On second variation of Wang-Yau quasi-local energy

0.34 MB·English

by  Pengzi Miao

ON SECOND VARIATION OF WANG-YAU
QUASI-LOCAL ENERGY
3
1 PENGZI MIAO AND LUEN-FAI TAM1
0
2
Abstract. We study a functional on the boundary of a com-
n
pact Riemannian 3-manifold of nonnegative scalar curvature. The
a
J functional arises as the second variation of the Wang-Yau quasi-
0 local energy in general relativity. We prove that the functional
2 is positive definite on large coordinate spheres, and more general
on nearly round surfaces including large constant mean curvature

]
spheres in asymptotically flat 3-manifolds with positive mass; it is
G
also positive definite on small geodesics spheres, whose centers do
D
not have vanishing curvature, in Riemannian 3-manifolds of non-
.
h negative scalar curvature. We also give examples of functions H,
t whichcanbe madearbitrarilycloseto 2,onthe standard2-sphere
a
m (S2,σ0) such that the triple (S2,σ0,H) has positive Brown-York
mass while the associated functional is negative somewhere.
[

1
v
6
5
6 1. Introduction
4
. In [10, 11], Wang and Yau introduced a new quasilocal mass. Briefly
1
speaking, its definition is as follows. Let Σ be a closed 2-surface, in
0
3 a spacetime N satisfying the dominant energy condition, such that
1
Σ bounds a compact, spacelike hypersurface Ω. Denote the induced
:
v
Riemannian metric on Σ by γ. Given a function τ on Σ such that
i
X γˆ = γ+dτ⊗dτ isametric ofpositiveGaussiancurvature, oneconsiders
r the isometric embedding
a
X : (Σ,γ) ֒→ R3,1
where X = (Xˆ,τ) and Xˆ = (Xˆ ,Xˆ ,Xˆ ) is an isometric embedding of
1 2 3
(Σ,γˆ) in R3 = {(x,0) ∈ R3,1}. Associated with each such a function τ
or equivalently each such an isometric embedding X, Wang and Yau
introduced a quantity, which we denote by E (Σ,τ), called the quasi-
WY
local energy of Σ in N with respect to τ. The Wang-Yau quasi-local
2010 Mathematics Subject Classification. Primary 53C20; Secondary 83C99 .
1ResearchpartiallysupportedbyHongKongRGCGeneralResearchFund#CUHK
403011.
1
2 Pengzi Miao and Luen-FaiTam
mass of Σ in N is then defined by
(1.1) m (Σ) = infE (Σ,τ)
WY τ WY
where the infimum is taken over all admissible functions τ (see [11] for
an exact formula of E (Σ,τ) and the definition of admissibility). It
WY
was proved in [11] that m (Σ) ≥ 0 and m (Σ) = 0 if the embedding
WY WY
Σ ֒→ N is isometric to R3,1 along Σ.
When Σ bounds a time-symmetric Ω and γ has positive Gaussian
curvature, there is a well-known Brown-York quasi-local mass of Σ
([1, 2]) given by
1
(1.2) m (Σ,Ω) = (H −H) dv
BY 8π 0 γ
ZΣ
where H is the mean curvature of the isometric embedding of (Σ,γ)
0
in R3 and H is the mean curvature of Σ in Ω. In this situation, one has
m (Σ,Ω) = E (Σ,τ ), where τ = 0 is an admissible function and
BY WY 0 0
is also a critical point of E (Σ,·) ([11]). The variational definition of
WY
m (Σ) suggests m (Σ) ≤ m (Σ,Ω). A natural question is whether
WY WY BY
m (Σ) = m (Σ,Ω). (Results regarding the global minimization of
WY BY
E (Σ,·) recently have been announced in [3].)
WY
Inthis paper, we consider thelocalminimality of m (Σ,Ω). Amain
BY
corollary of our result for surfaces in an asymptotically flat manifold
is:
Theorem 1.1. Let (M,g) be an asymptotically flat 3-manifold. Let
S = {x ∈ M | |z| = r} be a coordinate sphere in an admissible
r
coordinate chart {z } on a given end. Suppose the ADM mass of the
i
end is positive. For sufficiently large r, the Brown-York mass of S is
r
a strict local minimum of E (S ,·).
