1 7 7 6 ▁Jean ▁Bapt iste ▁Me us nier ▁showed ▁that ▁the ▁differential ▁equation ▁derived ▁by ▁Lag range ▁was ▁equivalent ▁to ▁the ▁van ishing ▁of ▁the ▁mean ▁curv ature ▁of ▁the ▁surface : ▁ ▁Min imal ▁surfaces ▁have ▁a ▁simple ▁interpretation ▁in ▁real ▁life : ▁they ▁are ▁the ▁shape ▁a ▁soap ▁film ▁will ▁assume ▁if ▁a ▁wire ▁frame ▁sh aped ▁like ▁the ▁curve ▁is ▁di pped ▁into ▁a ▁soap ▁solution ▁and ▁then ▁carefully ▁lifted ▁out . ▁The ▁question ▁as ▁to ▁whether ▁a ▁minimal ▁surface ▁with ▁given ▁boundary ▁exists ▁is ▁called ▁Pl ate au ' s ▁problem ▁after ▁the ▁Belg ian ▁phys ic ist ▁Joseph ▁Pl ate au ▁who ▁carried ▁out ▁experiments ▁on ▁soap ▁films ▁in ▁the ▁mid - n in ete enth ▁century . ▁In ▁ 1 9 3 0 ▁J esse ▁Douglas ▁and ▁Tib or ▁Rad ó ▁gave ▁an ▁aff irm ative ▁answer ▁to ▁Pl ate au ' s ▁problem ▁( D oug las ▁was ▁awarded ▁one ▁of ▁the ▁first ▁Field s ▁med als ▁for ▁this ▁work ▁in ▁ 1 9 3 6 ). ▁ ▁Many ▁explicit ▁examples ▁of ▁minimal ▁surface ▁are ▁known ▁explicitly , ▁such ▁as ▁the ▁c aten oid , ▁the ▁hel ico id , ▁the ▁Sch erk ▁surface ▁and ▁the ▁En ne per ▁surface . ▁There ▁has ▁been ▁extensive ▁research ▁in ▁this ▁area , ▁summar ised ▁in ▁. ▁In ▁particular ▁a ▁result ▁of ▁Os ser man ▁shows ▁that ▁if ▁a ▁minimal ▁surface ▁is ▁non - plan ar , ▁then ▁its ▁image ▁under ▁the ▁G auss ▁map ▁is ▁dense