Buser isoperimetric inequality

For a compact Riemannian manifold , let be the smallest positive eigenvalue of the Laplace–Beltrami operator (cf. also Laplace–Beltrami equation) of and define the isoperimetric constant of by

where varies over the compact hypersurfaces of which partition into two disjoint submanifolds , .

If the Ricci curvature of is bounded from below,

then the first eigenvalue has the upper bound

Note that a lower bound for the first eigenvalue, without any curvature assumptions, is given by the Cheeger inequality

References

[a1]	P. Buser, "Über den ersten Eigenwert des Laplace–Operators auf kompakten Flächen" Comment. Math. Helvetici , 54 (1979) pp. 477–493
[a2]	P. Buser, "A note on the isoperimetric constant" Ann. Sci. Ecole Norm. Sup. , 15 (1982) pp. 213–230
[a3]	I. Chavel, "Riemannian geometry: A modern introduction" , Cambridge Univ. Press (1995)