WY r
We will prove Theorem 1.1 by proving Theorem 4.1 in Section 4 for
a larger class of “large surfaces”, namely nearly round surfaces near
the infinity which were introduced in [9]. As mentioned in [9], besides
large coordinate spheres, notable examples of nearly round surfaces in
an asymptotically flat 3-manifold include the constant mean curvature
surfaces constructed in [5] and [12].
For small geodesic spheres in a manifold of nonnegative scalar cur-
vature, we have:
Theorem 1.2. Let (M,g) be a Riemannian 3-manifold of nonnegative
scalar curvature. Let p ∈ M be a point. For r > 0, let S be the
r
geodesic sphere of radius r centered at p and B be the corresponding
r
On second variation of Wang-Yauquasi-local energy 3
geodesic ball. If
(1.3) limr−5m (S ,B ) > 0,
r→0 BY r r
then m (S ,B ) is a strict local minimum of E (S ,τ) for r > 0
BY r r WY r
sufficiently small.
We note that condition (1.3) in Theorem 1.2 is equivalent to
(i) R(p) > 0, or
(ii) R(p) = 0 and |Ric(p)|2 > 0, or
(iii) R(p) = 0, |Ric(p)| = 0, and (∆R)(p) > 0
which follows from the asymptotic expansion of m (S ,B ) in [4] and
BY r r
the assumption R ≥ 0. Here R, Ric denote the scalar curvature, the
Ricci curvature of g.
For m (Σ,Ω) = E (Σ,τ ) to locally minimizes E (Σ,·), the
BY WY 0 WY
second variation of E (Σ,·) at τ is necessarily nonnegative. We
WY 0
recall the following result from [6].
Theorem 1.3 ([6]). Suppose m (Σ,Ω) is defined for a 2-surface Σ
BY
bounding a time-symmetric hypersurface Ω in a spacetime N. The
second variation of E (Σ,·) at τ = 0 (up to multiplication by 1 ) is
WY 0 8π
(∆η)2
(1.4) F (η) =: +(H −H)|∇η|2−II (∇η,∇η) dv
γ,H 0 0 γ
H
ZΣ(cid:20) (cid:21)
where II is the second fundamental form of (Σ,γ) when it is isomet-
0
rically embedded in R3. If there exists a constant β > 0 such that
(1.5) F (η) ≥ β (∆η)2dv , ∀ η ∈ W2,2(Σ),
γ,H γ
ZΣ
then m (Σ,Ω) is a strict local minimum of E (Σ,·).
BY WY
Therefore, to obtain the local minimality of m (Σ,Ω), it suffices
BY
to study the functional F (η). The induced metric and the mean
γ,H
curvature function on the surfaces {S } in Theorems 1.1 and 1.2 (after
r
rescaling) are close to the standard metric σ on the unit sphere S2
0
and the constant 2 respectively. Thus, one may ask whether F (η)
γ,H
satisfies (1.5) if the pair (γ,H) is sufficiently close to (σ ,2).
0
Our first task in this paper is to derive some sufficient conditions
on such a pair (γ,H) so that (1.5) is true. Applying these sufficient
conditions, we can prove Theorem 4.1, and part of Theorem 1.2 which
corresponds to cases (i) and (iii) above.
The other part of Theorem 1.2, which corresponds to case (ii), turns
out to be more subtle. We will prove it using more refined estimation
on (S2,σ ) (see Theorem 5.2 and Proposition 5.1). Motivated by our
0
4 Pengzi Miao and Luen-FaiTam
proof of this part of Theorem 1.2, we also construct examples to show
that on (S2,σ ), there are functions H which can be arbitrarily close
0
to 2, but F (η) < 0 for some η.
σ0,H
We remark that the general validity of (1.5) is of significance in
the study of boundary behaviors of compact manifolds of nonnegative
scalar curvature. If (1.5) is always true, it will impose a necessary con-
dition for a positive function H on Σ to arise as the mean curvature of
Σ in some compact Riemannian 3-manifold of nonnegative scalar cur-
vature, bounded by (Σ,γ). So far, a major known necessary condition
is (H − H)dv ≥ 0 by the result of [8]. It is worth to note that
Σ 0 γ
our examples of H above, with F (η) < 0 for some η, also satis-
R σ0,H
fies (2−H)dv > 0. Thus, if the Brown-York mass always locally
S2 σ0
minimizes the Wang-Yau quasi-local energy in the time-symmetric sit-
R
uation, then (1.5) will constitute a new necessary condition.
