Hypercompact stellar system

2013-04-26T00:25:05Z

174.60.54.189:

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}}.

==Theorem==
Let ''M'' be a [[Riemannian manifold]] with dimension ''n'', and let ''B''''M''(''p'', ''r'') be a geodesic ball centered at ''p'' with radius ''r'' less than the [[injectivity radius]] of ''p'' ∈ ''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 λ1(''B''''M''(''p'', ''r'')) of the Dirichlet problem in ''B''''M''(''p'', ''r'') with the first eigenvalue in ''B''''N''(''k'')(''r'') for suitable values of ''k''. There are two parts to the theorem:

* Suppose that ''K''''M'', 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'' = −1 and inj(''p'') = ∞, Cheng’s inequality becomes ''λ''*(''N'') ≥ ''λ''*(''H'' ''n''(−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=359–366}}.
*{{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]]

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Hypercompact stellar system