▁is : ▁ ▁Using ▁the ▁continu ity ▁of ▁the ▁exponential ▁map ▁on ▁Sob ole v ▁space ▁due ▁to ▁Neil ▁Tr ud inger , ▁this ▁non - linear ▁equation ▁can ▁always ▁be ▁solved . ▁ ▁Finally ▁in ▁the ▁case ▁of ▁the ▁ 2 - s phere , ▁ ▁and ▁the ▁equation ▁becomes : ▁ ▁So ▁far ▁this ▁non - linear ▁equation ▁has ▁not ▁been ▁anal ys ed ▁directly , ▁although ▁classical ▁results ▁such ▁as ▁the ▁Riemann - R och ▁theorem ▁imply ▁that ▁it ▁always ▁has ▁a ▁solution . ▁The ▁method ▁of ▁Ric ci ▁flow , ▁developed ▁by ▁Richard ▁S . ▁Hamilton , ▁gives ▁another ▁proof ▁of ▁existence ▁based ▁on ▁non - linear ▁partial ▁differential ▁equations ▁to ▁prove ▁existence . ▁In ▁fact ▁the ▁Ric ci ▁flow ▁on ▁conform al ▁metrics ▁on ▁ ▁is ▁defined ▁on ▁functions ▁ ▁by ▁ ▁After ▁finite ▁time , ▁Ch ow ▁showed ▁that ▁ ▁becomes ▁positive ; ▁previous ▁results ▁of ▁Hamilton ▁could ▁then ▁be ▁used ▁to ▁show ▁that ▁ ▁converges ▁to ▁+ 1 . ▁Prior ▁to ▁these ▁results ▁on ▁Ric ci ▁flow , ▁ ▁had ▁given ▁an ▁alternative ▁and ▁techn ically ▁simpler ▁approach ▁to ▁uniform ization ▁based ▁on ▁the ▁flow ▁on ▁Riemann ian ▁metrics ▁ ▁defined ▁by ▁. ▁ ▁A ▁simple ▁proof ▁using ▁only ▁elli ptic ▁operators ▁discovered ▁in ▁ 1 9 8 8 ▁can ▁be ▁found ▁in ▁. ▁Let ▁ ▁be ▁the ▁Green ' s ▁function ▁on ▁ ▁satisfying ▁, ▁where ▁ ▁is ▁the ▁point ▁measure ▁at ▁a ▁fixed ▁point ▁ ▁of ▁. ▁The ▁equation ▁, ▁has ▁a ▁smooth ▁solution