The researches of the eigenvalue problems of Laplace operator have many applications to physics，they relate to the levels of power belonging to the particles when particles across certain fields.By MaximumMinimum Principle，we can formulate the eigenvalues theoretically.Based on the eigenvalue problem of Laplace operator on a spherical band in three dimensional space with Robin boundary conditions，we formulated the first eigenvalue by virtue of Courant nodal domain theorem and MaximumMinimum Principle，then by this formulation，we proved that the first eigenvalue would be maximum when the band is symmetric about the equator (the area of the band is constant)，also，if the area of the band is smaller than or equals to 2π，then the first eigenvalue would monotonously increase when the band moves to the equator.