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 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 that the tangent space in relation to the fundamental form II(X, Y) The second = 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 II (X, But differentiation of Y) = (X df (Y) - - 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 Metric and Affine Conformal Proposition The 'Rdf mean Geometry curvature vector 'H dfR (*dR + RdR), 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 ldfTf Similarly for dR + dRR)Tf (*dR + RdR)Tf N Proposition 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 KIdf 12 K ldf 12 = = (< *dR, RdR -1 (< *dR, RdR > + < *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 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 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 (deg R + deg N) ofthe Gauss-Bonnet theorem by > 42 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 = * R' then Kj- K)Idfl2 * 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 * 7.2 The Mean Curvature We now discuss the characteristic dR *dR, RdR 1IRdRI2 + < 41 < RdRI - > *dR, RdR > Sphere in Affine Coordinates properties of S describe S relative to the frame i.e we in affine coordinates We write S = GMG-1, where f) G 01 First, SL C representation: L is equivalent EV to S H2 having the following matrix S= (1f) (' -R) -H 01 where N, R, H : M -+ H From S2 = N 2=-l=R The choice of 1f) (7.9) _I , 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) , = + = = sj 43 irSd = 1 Therefore *6 S6 = + we Ir = JS is equivalent to = *df and -df -7r 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 - HdN ' Hdf Hdf R + R dR) G-1 *dS - NdN *dH + HdfH - *dN + 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) - - - 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) Metric and Affine Conformal 44 Remark 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 4*A= G f (01), where G Using (7.12) and we can w w = w dR + R dH + H * dfH * 0) G-1, dR) G-', + R dH * dH + * 1H(NdN - We Proof H * - H * (7.15) H * dN *dN) to consider the reformulation of only have dfH - (7.14) rewrite dH + R = dN * -2dH + IH dN (dN * - - N * dN) H - 1H* (dN+N*dN) w * dN 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 < R - JRdR - *dR12 is the classical = (IHI2 - K integrand < A A *A >= (Ih 12 - K)Idfl2 - K-L)JdfJ2 7.3 The Willmore Condition in Affine Coordinates 45 Proof < A A *A > traceR(-A' Now see We Proposition now 1V) Re( - (*A)) = IV12 = 16 express the and * A = JdR + R * dR12 16 jRdR - *dR12 Euler-Lagrange equation write we * A = d * A 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 = 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 + - IdfH A (NdN =df AdH+dfRA*dH+ - *dN) *dN) df A dH - *df A *dH+ dfH A (NdN - *dN), io but *(NdN - *dfH = df (-R)H *dN) = (N * dN -df HN - N 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 "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 G = (dR)H * ((dR)H * H(2dfH) d*A= w R R - HdN) Proposition with - HdN - - =RH dH + R Therefore f * dH + (dw 0) !H(NdN 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 : 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, = This is -Y(S) -y be 0, a (r,2-ol = free elastic curve obtain, only if we if and ... Affine Conformal 44 Remark 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. .. 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,...40 Metric and Affine Conformal Proposition The 'Rdf mean Geometry curvature vector 'H dfR (*dR + RdR), 2 trace II is given by (*dN + dR + *dNdf (7. 4) NdN)df, (7. 5) NdN) (7- 3) Proof By definition