7 Metric and Affine Conformal
We consider the metric extrinsic geometry of quantities associated to
For
brevity
we
...
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7 Metric and Affine Conformal
We consider the metric extrinsic geometry of quantities associated to
For
brevity
we
write <
.,.
> instead of <
7.1 Surfaces in Euclidean Let
Proposition f is given by
II(X, Y) Proof. We
2
Space
Ndf
as
its fixed
point
(-I)-eigenspace
f
M
:
H,
-df R.
=
-+
v
i.e.
(X df (Y))--L of
-
dN(X)
*
N(x)vR(x)
df (Y)). is
an
(7.1)
involution with
set:
Ndf (Y)R Its
vector of
(*df (Y)dR(X)
know from Lemma 2 that
the tangent space
in. relation to the
fundamental form II(X, Y)
The second
7.
=
R
-+
>R-
N, R denote the left and right normal
*df
M
:
HPI.
M
L:=
f
Geometry
df (Y)
=
is the normal space,
so we
(7.2)
need to compute
1
II (X, But differentiation of
Y)
=
(X df (Y) -
2
-
NX
-
df (Y) R).
(7.2) yields
dN(X)df (Y)R +
NX
-
df (Y)R
+
Ndf (Y)dR(X)
=
X
-
df (Y),
or
X
-
df (Y)
-
NX
-
df (Y)R
=
dN(X)df (Y)R
=
-dN(X)
F. E. Burstall et al.: LNM 1772, pp. 39 - 46, 2002 © Springer-Verlag Berlin Heidelberg 2002
*
+
df (Y)
Ndf (Y)dR(X) +
*df (Y)dR(X).
40
7 Metric and Affine Conformal
Proposition
The
8.
'Rdf
mean
Geometry
curvature vector 'H
dfR
(*dR + RdR),
2
2
2
trace II is
given by
(*dN
+
dR +
*dNdf
(7.4)
NdN)df,
(7.5)
NdN).
(7-3)
Proof. By definition of the trace, 4'H jdfJ2
dN
=
*dfdR
=
-df (*dR
-
*
df
-
RdR)
+
df
*
(*dN
+
+
but
(*dN + NdN)df
=
*dNdf
=
-df
A
-
dN
dR
=
*
df
-dN A
=
df
=
-d(Ndf)
-df (*dR + RdR).
If follows that
27ildfI2
=
-df (*dR
RdR),
+
and
2 ldfTf Similarly for
dR +
dRR)Tf
(*dR + RdR)Tf
N.
Proposition
9. Let K denote the Gaussian curvature
and let K' denote the normal curvature
Kjwhere X E
=
and
TpM,
:=<
E-Lp
M
*
<
>R)
of f defined by
Rj- (X, JX) , N are
of (M, f
>R)
unit vectors. Then
1
KIdf 12 K
1
ldf 12
=
2 =
(< *dR, RdR
-1 (< *dR, RdR 2
> + <
*dN, NdN >)
(7.6)
>
*dN, NdN >)
(7.7)
-
<
Proof.
Kldfl4(X) Therefore
=<
II(X, X), I.T(jX, jX)
>
_III(X, jX)12.
7.1 Surfaces in Euclidean
4KIdf 14
=<
*df dR
*df
<
-
< N
=
<
dR
(df dR
+
<
N(df
<
df dR
+
df
dfdR
dfdR, dfR
*
dR >
<
dNdf, df R
*
dR >
df
+ <
*dNdf, dfRdR
ldf 12
dR,
ldf 12
<
I dyl2(<
21dfI2 (<
find, after
we
4K
1
jdfJ2
=< *dR
As
a
we use
(7.5)
10.
-
<
df
*
<
>
>
*
>
dN >
dR, NdNdf *dN,
>
dNdf
*
dN, N
>
NdN >
dN,N* dN
>
*dN, NdN >)
dR > + < dN,N
to obtain
we
dN
*
and the Ricci
equation
II(JX, JX)
-
>).
>,
The
1
N)
-
< *dN
>
NdN, NdN
-
-
<
(*dN
-
>
NdN)df, dfRdR
(7.7).
