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In [[Riemannian geometry]], '''Cheng's eigenvalue comparison theorem''' states in general terms that when a domain is large, the first [[Dirichlet eigenvalue]] of its [[Laplace–Beltrami operator]] is small.  This general characterization is not precise, in part because the notion of "size" of the domain must also account for its [[curvature]].<ref>{{harvnb|Chavel|1984|p=77}}</ref>  The theorem is due to {{harvtxt|Cheng|1975b}}. Using [[geodesic ball]]s, it can be generalized to certain tubular domains {{harv|Lee|1990}}.
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==Theorem==
Let ''M'' be a [[Riemannian manifold]] with dimension ''n'', and let ''B''<sub>''M''</sub>(''p'',&nbsp;''r'') be a geodesic ball centered at ''p'' with radius ''r'' less than the [[injectivity radius]] of ''p''&nbsp;∈&nbsp;''M''. For each real number ''k'', let ''N''(''k'') denote the [[simply connected]] [[space form]] of dimension ''n'' and constant [[sectional curvature]] ''k''. Cheng's eigenvalue comparison theorem compares the first eigenvalue λ<sub>1</sub>(''B''<sub>''M''</sub>(''p'',&nbsp;''r'')) of the Dirichlet problem in ''B''<sub>''M''</sub>(''p'',&nbsp;''r'') with the first eigenvalue in ''B''<sub>''N''(''k'')</sub>(''r'') for suitable values of ''k''. There are two parts to the theorem:
 
* Suppose that ''K''<sub>''M''</sub>, the [[sectional curvature]] of ''M'', satisfies
::<math>K_M\le k.</math>
:Then
::<math>\lambda_1\left(B_{N(k)}(r)\right) \le \lambda_1\left(B_M(p,r)\right).</math>
 
The second part is a comparison theorem for the [[Ricci curvature]] of ''M'':
 
* Suppose that the Ricci curvature of ''M'' satisfies, for every vector field ''X'',
::<math>\operatorname{Ric}(X,X) \ge k(n-1)|X|^2.</math>
:Then, with the same notation as above,
::<math>\lambda_1\left(B_{N(k)}(r)\right) \ge \lambda_1\left(B_M(p,r)\right).</math>
 
S.Y. Cheng used [[Barta's theorem]] to derive the eigenvalue comparison theorem. As a special case, if ''k''&nbsp;=&nbsp;−1 and inj(''p'')&nbsp;=&nbsp;∞, Cheng’s inequality becomes ''λ''<sup>*</sup>(''N'') ≥&nbsp;''λ''<sup>*</sup>(''H''<sup>&nbsp;''n''</sup>(−1)) which is [[McKean’s inequality]].<ref>{{harvnb|Chavel|1984|p=70}}</ref>
 
==See also==
*[[Comparison theorem]]
*[[Eigenvalue comparison theorem]]
 
==Notes==
{{reflist}}
 
==References==
*{{citation|title=On Cheng's eigenvalue comparison theorem|first1=G.P.|last1=Bessa|first2=J.F.|last2=Montenegro|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]|issn=0305-0041|volume=144|issue=3|year=2008|pages=673–682}}.
* {{citation|first=Isaac|last=Chavel|title=Eigenvalues in Riemannian geometry|series=Pure Appl. Math.|volume=115|publisher=[[Academic Press]]|year=1984}}.
*{{Citation | authorlink=Shiu-Yuen  Cheng | last1=Cheng | first1=Shiu Yuen | title=Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 2 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | mr=0378003  | year=1975a | chapter=Eigenfunctions and eigenvalues of Laplacian | pages=185–193}}
*{{citation|first=Shiu Yuen|last=Cheng|title=Eigenvalue Comparison Theorems and its Geometric Applications|journal=Math. Z.|volume=143|pages=289–297|year=1975b|doi=10.1007/BF01214381}}.
*{{citation|title=Eigenvalue Comparison for Tubular Domains|first=Jeffrey M.|last=Lee|journal=Proceedings of the American Mathematical Society|volume=109|year=1990|pages=843–848|jstor=2048228|doi=10.2307/2048228|issue=3|publisher=American Mathematical Society}}.
*{{citation|first=Henry|last=McKean|authorlink=Henry McKean|title=An upper bound for the spectrum of △ on a manifold of negative curvature|journal=[[Journal of Differential Geometry]]|volume=4|year=1970|pages=359&ndash;366}}.
*{{citation|first1=Jeffrey M.|last1=Lee|first2=Ken|last2=Richardson|title=Riemannian foliations and eigenvalue comparison|journal=Ann. Global Anal. Geom.|volume=16|year=1998|pages=497–525|doi=10.1023/A:1006573301591}}/
 
[[Category:Theorems in Riemannian geometry]]

Revision as of 00:03, 12 February 2014

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