Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued functions. Then the length of α is the integrl α α (t) dt x (t) 2 + y (t) 2 dt. Note tht if α is only piecewise smooth we cn still define α. In prticulr if α is piecewise smooth the derivtive α will be defined t ll but finitely mny points in the intervl [, b] so the bove integrl still mkes sense. Mny formuls become simpler by using complex nottion. Tht is we think of α s mp to C by setting α(t) x(t) + ıy(t). Then α (t) x (t) + ıy (t) is lso complex number. Thought of s complex number the bsolute vlue of α(t) gives us the sme nswer: α (t) x (t) 2 + ıy (t). Note tht the using the books nottion we hve α α dz α (t) dt. Let Ω be n open subset of R 2 tht contins the imge of α nd let f : Ω R 2 be smooth function. We then hve new pth define by ᾱ f α. To clculte the length of ᾱ we use the chin rule. In prticulr, if f(x, y) (u(x, y), v(x, y)) then ᾱ (t), written s column vector, is ( ᾱ ux (α(t)) u (t) y (α(t)) v x (α(t)) v y (α(t)) ) ( x (t) y (t) We cn think of f hs complex function by setting z x + ıy nd f u + ıv. If f is holomorphic we relly see the dvntge of using complex nottion. The Cuchy- Riemnn equtions tell us tht u x v y nd v x u y. Furthermore the complex derivtive of f is f u x + ıv x. If we tret ᾱ (t) s complex number we see tht ᾱ u x x v x y + ı(v x x + u x y ) (u x + ıv x )(x + ıy ). Tht is we hve ᾱ (t) f (α(t))α (t). This gives very simple formul for the length of ᾱ: ᾱ f (α(t)) α (t) dt. We sy tht f is n isometry of the Eucliden metric if the length of every pth α is equl to the length of the pth ᾱ f α. Clerly f is n isometry if f. In fct ).
it is not hrd to see tht this is lso necessry condition since if f (z) < t z then by continuity this will be true in neighborhood U of z. For ny pth α whose imge is contined in U we will then hve tht ᾱ is shorter thn α. We cn mke similr rgument if f (z) > t z. In homework problem we sw tht ny holomorphic function tht hd constnt bsolute vle must be constnt. In clss we will soon see tht the derivtive, f, of holomorphic function is lso holomorphic. For now we tke this s n ssumption. Therefore if f (z) then f (z) c where c nd f must be of the form f(z) where d is n rbitrry complex number. It is often useful to use lterntive definitions of distnce. In prticulr if Ω is gin n open subset of R 2 let λ : Ω R be positive function. We cn define the length of α with respect to λ by α λ α (t) λ(α(t))dt. If we hve two different metrics defined by functions λ nd ρ we cn then discuss whether f is n isometry from the λ-metric to the ρ-metric. To mesure the length ᾱ in the ρ-metric we hve the formul ᾱ ρ ᾱ (t) ρ(ᾱ(t))dt f (α(t)) α (t) ρ(f(α(t)))dt. For this to be the sme s the λ-length of α for ll pths α we need to hve or f (α(t)) ρ(f(α(t))) λ(α(t)) f (z) ρ(f(z)) λ(z). Note tht this formul gives us wy for defining metric. In prticulr if ρ then the ρ-metric is just the stndrd Eucliden metric. If we define λ by setting λ(z) f (z) then f will be n isometry from the λ-metric to the Eucliden metric. If we define λ by λ(z) f (z) ρ(f(z)) then f is n isometry from λ-metric to the ρ-metric. One very useful metric tht we will work with is the hyperbolic metric. It is defined on the upper hlf plne of C which we define s H 2 {z C : Im z > 0}. 2
The hyperbolic metric is λ H 2(z) Im z. The isometries of the hyperbolic metric re liner frctionl trnsformtions tht preserve the upper hlf plne. Nmely let where, b, c, d R nd d bc. Then We lso need to clculte Im T (z): nd therefore We then hve T (z) z + b T (z) 2ı Im T (z) T (z) T (z) z + b () 2. ( z + b z + b z + b (z + b)() (z + b)() (d bc)(z z) 2ı Im z Im T (z) ) Im z. T (z) λ H 2(T (z)) Im T (z) Im z Im z λ H 2(z) so T (z) is n isometry for the hyperbolic metric. We cn use the metric λ to define distnce function on the region Ω. Let P(z 0, z ) be the set of piecewise smooth pths in Ω from z 0 to z. We then define d λ (z 0, z ) inf α λ. γ P(z 0,z ) It is esy to check tht d λ stisfies the properties of distnce function: 3
. Clerly d λ (z 0, z ) d λ (z, z 0 ) since by reversing directions ny pth from z 0 to z becomes pth from z to z 0 of the sme length. 2. It is lso esy to check the tringle inequlity. (Here it is importnt tht we re llowing piecewise smooth pths.) If we conctente pth from z 0 to z with pth from z to z we obtin pth from z 0 to z 2. In prticulr if there is pth of length l 0 from z 0 to z nd pth of length l from z to z 2 then there is pth of length l 0 + l from z 0 to z. This implies tht d λ (z 0, z 2 ) d(z 0, z ) + d(z, z 2 ). 3. Finlly we need to see tht d(z 0, z ) 0 iff z 0 z. The function λ is continuous nd positve so for ny z 0 there is n ɛ > 0 nd n r > 0 so tht on the Eucliden disk of rdius r such tht λ > ɛ on the disk. Let α be pth from z 0 to z. If α is contined in this Eucliden disk then α λ > ɛ α ɛd(z 0, z ) > 0 if z 0 z. If α is not contined in the disk there is sub-pth α connecting z 0 to the boundry of the disk so α λ α λ ɛr > 0. In prticulr if z z is in the disk then d λ (z 0, z ) ɛd(z 0, z ) > 0 nd if z is not in the disk then d(z 0, z ) ɛr > 0 so d(z 0, z ) > 0 if z 0 z. It is cler tht d(z 0, z ) 0 if z z 0. The distnce function mkes (Ω, d λ ) into metric spce nd we will be ble to use ll the properties of metric spces to study it. We lso note if ρ λ defines nother metric on Ω then d ρ (z 0, z ) d λ (z 0, z ) for ll points z 0, z Ω. Problems. Let be the unit disk in C. Construct liner frction trnsformtion S : Ĉ Ĉ tht tkes to the upper hlf plne. 2. Define metric ρ on by the formul ρ(z) 2 z 2. Show tht S is n isometry from the ρ-metric to the hyperbolic metric λ H 2. In prticulr, the metric ρ on is nother representtion of the hyperbolic metric. To emphsize this we write ρ s ρ H 2. 3. The f(z) z 2 tke to itself. Show tht for ny two points z 0 z in we hve d ρh 2 (f(z 0), f(z )) d ρh 2 (z 0, z ). 4
4. Define metric on C by σ(z) 2. Given point z C find liner frctionl + z 2 trnsformtion R with R(0) z, R( ) z nd such tht R is n isometry for σ-metric. Comments: Problem 3 is n exmple of very importnt nd much more generl phenomenom. In prticulr ny holomorphic mp tht tkes into itself will be contrction of the hyperbolic metric. This is essentilly the Schwrz Lemm which we will (soon!) prove in clss. 5