This paper is organized as follows. In Section 2, we collect some
lemmas which are to be used frequently in later sections. In Section 3,
we obtain sufficient conditions for (1.5) to hold. In Section 4, we apply
the derived sufficient conditions to prove Theorem 4.1 which implies
Theorem 1.1. In Section 5, we establish the positivity of F on small
γ,H
geodesic spheres in Theorem 5.1 which implies Theorem 1.2. There
whether the scalar curvature vanishes at the center of a geodesic sphere
makesanimportantdifferenceintheproof. Amainresultrelatedtothe
case of vanishing scalar curvature at the sphere center is Theorem 5.2,
which we prove using a functional inequality on the standard sphere
(S2,σ ) (Proposition 5.1) which may have independent interest. In
0
Section 6, we give examples of H on the standard unit sphere so that
F (η) < 0 for some η while (2 −H)dv > 0. In the Appendix,
σ0,H S2 σ0
we list some elementary computational results, which are needed in
R
Section 5.
2. Preliminaries
Throughout this paper, Σ always denotes a closed 2-surface that is
diffeomorphic to a 2-sphere. Given a metric γ of positive Gaussian
curvature and a positive function H on Σ, we let
(∆η)2
(2.1) F (η) = +(H −H)|∇η|2−II (∇η,∇η) dv
γ,H 0 0 γ
H
ZΣ(cid:20) (cid:21)
for any η ∈ W2,2(Σ). Here ∆ and ∇ denote the Laplacian and the
gradient on (Σ,γ), H and II are the mean curvature and the second
0 0
fundamental form of (Σ,γ) when it is isometrically embedded in R3,
and dv is the volume form on (Σ,γ). We also denote the symmetric
γ
On second variation of Wang-Yauquasi-local energy 5
bilinear form associated to F by Q . Namely
γ,H γ,H
(2.2)
∆η ·∆η
Q (η ,η ) = 1 2 +(H −H)h∇η ,∇η i−II (∇η ,∇η ) dv .
γ,H 1 2 0 1 1 0 1 2 γ
H
ZΣ(cid:20) (cid:21)
All metrics on Σ below will be assumed to be smooth for simplicity.
We recall some basic results from [6].
Lemma 2.1. Let γ be a metric of positive Gaussian curvature on Σ.
Let X = (X ,X ,X ) : (Σ,γ) → R3 be an isometric embedding of
1 2 3
(Σ,γ) in R3. The functional
(∆η)2
F (η) = −II (∇η,∇η) dv
γ,H0 H 0 γ
ZΣ(cid:20) 0 (cid:21)
satisfies:
(i) F (η) ≥ 0, ∀ η ∈ W2,2(Σ) .
γ,H0
(ii) F (η) = 0 if and only if η ∈ L(γ), where
γ,H0
3
L(γ) = a + a Xi | a ,a ,a ,a are arbitrary constants .
0 i 0 1 2 3
( )
i=1
X
(iii) If η ∈ L(γ), then Q (η,φ) = 0, ∀ φ ∈ W2,2(Σ).
γ,H0
Remark 2.1. (i) and (ii) are proved in [6, Corollary 3.1]. (iii) is a direct
consequence of (i) and (ii) by considering the first variation of F .
γ,H0
Remark 2.2. Since any two isometric embeddings of (Σ,γ) differ by a
rigid motion in R3, the space L(γ) defined above is independent on the
choice of X.
Lemma 2.1 shows F (·) vanishes on L(γ) when H = H . For an
γ,H 0
arbitrary H, we have the following from (3.12) in [6, Proposition 3.2].
Lemma 2.2. Suppose H > 0. For any η = a + 3 a Xi ∈ L(γ),
0 i=1 i
(H −H)2
F (η) = |a|2 (H −H)dv + ha,ν i2P0 dv ,
γ,H 0 γ 0 γ
H
ZΣ ZΣ
where a = (a ,a ,a ) and ν is the unit outward normal to (Σ,γ) when
1 2 3 0
it is isometrically embedded in R3.
If (H −H)dv > 0, then Lemma 2.2 implies
Σ 0 γ
R F (η) ≥ β (∆η)2dv
γ,H γ
ZΣ
for some β > 0 for all η ∈ L(γ).
