Difference between revisions of "Buser isoperimetric inequality"
From Encyclopedia of Mathematics
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| + | For a compact [[Riemannian manifold|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|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|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 | 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 | 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==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> P. Buser, "Über den ersten Eigenwert des Laplace–Operators auf kompakten Flächen" ''Comment. Math. Helvetici'' , '''54''' (1979) pp. 477–493</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> P. Buser, "A note on the isoperimetric constant" ''Ann. Sci. Ecole Norm. Sup.'' , '''15''' (1982) pp. 213–230</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> I. Chavel, "Riemannian geometry: A modern introduction" , Cambridge Univ. Press (1995)</td></tr></table> |
Latest revision as of 16:59, 1 July 2020
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) |
How to Cite This Entry:
Buser isoperimetric inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Buser_isoperimetric_inequality&oldid=50321
Buser isoperimetric inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Buser_isoperimetric_inequality&oldid=50321
This article was adapted from an original article by H. Kaul (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article