EQUATIONS OF ELASTIC HYPERSURFACES

3. CALCULUS OF DIFFERENTIAL OPERATORS ON HYPERSURFACES 93
valid at points on S , because ∑ k
ν k D k = 0, ∑ j
ν j U j = 0 and D k ν j = D j ν k . There are
four such terms in (3.115), i.e. containing either D U (ν j ν k ) or D V (ν j ν k ). An inspection of
the above calculation shows that, on S , they are all equal to −〈WS 2 U, V 〉. We still have to
compute the last integrand in (3.115):
n∑
D U (ν j ν k )D V (ν j ν k )
j,k=1
=
n∑
j,k,r,l=1
= 2〈W 2 S U, V 〉 + 2
[U r (D r ν j )ν k + U r (D r ν k )ν j
][V l (D l ν j )ν k + V l (D l ν k )ν j
]
n∑
k,r,l=1
on S . At this point we combine all the above to get
4
n∑
∫
j,k=1
S
1
( n∑
U r (D r ν k )V l ν k
2 D l
j=1
ν 2 j
)
= 2〈WS 2 U, V 〉,
∫
D jk (U)D jk (V ) dS = 2 〈−∆ S U − ∇ S Div S U − HS 0 W S U, V 〉 dS . (3.121)
S
Having deduced (3.121), we may now compute
∫
∫
4 〈Def ∗ S Def S (U), V 〉 dS = 〈Def S (U), Def S (V )〉 dS
S
S
n∑
∫
∫
= 4 D jk (U)D jk (V ) dS = 2 〈−∆ S U − ∇ S Div S U − HS 0 W S U , V 〉 dS.
∫
=
j,k=1
S
S
S
〈−π S ∆ S U − ∇ S Div S U − H 0
S W S U , V 〉 dS, (3.122)
since the original operator Def ∗ S Def S
: V (S ) → V (S ) is tangential. Thus,
4 Def ∗ S Def S = −2π S ∆ S − 2 ∇ S Div S − 2 H 0
S W S , (3.123)
since the tangential vectors fields U, V are arbitrary (note that here we use the fact that the
image of W S is a subspace of TS ).
The first identity in (3.113) now follows easily from (3.123) and (3.104). The remaining
identity in (3.113) then follows from what we have just proved and from Theorem 3.18.
Next recall the definition of the Hodge-Laplacian acting on 1-forms, i.e.
∆ HL := −d S d ∗ S − d ∗ S d S : Λ 1 TS −→ Λ 1 TS (3.124)
where d S is the exterior derivative operator on S , and d ∗ S its formal adjoint. As explained
in §2, 1-forms on S are naturally identified with tangential fields to S so, from now on, we
shall think of ∆ HL as mapping TS into itself.