have
T7r =
>
RdR), NdNdf
pull-back of the 2-sphere
Integrating this for compact
3-space (R
dfdR, N
-ldf 12
II(X, JX), II(X, X)
R*dA
In
*
<
dR,NdNdf
*dNdf, NdNdf <
-
> + <
Using (7.1)
RdR, RdR
-
df (*dR
corollary
Proposition
df
> + <
*
>
similar computation,
a
+ <
On this
-
dR,R
This proves the formula for K. =< N
dNdf
> + <
>
>
>
dNdf, N
dR,R*dR
*dR, RdR
<
dNdf
<
+jdf 12
,
*
>
dNdf
Ndf dR >
*dNdf
<
*dNdf)
dfdR, N
> + <
*dR, RdR
>
<
-
dR >
>
>
NdNdf
+
>
dNdf
dNdf)
+
dR + N
-
RdR >
dNdf, Ndf
+ < +
dR,dfRdR
<
dR +
*dNdf, dfRdR
<
*
*
*dNdf
-df dR + *
>
dNdf
+
dR +
*
*dNdf, N(-dfdR dNdf, dfR
+
*dNdf
df, -df dR
*
*dNdf),
+ <
-
Kj-
*dN
dNdf ), -df
dR +
df
+ <
-
dR +
*
dNdf, N(-df
dR +
*
df, -df
*
dR +
*
<
=-
*
dN
-
41
Space
f
M
=<
Kjdf 12 a
under R is given
>
yields
A M
this is
*dR, RdR
area
version
2
(deg R
+
deg N).
ofthe Gauss-Bonnet
theorem.
by
>
42
7 Metric and Affine Conformal
Proposition
Geometry
We obtain
11.
(J-H12 particular, if f : M integrand is given by
In
K
-
-+
-
Im H
(I Ij 12
-
Kj-) ldf12 =
4
1
*
R' then Kj-
K)Idfl2
1
4
*
dR
=
dR
-
RdRJ2
0, and the classical Willmore
-
RdRJ2.
(7.8)
Proof. Equations (7.3), (7.6), (7.7) give
(I,HI2
-
K
-
K-L)Idfl2
11
*
dR +
11
*
dR12
RdRj2_
4
4
*
7.2 The Mean Curvature We
now
discuss the characteristic
dR
*dR, RdR
1IRdRI2
+
<
4
1
41
<
RdRI
-
2
>
*dR, RdR
>
2
Sphere in Affine Coordinates properties of S
describe S relative to the frame
i.e.
we
in affine coordinates. We write S
1
0
=
GMG-1,
where
f)
G
01
First, SL C representation:
L is
equivalent
EV
to S
H2 having the following matrix
--
0
S=
(1f) (' -R) -H
01
where N, R, H
:
M
-+
H. From
S2
=
N 2=-l=R
The choice of
0
1f)
(7.9)
_I 2 ,
RH
(7.10)
HN.
symbols is deliberate: N and R turn out to be the left and right f while H is closely related to its mean curvature vector
normal vectors, of
,
X The bundle L, has the nowhere
section,
we
compute
vanishing
section
(fl)
E V
(L). Using
this
7.2 The Mean Curvature
(f) is
in Affine Coordinates
Sphere
(*df) R)) 7r((-dfR) ird(S (f 7rd( (f (f) (-dR)) (f) Ir((Ndf) (f) (-Hdf)) (Ndf (f) ,
=
0
+
=
=
0
sj
43
irSd
=
1
1
Therefore *6
S6
=
+
0
we
0
Ir
=
1
0
JS is equivalent to
=
*df and
-df -7r
1
Ndf
=
-df R,
=
have identified N and R.
For the
computation of the Hopf fields, we need dS. This is a straightlengthy computation, somewhat simplified by the fact that dG G-dG. We skip the details and give the result:
forward but
GdG
=
=
dS
SdS
Rom this
Q
4A
SdS
=
G
( (
.
The condition so
-dH
-
-dR +
Ndf) P7
+ NdN
-NdfH
+ RdH
-
0
HdN
'
Hdf
Hdf R + R
dR)
G-1.
*dS
-
NdN *dH +
HdfH
-
*dN
0
+ RdH
-
HdN
2HdfR
+ RdR +
*dR)
G-1
SdS + *dS G
used
G
G
obtain
we
=
=
=
-dfH + dN -dfR
( (HdfH
=
far-,
NdN + *dN -
dH +
*
QIL
=
have the
HdfH
-
2dfH with
2Hdf
=
dR
2dfH
=
dN
equations (7.3)
-
-
-
0
HdN RdR
0, and the corresponding AH following equivalents:
2Hdf
Together
2Ndf H
+ RdH
R N
dR
=
dN
dR dN
we
-
-
-
C
*dR) L,
G-1.
which
we
have not
R
*
dR,
(7.11)
N
*
dN.
(7.12)
find
-2RTtdf, -R(*dR + RdR) + 2NdfR -N(*dN NdN) =
=
=
=
-2dfRR,
and therefore H
=
-RN
=
-RR.
(7.13)
R)
7 Metric and Affine Conformal
44
Remark 9. Given vature vector of
is the
have,
mean
immersed
an
f
at
fact, the same sphere.
in
M
E
x
mean
holomorphic
the
L
curve
mean cur-
is determined
of
sphere
curvature
Geometry
Sx,
by Sx. On the other hand, S" Example 17. Therefore S,' and f vector at x, justifying the name mean
see
curvature
curvature
Equations (7.11), (7.12) simplify the coordinate expressions for the Hopf fields, which we now write as follows
Proposition
12.
dN + N
( (w
4*Q=G
0
4*A= G
f
(01),
where G
Using (7.12)
and
we can
w
w
=
w
0
dR + R
dH + H
*
dfH
*
0) G-1, dR) G-',
+ R
dH
*
dH +
*
1H(NdN
-
2
We
Proof. H
*
-
H
*
(7.15)
H
*
dN.
