I I Let 5 Sgbea closedsurfaceofgenus922

冥

endowwitnonfomann.peifstructuresUnionization EveryRiemannsurfaceof genusg canberepresented的

仁叫个Hele叫 istheupperhalf plane tea InEzo endowedwith

the Poincaré hyperbolemetric仁器 and T is a Fuchsiasubgroupof psuz.IR isomorphic to

TLS.liample a hyperbolais

surfaceof genus 2 iLL

v d nb

Nd I

I 2 TheTeichmülktspaceof S is defined to bethesetof isotopyclassesofmarkedconformal hyperbolic structures on

S.DefiiAmatkedhypetbolusurfaelx.fi isa hyperbolicsurface x together with a orientation_

preserving homeomorphism fi S XTwo marked hyperbolic surfaces Xi fil.it I ate

equivalent if there is an isometry hi Xxzsth.fiis homotopic to fz

The Teichmillet space Tg X f h

弱⼼尷 ⾯

1 IDehntwi.tt

⼀掏

TheDehntwist his consideras an elementof themappingclassgroup

Let Modi Dif SID𠵎 的 ThenModgacts on

Tgbychanging the markings i.e forany t EMdg

x f 1 X fool1 3 In this course I will explain

l Tg has a topology homeomorphic to BitI Themappingclassgroupacts properlydiscontinuously onTg The quotientTglmodg Mg is themodulispace

引导admits a natural Complex manifoldstructure and

4 several related geometric structures such asthe Teichmüwr metric Wet Petersson metric

5 with applications to mapping class group了⼀manifolds billiard dynamics etc

1.4 Foranycloudcurve de S and X H there is a uniquecloudgeodesic freely isotopic to fix on X We denote

its hyperbolic length by友⼼ Then we call

li Tgs Bthe geodesic length function of X

We endow T with the topology stall lengthfunctionate continuousTheFuchsian representation X 叫个 induces an

homomorphism

fi 𥘅 5 ㄗ PSLR.IR

9 f 1 which is well defined up to conjugacyThus the tichmüuet space admits an embedding

Tu Hom 𥘅 S PSUZ.PH psuaR

note that lengths are recovered by traces了

The F.nohetNielsencoordinatesTheTeichmilletspaceTg is homeomorphicto Rfi R

pnofletfi.in bea pantsdecompositionof Si

noise

X

11_fengthparametetsxc

Tgmll.cn ⼼ t.gs刚 ER3

twist parameters

rig iiil

t.nl

a pants decomposition P of S we obtain theFennelWidelengthtwist coordinates Hi TP for Tg In this coordinates a

Dehntwistaround facts by Tim titi

仙盟 Wolpert proved that the symplectic format Idf nd Eis invariant under the actionof Mdg

Hence w descends to a symplectic form on MyThere is a beautiful idea to integrate certainfunction over

Mg dueto Mirzakhani

let us fix a simple nonseparating closedcurves r on Sand let fi Ri Rt

Define fix f l l 刚dEModg r

Denotby Nd the symplectic volumeform Ther

f fr x Nd r_

⼆

䋆以刚 Ndt

iiiiiiiryiiefuivollMg.nu⽐ 不

ME X x XEMg 2EModg.rs

1.5 Fix a basispointXETg.rs Denoteby S thespaceofsimpleclosed geodesics on XFor any ct.DE RtXS let tf M be the his

deformation of X along

T.twR 8 TgThe image of tw is dense in analogueto rational

rays in R In fact

theorem Thurston There is a completion ofPiS calledmeasured lamination space M.li itand the twist deformation extends to a homeomorphism

Mh 3 Tg

i

Nielsen realization Anyfinit subgroup G of Mdghasa fixedpoint in Tg

P ibyketckh.fiI The lengthfunction fi Tai Rt forany cloudcurvein strictly convexalong twopaths

I Take a finite collection of curvesㄗ n 不 s t

it is G invariantand filling Then

eiitridefines a proper G invariant function on T3 it XETg is a point of minimumftp.thenxis a fixed point of G

Uniformization

Teichmuller metric

Hyperbolicsurfaces

Riemannsurfaces

Fenchel-Nielsen

coordinates

Earthquakeflow

Teichmullermapping

Bersembedding

Quadraticdifferentials

Teichmullergeodesic flow

Weil-PeterssonKahler metric