Buser isoperimetric inequality

For a compact Riemannian manifold $M = M ^ { n }$, let $\lambda _ { 1 } = \lambda _ { 1 } ( M )$ be the smallest positive eigenvalue of the Laplace–Beltrami operator (cf. also Laplace–Beltrami equation) of $M$ and define the isoperimetric constant of $M$ by

\begin{equation*} h = h ( M ) = \operatorname { inf } _ { \Gamma } \frac { \operatorname { Vol } ( \Gamma ) } { \operatorname { min } \{ \operatorname { Vol } ( M _ { 1 } ) , \text { Vol } ( M _ { 2 } ) \} }, \end{equation*}

where $\Gamma$ varies over the compact hypersurfaces of $M$ which partition $M$ into two disjoint submanifolds $M _ { 1 }$, $M _ { 2 }$.

If the Ricci curvature of $M$ is bounded from below,

\begin{equation*} \operatorname { Ric } \geq - ( n - 1 ) \delta ^ { 2 } , \quad \delta \geq 0, \end{equation*}

then the first eigenvalue has the upper bound

\begin{equation*} \lambda _ { 1 } \leq 2 ( n - 1 ) \delta h + 10 h ^ { 2 }. \end{equation*}

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

\begin{equation*} \frac { 1 } { 4 } h ^ { 2 } \leq \lambda _ { 1 }. \end{equation*}

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)