fTwo

gewan Polyhedron in

yis a topologicalsurfaceconsistingof faces edged

Polyhedralsurface and vertices

Wewill showthat a polyhedrondetermines a Reimannsurface Thisconstructiongives a large number

ofexamples unlikepreviousconstructions includingexamplesof Reimannsurface structures on

oriental surfaces of every genus Theseconstructionseachbuoe a finite ofparametersthat canbe adjusted

Step 1 We will start byconstructing dartsat interior points of facesOur surface comes with an outward pointingunit normal

Hit Cx Yox

idkePolyhedralsurface

Conatuetpolarcoordinatesin a rebel ofp Our aSo 0 so I 21T

If two points are in theamuefucelton rublecan overlap and transition functions leavethe four 2 a 2 to

ftp.z Coordinates on edgesat a pointp on theboundary

of a face timei off can construct a

half dials coordinate

If we puttogether apairofthese

Edge points wood we get a dish coordinateDecidewhich

Tt halfdishbecomes

theupperhalf

feesa amountwqn.tw

O

El EF

At this point we have an altarfor P outeciawhereall of the transition umpshave the form0 z vs et t c j This is a holomorphic altar

of a special form and we will see thisagain

Steps Coordinates at certain

Remarks about the function Z vs za

We can make sense of this bywritingA exp log E exp a log 7

We interpretthisbychoosing a branch ofthe

logarithm Now lets assume that xia real

and positive We get the followingpictureate

attic

o o

keep feyp

gU

Choosing a different branch of the logarithmchanges the lift ofthemapby a multipleofa it and changes the map by a rotation

of expcutinNote that choosing x and y for our coordinates

upstairs corresponds to using polarcoordinates a et f e downstairs

eypitiy e et EY r f

In pokercoordinates Zaza becomes G other hotit il alogutia2 AZ

tea texp

fexp t ra ri explalogu

5 EE To

Emma

F t

aa a

F

31 f UT

L at4 fz z4GIgo a IT

zita 0eat

CraisLra att const TutEe

2 Zx saz at o'T

adding in charts afteria formgive natraisitionfunctions of the form 2 azote where E isa branch of the powerfunction

3 pointsThere is a Riemann surface attus afor P according to the strict definition

of what a Reimann surfaceatlas is

Thisdefinitioncreates a smoothsurfaceCwith a tangent bundle which ishomeomorphic to P

Our intuition does not exactlyagreewith the stint definition We would like

some of the attucture of P beyond just

thetopologyto be reflected in a In

particular where P has a recognizableconformalstructure we would like this

to agree with theconformalstructure

givenby aClaim that

any two attames for Pwhich are conformal away from thevertices are equivalentThissays that even though Piometamooth

itself P the conformalstructures on P Ecortices3determines a unique conformal structureson all of Po

simplestexample come in 1123 with

cone angle x

buy we have

Recallthetheoremabout eniolatedsingularities

3 cases

Removablesingularitypole

Essential singularity

Theorem Removable anigulositiesfor atlassen

Tet R and R be Riemannsurfaces tet2 and I be discrete subsetsof Rand Rand let f R R be a homeomorphism

tutoring 2 to 2 which is a holomorphic

mapfrom R E to R E ther f is in facta holomorphicmap from R to R

Prof f

dit t di

to

Need to check that fin is holomorphicfji is a homeomorphism which is holomorphic

away from an isolated singularpointIt followerfrom the classical removeable

singularities theorem that f is holomorphicat p