rivial ▁eigen value ▁of ▁the ▁Lap lace – Bel tr ami ▁operator ▁on ▁M ▁to ▁h ( M ). ▁This ▁proved ▁to ▁be ▁a ▁very ▁influ ential ▁idea ▁in ▁Riemann ian ▁geometry ▁and ▁global ▁analysis ▁and ▁inspired ▁an ▁analog ous ▁theory ▁for ▁graphs . ▁ ▁Definition ▁▁ ▁Let ▁M ▁be ▁an ▁n - dimensional ▁closed ▁Riemann ian ▁manifold . ▁Let ▁V ( A ) ▁denote ▁the ▁volume ▁of ▁an ▁n - dimensional ▁sub man if old ▁A ▁and ▁S ( E ) ▁denote ▁the ▁n − 1 - dimensional ▁volume ▁of ▁a ▁sub man if old ▁E ▁( comm only ▁called ▁" area " ▁in ▁this ▁context ). ▁The ▁Che eg er ▁is oper imet ric ▁constant ▁of ▁M ▁is ▁defined ▁to ▁be ▁▁▁▁ ▁where ▁the ▁inf imum ▁is ▁taken ▁over ▁all ▁smooth ▁n − 1 - dimensional ▁sub man if olds ▁E ▁of ▁M ▁which ▁divide ▁it ▁into ▁two ▁dis joint ▁sub man if olds ▁A ▁and ▁B . ▁Is oper imet ric ▁constant ▁may ▁be ▁defined ▁more ▁generally ▁for ▁non comp act ▁Riemann ian ▁manif olds ▁of ▁finite ▁volume . ▁ ▁Che eg er ' s ▁inequality ▁▁ ▁The ▁Che eg er ▁constant ▁h ( M ) ▁and ▁ ▁the ▁smallest ▁positive ▁eigen value ▁of ▁the ▁Lap la cian ▁on ▁M , ▁are ▁related ▁by ▁the ▁following ▁fundamental ▁inequality ▁proved ▁by ▁Jeff ▁Che eg er : ▁▁▁▁ ▁This ▁inequality ▁is ▁optimal ▁in ▁the ▁following ▁sense : ▁for ▁any ▁h ▁> ▁ 0 , ▁natural ▁number ▁k ▁and ▁ ε ▁> ▁