*dN).
to consider the reformulation of
only have
dfH
-
(7.14)
rewrite
dH + R
=
dN 0
*
-2dH +
IH
dN
(dN
*
2 -
-
N
*
dN)
H
-
1H* (dN+N*dN)
w.
*
dN
2
2
But
H(NdN
-
*dN).
7.3 The Willmore Condition in Affine Coordinates We
use
the notations of the
previous Proposition 12, and
in addition abbre-
viate v,
=
dR+R*dR.
Note that V
Proposition
13.
-dR +. *dRR
=
The Willmore
-dR
--
integrand
A A *A >
=
16
For
f
:
M
-4
R, this
is
*
dR
=
-v.
given by
1
1 <
R
-
JRdR
-
*dR12
is the classical
=
4
(IHI2
-
K
integrand 1
< A A *A >=
4
(Ih 12
-
K)Idfl2.
-
K-L)JdfJ2.
7.3 The Willmore Condition in Affine Coordinates
45
Proof. < A A *A >
8
traceR(-A' 1 4
Now
see
We
Proposition
now
1V)
4 Re(
-
(*A))
=
IV12
=
2
16
4
express the
and
*
A
=
JdR + R * dR12
16
jRdR
-
*dR12.
Euler-Lagrange equation write 4
we
*
A
=
d
*
A
0 for Willmore
=
GMG-1,
then
G(G-1 dG A M + dM + M A G-1 dG)G-1,
again using G-'dG 4d
16
traceR(A2)
and, for the second equality, (7.8).
11
surfaces in affine coordinates. If 4d
4
=
A
*
dG G
=
easily find
we
(
df
A
df
w
dv +
dw
A
v
w
A
df)
G-1.
Most entries of this matrix vanish:
Proposition
14.
We have
df
Aw=O
df
Av=O
(7.16) (7.17)
dv+wAdf =-(2dH-W)Adf Proof.
(7.18)
=0.
We have
df
A
w
=
df
A dH +
df
A R
*
Idf A H(NdN
dH +
-
2
IdfH A (NdN
=df AdH+dfRA*dH+
-
2
*dN)
*dN)
1
df
A dH
-
*df
A *dH+
2
dfH
A
(NdN
-
*dN),
io but
*(NdN
-
*dfH
=
df (-R)H
*dN)
=
(N
*
dN
-df HN
-
N
2
dN)
=
-N(NdN
Hence, by type, the second term vanishes as well, and A similar, but simpler, computation shows (7.17) Next, using (7.11), we consider
we
-
*dN).
get (7.16).
dv+wAdf =d(dR+R*dR)+wAdf =
d(-2Hdf)
=
(-2dH
=
(-dH
+
+
w
A
w)
A
df
+ R
*
df
dH +
1H(NdN 2 "o
-
*dN))
A
df.
7 Metric and Affine Conformal
46
Again we show Clearly
*a
*(NdN showing *0 *a
-
-
*dN)
N
dN + NdNN
*
=
(NdN
-
*dN)N,
Further
flN.
aN
ON. Then .(7.18) will follow by type.
aN,
=
Geometry
*
dH
*
dH
RdH + dHN
-
d(RH)
-
R(*dH)N
-
(dR) H
+
+
d(HN)
-
HdN
R(d(HN)
-
=RH
+R 2* dH +
(dR)H (dR
-
2HdfH =
As
a
HdN
-
R
(dR)H
*
-
dR)H
corollary
we
H(dN
-
+ RH
N
-
*
*
+ RdH
-
HdN)
dN)
dN)
get:
15. 1
0
G 4
=
(dR)H
*
((dR)H
*
H(2dfH)
d*A=
w
R
R
-
HdN)
0.
Proposition
with
-
HdN
-
-
=RH
dH + R
Therefore f
*
dH +
0
(dw 0)
4
!H(NdN 2
is Willmore
if
fdw
G-1
and
-
(dw
-fdwf) -dw
*dN).
only if dw
Example 20 (Willmore Cylinder). Let -y f : R2 -+ H the cylinder defined by
:
0.
RIm H be
a
unit-speed
curve,
and
f (S' t) with the conformal structure
after
some
computation,
J-L as
that
f
=
is
=
-2-. at
Ir.3+ K11 -r.7, exactly the condition that
+ t
Then
using Proposition 15,
(non-compact) Willmore, 2 =
2
This is
-Y(S)
-y be
0, a
(r,2-ol
=
free elastic
0.
curve.
obtain, only if
we
if and