Next we estimate F (η) for η that are γ-L2 orthogonal to L(γ).
γ,H0
6 Pengzi Miao and Luen-FaiTam
Lemma 2.3. Let σ be a metric of positive Gaussian curvature on Σ.
There exist positive constants δ and β such that if γ is a metric on Σ
satisfying ||γ −σ|| < δ, then
C2,α(Σ,σ)
F (η) ≥ β (∆ η)2 dv
γ,H0 γ γ
ZΣ
for all η ∈ W2,2(Σ) that is γ-L2 orthogonal to L(γ). Here ∆ denotes
γ
the Laplacian on (Σ,γ).
Proof. We argue by contradiction. Suppose it is not true, then there
exists a sequence of metrics {γ } on Σ and a sequence of functions
k
{η } ⊂ W2,2(Σ) such that
k
(2.3) lim ||γ −σ|| = 0,
k C2,α(Σ,σ)
k→∞
(2.4) η φ dv = 0, ∀φ ∈ L(γ ), k = 1,2,3,...,
k γk k
ZΣ
and
1
(2.5) F (η ) ≤ (∆ η )2 dv .
γk,H0k k k k k γk
ZΣ
HereHk istheH associatedtoγ and∆ standsfor∆ . Wenormalize
0 0 k k γk
η such that
k
(2.6) η2 dv = 1.
k γk
ZΣ
Let X : (Σ,σ) → R3 be an isometric embedding of (Σ,σ). By (2.3)
and the result of Nirenberg [7, p.353], for each large k, there exists an
isometric embedding Xk of (Σ,γ ) in R3 such that
k
(2.7) ||Xk −X|| ≤ C||γ −σ||
C2,α(Σ,σ) k C2,α(Σ,σ)
where C is some constant depending only on σ. Let IIk, II be the
0 0
second fundamental form of Xk(Σ), X(Σ) respectively. (2.7) implies
thatIIk andII (viewedas(0,2)tensorfieldsonΣthroughthepullback
0 0
by Xk and X) satisfy
(2.8) lim ||IIk −II || = 0.
0 0 C0,α(Σ,σ)
k→∞
Consequently, {Hk} converges to H uniformly on Σ, where H is the
0 0 0
mean curvature of X(Σ).
On second variation of Wang-Yauquasi-local energy 7
It follows from (2.3) - (2.6), (2.8), the interpolation inequality for
Sobolev spaces and the L2-estimates that
(2.9)
(∆ η )2 1
k k dv ≤ IIk(∇ η ,∇ η ) dv + (∆ η )2 dv
Hk γk 0 k k k k γk k k k γk
ZΣ 0 ZΣ ZΣ
1 (∆ η )2 1
≤ k k dv +C + (∆ η )2 dv
2 Hk γk 1 k k k γk
ZΣ 0 ZΣ
where ∇ is the gradient on (Σ,γ ). Here and below, {C } always
k k i
denote positive constants that are independent on k. Now (2.9) shows
(2.10) (∆ η )2 dv ≤ C ,
γk k γk 2
ZΣ
which combined with (2.3), (2.6) and the L2-estimates implies
(2.11) ||η || ≤ C .
k W2,2(Σ,γk) 3
By (2.3), this in turn shows ||η || ≤ C . Hence ∃ η ∈ W2,2(Σ)
k W2,2(Σ,σ) 4
such that, passing to a subsequence, {η } converges to η weakly in
k
W2,2(Σ,σ) and strong in W1,2(Σ,σ). By (2.3) - (2.4) and (2.6) - (2.7),
η is σ-L2 orthogonal to L(σ) with η2dv = 1.
Σ σ
We claim
R
(∆η)2
(2.12) F (η) = −II (∇η,∇η) dv ≤ 0.
γ,H0 H 0 σ
ZΣ 0
If this is true, then we have a contradiction by Lemma 2.1.
To prove (2.12), we apply Lemma 2.1 to obtain F (η −η) ≥ 0.
γk,H0k k
By (2.3), (2.5) (2.7), (2.8) and (2.10), we have
C
2
≥ F (η )
k γk,H0k k
(2.13)
≥ 2Q (η ,η)−F (η)
γk,H0k k γk,H0k
= 2Q (η ,η)−F (η)+o(1)
γ,H0 k γ,H0
as k → ∞. Using the fact that η converge to η weakly in W2,2(Σ) and
k
strongly in W1,2(Σ), we conclude from (2.13) that
0 ≥ 2Q (η,η)−F (η)
γ,H0 γ,H0
= F (η)
γ,H0
(cid:3)
which verifies (2.12). The Lemma is proved.
