s,t)dt= .
2π IS1 √1+s2
Since p2(s, ) equals one on a domain of measure at least 2π 4δ on each period we get
· −
1 2δ ε
p2(s,t)dt>1 >1 .
2π IS1 − π − 1+s2
The function L:R S1 R
× →
t
L(s,t):= 1+s2 p(s,u) 1 du
· −
Z0 (cid:16)p (cid:17)
then has all the desired properties. (cid:3)
Lemma 4.2 (Singular Corrugation)
For everyε>0and c>0thereexistsamapL C ([0,c] S1),(s,t) L(s,t)satisfying
∞
∈ × 7→
the following conditions:
(4.9) e ∂ L2 6s2 e
t
1
(4.10) s2 ∂tL2 edt<ε
2π IS1(cid:16) − (cid:17)
1
(4.11) e∂ Ldt=0
t
2π IS1
Proof. Consider the convolution of the 2π-periodiec extension of the map
s, t [π,3π]
[0,2π] t − ∈ 2 2
∋ 7→( s else
with ϕ , where δ < επ. This convolution gives rise to a map p(s,t). The map
δ 2c2
t
L(s,t):= p(s,u)du
Z0
then has all the desired propertiese. (cid:3)
7
S2 (0) S1 ( 1)
r1 r2 −
S0 (0) S0 ( 1)
r1 r2 −
Figure 1. From left to right and from top to bottom the picture illus-
trates the Corrugation map used by Nash, the one used by Conti, de
Lellis and Sz´ekelyhidi, and the maps L and L from the Lipschitz case.
The radii of the spheres are given by r =s and r =√1+s2.
1 2
e
The conditions (4.7) and (4.10) oppose a C1-estimate for L and L analogous to (4.4),
hencetheC1-normsofthemapscannotbecontrolledduringtheiteration,butwewilluse
asuitableL2-estimateduetoSz´ekelyhidiusinganidentity relatingtehe firstderivativesto
the trace of the metric defect and integration by parts later on (see Proposition 5.2).
5. Iteration
Let ℓ > 0 and let u be a piecewise affine subsolution adapted to (f,g) in the sense of
Definition 3.4 and decompose the metric defect as
m
g−u∗g0 = a2kνk⊗νk.
k=1
X
Here the a are nonnegative functions that are piecewise constant on Ωc . Since u is
k ℓ/2
already isometric on B, we will add a “cut-off” error η2(g u g ) using the building
ℓ − ∗ 0
blocks L and L.
5.1. k-th Step. Let u be a piecewise affine subsolution adapted to (f,g) s.t. u is
piecewise affinee on Ωc k.−1We introduce a map Θ C (Ω¯,[0,1]) in the k-th step tkh−a1t is
ℓ/2 k ∈ ∞
associated to u as follows: Let S be the simplicial decomposition of Ω¯ according
k 1 i i
− { }
to u and let U be an open neighborhood of
k 1 k
−
K :=Ω¯ S˚
k i
\
i
[
andsetΘ =1onUc andΘ =0onK . ObservethatU canbechosentohavearbitrary
k k k k k
small Lebesgue-measure. Let 0 < δ < 1 (the exact value will be determined later). We
will now discuss the step for a simplex S Ωc . The restrictionof u to S is an affine
⊂ ℓ/2 k−1
function. If u is regular on S, let
k 1
∇ −
ξ := u ( uT u ) 1 ν
k ∇ k−1 ∇ k−1∇ k−1 − · k
ξek :=ξk|ξk|−2
s :=(1 δ)1/2Θ η a ξ
k e −e k ℓ k| k|
L (x):=L(s ,λ x,ν ).
k k k k
h i e
8
and define uk :=uk−1+ λ1kLkξk. We have uk ∈C∞(S,Rn) and kuk−1−ukkC0(S) can be
made arbitrarilysmallprovidedthe free parameterλ is largeenough. For the Euclidean
k
metric pulled back by u , we find
k
1
∇uTk∇uk =∇uTk−1∇uk−1+ ξk 2 2∂tLk+∂tL2k νk⊗νk+O λ−k1 .
| |
(cid:0) (cid:1) (cid:0) (cid:1)
If u is singular, choose ξ ker( uT ) to be a unit vector, let
∇ k−1 k ∈ ∇ek−1
s :=(1 δ)1/2Θ η a
k k ℓ k
−
L (x):=L(s ,λ x,ν )
k k k k
h i
and define u := u + 1 L ξ . As in the regular case, u C (S,Rn) and u
k k−1 λk k ke e k ∈ ∞ k k−1−
uk C0(S) can be made arbitrarily small. For the Euclidean metric pulled back by uk we
k
find e
∇uTk∇uk =∇uTk−1∇uk−1+∂tL2kνk⊗νk+O λ−k1 .
(cid:0) (cid:1)
5.2. Stage. We approximate the resulting map eafter each step by an adapted piecewise
affine subsolution. This introduces a further subdivision of the simplicial decomposition
ofΩ¯ aftereachstep. Ineachstepwewillleavethemapfromtheforegoingstepunchanged
nearB (due to the cut-offby η ) andnearthe (n 1)-skeletonofthe simplicialdecompo-
ℓ
−
sition (thanks to Θ ). This procedure does not allow for a pointwise control of the new
k
metric error, but we are still able to bound it in an integral sense.
