▁on ▁this ▁conject ure ▁was ▁in ▁ 1 9 2 4 ▁by ▁Hans ▁Hamb urger , ▁who ▁noted ▁that ▁it ▁follows ▁from ▁the ▁following ▁stronger ▁claim : ▁the ▁half - integer ▁val ued ▁index ▁of ▁the ▁principal ▁curv ature ▁fol iation ▁of ▁an ▁isolated ▁um bil ic ▁is ▁at ▁most ▁one . ▁Zero ▁Gaussian ▁curv ature : ▁a ▁complete ▁surface ▁in ▁ ▁with ▁zero ▁Gaussian ▁curv ature ▁must ▁be ▁a ▁cyl inder ▁or ▁a ▁plane . ▁Hil bert ' s ▁theorem ▁( 1 9 0 1 ): ▁no ▁complete ▁surface ▁with ▁constant ▁negative ▁curv ature ▁can ▁be ▁imm ers ed ▁is omet r ically ▁in ▁. ▁ ▁The ▁Will more ▁conject ure . ▁This ▁conject ure ▁states ▁that ▁the ▁integral ▁of ▁the ▁square ▁of ▁the ▁mean ▁curv ature ▁of ▁a ▁tor us ▁imm ers ed ▁in ▁ ▁should ▁be ▁bounded ▁below ▁by ▁. ▁It ▁is ▁known ▁that ▁the ▁integral ▁is ▁Mo eb ius ▁invariant . ▁It ▁was ▁solved ▁in ▁ 2 0 1 2 ▁by ▁Fernando ▁C od á ▁Mar ques ▁and ▁André ▁Ne ves . ▁Is oper imet ric ▁in equal ities . ▁In ▁ 1 9 3 9 ▁Schmidt ▁proved ▁that ▁the ▁classical ▁is oper imet ric ▁inequality ▁for ▁curves ▁in ▁the ▁E uclidean ▁plane ▁is ▁also ▁valid ▁on ▁the ▁sphere ▁or ▁in ▁the ▁hyper bol ic ▁plane : ▁namely ▁he ▁showed ▁that ▁among ▁all ▁closed ▁curves ▁bound ing ▁a ▁domain ▁of ▁fixed ▁area , ▁the ▁per imeter ▁is ▁minim ized ▁by ▁when ▁the ▁curve ▁is ▁a ▁circle ▁for ▁the ▁metric