by - warwick.ac.uk · saz in form branch Emma F t aa F 31 f UT L at 4 fz z4G Igoa IT a0eat...
Transcript of by - warwick.ac.uk · saz in form branch Emma F t aa F 31 f UT L at 4 fz z4G Igoa IT a0eat...
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