i

gQ

g

Hutu definitionof the surfacestructureweaeurallowmultiplecopiesofQ

µ

Hinnant t.fiQxiuywscxil.co

nintheroftwaeweeps

1is holomorphic hequestionof

gHolomorpdiiutffirataraia

xinyma 4,4 withereapeattertheoverlapfunctions

productive

Fateg

p uaa.KTti.o.mimaaupa

ai

ii

Feoffee

Recallltuttheiptcompactificationof alocallycompactspace

isobtainedbyadding a pointand declaring troublegas tree complements

of compactsets

extendotrahomeomorphism

asYt ft

94,4Corusiderlaie

PE i uiY SX Sy a Dist fetto

Csx Sy Cts hormonal

ascompartaetrofecorrespondto

computt syenite 25 0

setsftpse YYP3gIaArwbouuo.soloefous sittin toLs z't

fTwo

game polyhedron inis a topologicalsurfaceconsistingof faces edges

Polyhedralsurface and vertices

Wewill showthat a polyhedrondetermines a Reimannsurface Thisconstructiongives a large number

ofexamples unlikepreviousconstructions includingexamplesof Riemannsurface structures on

oriental surfaces of every genus Theseconstructionseachbuoe a finite ofparametersthat canbe adjusted

Step't We will start byconstructing abortsat interior points of facesOur surface comes with an outward pointingunit normal

Hit Cx4,01a

ePolyhedralsurfaceComtructpolarcoordinatesin a rebel ofp our too 0 0 I 21T

If two points are in theamuefucelton ablecan overlap and transition functions leavethe form 2 a 2 to

ftp.z Coordinates on edgesat a pointp on theboundary

of a facets we

off can construct a

half dials coordinate

If we puttogether apairofthese

Edge points wood we get a dials coordinate4 Decidewhich

Tt halfdishbecomes

theupperhalfJ.fi i aauadwlq.neEe

1 EH

At this point we have an atlasfor P outeciowhereall of the transition unpalumette formQuiz etitttcj.ie This is a holomorphic atlas

of a special form and we will see this again

Steps Coordinates at certain

E

f

anglesmallerthank II HUT 4g

a 4g 2 2 I

IEE a ITEYEDe3Iz

crankshaftconst IutEe

2 Zod saz at o'T

addinginduertaafteriaformgrieuatraisitionfunctions of the form z xEtc where Isisa branch of the powerfunction euddhere

Remark Construction can be applied toa collectionof polyous in LTE with identificationsof sides which need not be realigable in 1123

Parallelogram

I c Identify oppositedi tu sided

In this case the vertex bus cone angled t t r t d t dy IT

Orientationreversingmapsofsides to create orientation

preserving clients

Octagon yo0t t

x sI o

future case all oftheverticesgetidentifiedto a single point and the core angle at that pointis I

There are two typesof pathologicalbehavior tteat can occur

a priori a seafarer need not beHausdorfa surface need not have a countable

base for its topologyRiemann

Here is a simpleexampleofayaurfacewhich is neither Hausdorff was 2ndcountable GiaelRsthetop

it tetpa exy z a Gqditjztpg.im

8ounthequutieitapncecry by identifying exit 7 and

H Y 27 if Yo

C oraruoreaopluoticatedaompkaeeBeeerdou.J

Thefirst property is compatiblewith a Riemann surfacestructureand we will assume that our

Riemannsurfaces are Hausdorff

It is altman which will be a corollary

of something that we

connected

prove ltat aHausdorff Riemann

surface is 2nd countable

Example CHP

q2503

I Cz w tired

W I co073

Define an equivalence relationouw byCz w UN riff for some tox G w Ctu.tv

note to a 03

Writethe equivalenceclassofGut as Kiwifeta p be the set of equivalence classes

w clip

X toas

Hz w E if wt oCs w O

p Z WJ Iw Wto

as w O

Q 5 If we pull back the atlas on 5

to clip we got U Et i w wt03

Ur ft W 2 to

qiu 4 law E

if uz 94ft w Ee

ThisgivesQP a Reimann surface alterative