Often we need estimates of |∇η|2dv by (∆η)2dv which depend
Σ γ Σ γ
explicitly on the metric γ. This can be given in terms of eigenvalues of
R R
(Σ,γ).
8 Pengzi Miao and Luen-FaiTam
Lemma 2.4. Let (M,g) be a compact Riemannian manifold without
boundary. Let E be the space of eigenfunctions with the k-th nonzero
k
eigenvalue µ of (M,g). Let E be the space of constant functions.
k 0
Suppose φ ∈ W2,2(M) is g-L2 orthogonal to E ,E ,...,E , then
0 1 k−1
µ |∇φ|2dv ≤ (∆φ)2dv .
k g g
ZM ZM
In particular,
µ |∇φ|2dv ≤ (∆φ)2dv , ∀ φ ∈ W2,2(M).
1 g g
ZM ZM
Proof. Since
1 1
2 2
|∇φ|2dv = − φ∆φdv ≤ φ2dv (∆φ)2dv ,
g g g g
ZM ZΣ (cid:18)ZM (cid:19) (cid:18)ZΣ (cid:19)
we have
|∇φ|2dv (∆φ)2dv
M g ≤ M g,
φ2dv |∇φ|2dv
R M g RM g
(cid:3)
from which the claim follows.
R R
3. Sufficient conditions for the positivity of F
γ,H
In this section, we provide some sufficient conditions guaranteeing
the positivity of F . Note that Lemma 2.3 implies
γ,H
F (η)
inf γ,H0 > 0
η∈L(γ)⊥ Σ(∆η)2dvγ
where L(γ)⊥ is the space of funRctions that are γ-L2 orthogonal to L(γ).
Proposition 3.1. Let γ be a metric of positive Gaussian curvature on
Σ. Let β be a positive constant such that
(3.1) F (η) ≥ β (∆η)2dv , ∀ η ∈ L(γ)⊥.
γ,H0 γ
ZΣ
Suppose the first nonzero eigenvalue of (Σ,γ) is at least λ > 0. Then
∃ δ > 0, depending only on β, λ, H and a given constant α > 0, such
0
that if H ≥ α is a function on Σ satisfying
(a) (H −H) dv > 0
0 γ
ZΣ
|H −H|2 dv
(b) sup|(H −H) | < δ and Σ 0 γ < δ,
0 −
(H −H) dv
Σ RΣ 0 γ
R
On second variation of Wang-Yauquasi-local energy 9
where (H −H) = min{H −H,0}, then F (η) ≥ β˜ (∆η)2 dv , ∀
0 − 0 γ,H γ
ZΣ
η ∈ W2,2(Σ), where β˜ > 0 is a constant depending only on H and β.
0
Proof. Given any constant α > 0, let H ≥ α be a function on Σ
satisfying (a) and (b) with δ > 0 to be chosen later.
Let X = (X ,X ,X ) be an isometric embedding of (Σ,γ) in R3.
1 2 3
Given any η ∈ W2,2(Σ), decompose η = η + η where η = a +
1 2 1 0
3 a X ∈ L(γ) and η ∈ L(γ)⊥. Let a = (a ,a ,a ). If a = 0, then
i=1 i i 2 1 2 3
(∆η )2
P F (η) = F (η )+ (H −H) 2 +|∇η |2 dv
γ,H γ,H0 2 0 HH 2 γ
ZΣ (cid:18) 0 (cid:19)
1 1
(3.2) ≥ β (∆η )2dv −δ + (∆η )2dv
2 γ 2 γ
αinf H λ
ZΣ (cid:18) Σ 0 (cid:19)ZΣ
1 1
= β −δ + (∆η)2dv
γ
αinf H λ
(cid:20) (cid:18) Σ 0 (cid:19)(cid:21)ZΣ
where we have used (3.1) and Lemma 2.4.
