. ▁In ▁one ▁dimension ▁higher , ▁it ▁is ▁known ▁that ▁among ▁all ▁closed ▁surfaces ▁in ▁ ▁ar ising ▁as ▁the ▁boundary ▁of ▁a ▁bounded ▁domain ▁of ▁unit ▁volume , ▁the ▁surface ▁area ▁is ▁minim ized ▁for ▁a ▁E uclidean ▁ball . ▁Syst ol ic ▁in equal ities ▁for ▁curves ▁on ▁surfaces . ▁Given ▁a ▁closed ▁surface , ▁its ▁syst ole ▁is ▁defined ▁to ▁be ▁the ▁smallest ▁length ▁of ▁any ▁non - cont ract ible ▁closed ▁curve ▁on ▁the ▁surface . ▁In ▁ 1 9 4 9 ▁Lo ew ner ▁proved ▁a ▁tor us ▁inequality ▁for ▁metrics ▁on ▁the ▁tor us , ▁namely ▁that ▁the ▁area ▁of ▁the ▁tor us ▁over ▁the ▁square ▁of ▁its ▁syst ole ▁is ▁bounded ▁below ▁by ▁, ▁with ▁equality ▁in ▁the ▁flat ▁( constant ▁curv ature ) ▁case . ▁A ▁similar ▁result ▁is ▁given ▁by ▁Pu ' s ▁inequality ▁for ▁the ▁real ▁project ive ▁plane ▁from ▁ 1 9 5 2 , ▁with ▁a ▁lower ▁bound ▁of ▁ ▁also ▁att ained ▁in ▁the ▁constant ▁curv ature ▁case . ▁For ▁the ▁Klein ▁bott le , ▁Bl atter ▁and ▁Bav ard ▁later ▁obtained ▁a ▁lower ▁bound ▁of ▁. ▁For ▁a ▁closed ▁surface ▁of ▁genus ▁, ▁Heb da ▁and ▁Bur ago ▁showed ▁that ▁the ▁ratio ▁is ▁bounded ▁below ▁by ▁. ▁Three ▁years ▁later ▁Mik h ail ▁G rom ov ▁found ▁a ▁lower ▁bound ▁given ▁by ▁a ▁constant ▁times ▁, ▁although ▁this ▁is ▁not ▁optimal . ▁As ym pt ot ically ▁sharp ▁upper ▁and ▁lower ▁bounds ▁given ▁by ▁constant ▁times ▁ ▁are ▁due