In Riemannian geometry, Cheng's eigenvalue comparison theorem states in general terms that when a domain is large, the first

Dirichlet eigenvalue In mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can hear the shape of a drum is: given the Dirichlet eigenvalues, what features of the shape of t ...

of its

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is small. This general characterization is not precise, in part because the notion of "size" of the domain must also account for its

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...

. The theorem is due to by Shiu-Yuen Cheng. Using geodesic balls, it can be generalized to certain tubular domains .

Theorem

Let ''M'' be a

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

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

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space form Space is the boundless Three-dimensional space, three-dimensional extent in which Physical body, objects and events have relative position (geometry), position and direction (geometry), direction. In classical physics, physical space is often ...

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 ::

K_M\le k.

:Then ::

\lambda_1\left(B_(r)\right) \le \lambda_1\left(B_M(p,r)\right).

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'', ::

\operatorname(X,X) \ge k(n-1), X, ^2.

:Then, with the same notation as above, ::

\lambda_1\left(B_(r)\right) \ge \lambda_1\left(B_M(p,r)\right).

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.

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* . * . * * . * . * . * {{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/ Theorems in Riemannian geometry Chinese mathematical discoveries

Theorem

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