Now suppose a 6= 0, we may normalize a so that |a| = 1. Then for
any ǫ ,ǫ > 0,
1 2
F (η)−F (η)
γ,H γ,H0
1 1
= − (∆η)2 +(H −H)|∇η|2 dv
0 γ
H H
ZΣ(cid:20)(cid:18) 0(cid:19) (cid:21)
1 1
= − (∆η )2 +(H −H)|∇η |2 dv
1 0 1 γ
H H
ZΣ(cid:20)(cid:18) 0(cid:19) (cid:21)
1 1
+ − (∆η )2 +(H −H)|∇η |2 dv
2 0 2 γ
H H
ZΣ(cid:20)(cid:18) 0(cid:19) (cid:21)
1 1
+2 − ∆η ·∆η +(H −H)h∇η ,∇η i dv
1 2 0 1 2 γ
H H
ZΣ(cid:20)(cid:18) 0(cid:19) (cid:21)
(3.3) 1 1
≥ (H −H)dv −δ + (∆η )2dv
0 γ 2 γ
αinf H λ
ZΣ (cid:18) Σ 0 (cid:19)ZΣ
2
1 1
−ǫ−1 − (∆η )2dv −ǫ (∆η )2dv
1 H H 1 γ 1 2 γ
ZΣ(cid:18) 0(cid:19) ZΣ
−ǫ−1 (H −H )2|∇η |2dv −ǫ |∇η |2dv
2 0 1 γ 2 2 γ
ZΣ ZΣ
≥ 1−ǫ−1α−2δ −ǫ−1δ (H −H)dv
1 2 0 γ
ZΣ
(cid:0) 1 1(cid:1)
− δ + +ǫ +λ−1ǫ (∆η )2dv
1 2 2 γ
αinf H λ
(cid:18) (cid:18) Σ 0 (cid:19) (cid:19)ZΣ
10 Pengzi Miao and Luen-FaiTam
where we have used Lemma 2.2, the fact (∆η )2 = ha,ν i2H2 and
1 0 0
|∇η |2 = 1 − ha,ν i2. On the other hand, it follows from Lemma 2.1
1 0
and (3.1) that
F (η) = F (η ) ≥ β (∆η )2dv .
γ,H0 γ,H0 2 2 γ
ZΣ
Hence
F (η) ≥ 1−ǫ−1α−2δ −ǫ−1δ (H −H)dv
γ,H 1 2 0 γ
ZΣ
(cid:0) 1 (cid:1) 1
+ β − δ + +ǫ +λ−1ǫ (∆η )2dv .
1 2 2 γ
αinf H λ
(cid:20) (cid:18) (cid:18) Σ 0 (cid:19) (cid:19)(cid:21)ZΣ
Let ǫ = λ−1ǫ = β and choose δ > 0 such that 1−(ǫ−1α−2+ǫ−1)δ ≥ 1,
1 2 4 1 2 2
and δ 1 + 1 ≤ β, then
αinfΣH0 λ 4
(cid:16) (cid:17)
1 β
F (η) ≥ (H −H)dv + (∆η )2dv .
γ,H 0 γ 2 γ
2 4
ZΣ ZΣ
Combining this with (3.2) and the fact (∆η )2 ≤ H2, we conclude that
1 0
(cid:3)
the Proposition is true.
In [6, Theorem 3.1], it was proved that F is positive definite if
γ,H
H ≤ H and (H −H)dv > 0. By arguments similar to the proof of
0 Σ 0 γ
Proposition 3.1, it can be shown that F remains positive definite if
γ,H
R
H is allowed to be slightly bigger thanH . First, we give a quantitative
0
estimate of the case H ≤ H .
0
Lemma 3.1. Let γ be a metric of positive Gaussian curvature on Σ.
Let β > 0 be a constant such that
F (η) ≥ β (∆η)2dv , ∀ η ∈ L(γ)⊥.
γ,H0 γ
ZΣ
Let λ > 0 be a lower bound for the first nonzero eigenvalue of (Σ,γ).
Given any positive function H on Σ with H ≤ H , let α > 0 be a lower
0
bound of H. Then
β
β
F (η +η ) ≥ 2 (H −H)dv + (∆η )2dv
γ,H 1 2 α−1 +λ−1sup H + β 0 γ 2 2 γ
Σ 0 2 ZΣ ZΣ
foranyη ∈ L(γ)⊥ and η = a + 3 a Xi ∈ L(γ) with a = (a ,a ,a )
2 1 0 i=1 i 1 2 3
being a unit vector.
P