Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to po ...

, 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

Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named ...

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 can ...

. The theorem is due to by

Shiu-Yuen Cheng Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from ...

. 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, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...

with dimension ''n'', and let ''B''_''M''(''p'', ''r'') be a geodesic ball centered at ''p'' with radius ''r'' less than the

injectivity radius This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or provid ...

of ''p'' ∈ ''M''. For each real number ''k'', let ''N''(''k'') denote the

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...

space form Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consid ...

of dimension ''n'' and constant

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...

''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

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 In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...

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.

References

Citations

Bibliography

* . * . * * . * . * . * {{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

See also

References

Citations

Bibliography