Proposition 5.1 (Stage)
Letu beasubsolution adapted to(f,g). Then for any ε>0, thereexistsa piecewise affine
subsolution u adapted to (f,g) satisfying
(5.1) e ku−ukC0(Ω¯) <ε
(5.2) tr g uT u dx<ε
ZΩ¯ −∇ ∇e
(cid:0) (cid:1)
Proof. Since tr(g uT u)dx convergees toezero, as ℓ 0, there exists ℓ > 0 small
Ω¯ℓ −∇ ∇ →
enough such that after approximating u by a piecewise affine subsolution u adapted to
R 0
(f,g) that is piecewise affine on Ωc we get
ℓ/2
e
ε
(5.3) tr(g uT u )dx< .
ZΩ¯ℓ −∇ 0∇ 0 7
Choose (for ℓ now fixed) δ such that e e
(5.4) δid<(g−u∗0g0)|Ωcℓ/2 Ln-a.e.
1
ε −
(5.5) δ < e tr(g uT u )dx .
7 ZΩcℓ −∇ 0∇ 0 !
We use the step iteratively to produce a sequeenceeof maps starting with u . After a
0
step, say the k-th, we approximate the resulting map u by an adapted piecewise affine
k
subsolution u that is piecewise affine on Ωc and leave it unchanged on Ω¯ . After
k ℓ/2 eℓ/2
m steps, we set u := u . Choosing the free parameter λ sufficiently large in each step
m k
together withe suitable approximations by piecewise affine subsolutions proves (5.1). On
each simplex (ineΩcℓ/2e) from to the simplicial decomposition corresponding to the map
u we performed m steps and since after each step, the resulting map was affine on that
simplex, we used (depending on whether u was regular or singular)the “regular”or
k 1
∇ −
tehe “singular” k-th step. This splits the set 1,...,m into R and S corresponding to
{ }
e9
indices k where u was regular and singular respectively. A direct computation on a
k 1
∇ −
fixed simplex shows
g ueT u =g uT u + uT u uT u
−∇ m∇ m −∇ 0∇ 0 ∇ 0∇ 0−∇ m∇ m
m
e e = ae2kνk⊗eνk+∇euTk−e1∇uk−e1−∇euTk∇uk +A
k=1
(5.6) X(cid:0) (cid:1)
= a2k−|ξk|−2 2e∂tLk+e∂tL2k νk⊗νk+O(λ−k1)
kX∈R(cid:16)(cid:16) (cid:0) (cid:1)(cid:17) (cid:17)
+ a2k−∂teL2k νk⊗νk+O(λ−k1) +A,
Xk∈S(cid:16)(cid:16) (cid:17) (cid:17)
where A := mk=1 ∇uTk∇uk−∇uTk∇uk esatisfies kAkC0(Ω¯) < εˆand εˆ> 0 will be fixed
later(this ispossibleforeveryεˆbythe useofsuitableapproximations). Inorderto prove
P (cid:0) (cid:1)
that u is a piecewise affine subsoelutioen adapted to (f,g), first observe that every uk is
piecewise smooth and continuous. This follows from the infinite differentiability in the
interior of every simplex and the fact that u agrees with u on K . Now we prove
e k k 1 k
shortness on Ωc : We use the computation (5.6) and the point−wise estimates (4.6), (4.9)
ℓ/2
and (5.4) to obtain n-a.e.
L
g−∇uTm∇um = a2k−|ξk|−2 2∂tLk+∂tL2k νk⊗νk+O(λ−k1)
kX∈R(cid:16)(cid:16) (cid:0) (cid:1)(cid:17) (cid:17)
e e + a2k−∂teL2k νk⊗νk+O(λ−k1) +A
Xk∈S(cid:16)(cid:16) (cid:17) (cid:17)
m e
> 1−(1−δ)Θ2kηℓ2 a2kνk⊗νk+O(λ−k1)+A
k=1
X(cid:0) (cid:1)
m
>δ(g−u∗0g0)+ O(λ−k1)−εˆid
k=1
X
m
e
>δ2id+ O(λ−k1)−εˆid>0
k=1
X
providedεˆissmallenoughandthefrequenciesλ arelargeenough. Notethatthepullback
k
oftheEuclideanmetricisnotdefinedonK . Inordertoprove(5.2),weusethefollowing
m
estimates on a simplex S in the regular and singular case respectively:
ε
(5.7) s2k−|ξk|−2(2∂tLk+∂tL2k) dx< 7mvolΩc
ZS(cid:16) (cid:17) ℓ/2
e ε
(5.8) s2 ∂ L2 dx<
k− t k 7mvolΩc
ZS(cid:16) (cid:17) ℓ/2
These estimates are direct consequences of (4.7e) and (4.10) and the fact that for every
f C0(Ω¯ S1)
∈ ×
1
λ
f(x,λ x,ν )dx →∞ f(x,t)dtdx.
ZΩ¯ h i −→ ZΩ¯ 2π IS1
This is the content of Proposition A and will be proved in the appendix. Let
N
tr g uT u dx= tr g uT u dx+ tr g uT u dx,
ZΩ¯ −∇ m∇ m ZΩ¯ℓ/2 −∇ m∇ m i=1ZSi −∇ m∇ m
(cid:0) (cid:1) (cid:0) (cid:1) X (cid:0) (cid:1)
e e =:Ie1 e =:I2e e
whereN isthetotalnumbero|fsimplicesin{zthesimplic}iald|ecomposition{ozfΩc accord}ing
ℓ/2
tothemapu . Sinceu andu agreeonΩ¯ ,weuse(5.3)toobtainI 6 ε. Weuse(5.6),
m m 0 ℓ/2 1 7
10
e e e