▁ ▁is ▁a ▁finit ely ▁presented ▁group . ▁The ▁gener ators ▁and ▁relations ▁are ▁encoded ▁in ▁a ▁ge odes ically ▁convex ▁fundamental ▁ge odes ic ▁polygon ▁in ▁ ▁( or ▁) ▁corresponding ▁geomet r ically ▁to ▁closed ▁ge odes ics ▁on ▁. ▁ ▁Ex amples . ▁ ▁the ▁Bol za ▁surface ▁of ▁genus ▁ 2 ; ▁ ▁the ▁Klein ▁quart ic ▁of ▁genus ▁ 3 ; ▁ ▁the ▁Mac be ath ▁surface ▁of ▁genus ▁ 7 ; ▁ ▁the ▁First ▁Hur witz ▁triple t ▁of ▁genus ▁ 1 4 . ▁ ▁Un iform ization ▁ ▁Given ▁an ▁orient ed ▁closed ▁surface ▁ ▁with ▁Gaussian ▁curv ature ▁, ▁the ▁metric ▁on ▁ ▁can ▁be ▁changed ▁conform ally ▁by ▁scaling ▁it ▁by ▁a ▁factor ▁. ▁The ▁new ▁Gaussian ▁curv ature ▁ ▁is ▁then ▁given ▁by ▁ ▁where ▁ ▁is ▁the ▁Lap la cian ▁for ▁the ▁original ▁metric . ▁Thus ▁to ▁show ▁that ▁a ▁given ▁surface ▁is ▁conform ally ▁equivalent ▁to ▁a ▁metric ▁with ▁constant ▁curv ature ▁ ▁it ▁suff ices ▁to ▁solve ▁the ▁following ▁variant ▁of ▁Li ou ville ' s ▁equation : ▁ ▁When ▁ ▁has ▁E uler ▁characteristic ▁ 0 , ▁so ▁is ▁dif fe omorphic ▁to ▁a ▁tor us , ▁, ▁so ▁this ▁amounts ▁to ▁solving ▁ ▁By ▁standard ▁elli ptic ▁theory , ▁this ▁is ▁possible ▁because ▁the ▁integral ▁of ▁ ▁over ▁ ▁is ▁zero , ▁by ▁the ▁G auss – B on net ▁theorem . ▁ ▁When ▁ ▁has ▁negative ▁E uler ▁characteristic , ▁, ▁so ▁the ▁equation ▁to ▁be ▁solved