Cheng's eigenvalue comparison theorem

In

• Suppose that K_M, the sectional curvature of M, satisfies

${\displaystyle K_{M}\leq k.}$

Then

${\displaystyle \lambda _{1}\left(B_{N(k)}(r)\right)\leq \lambda _{1}\left(B_{M}(p,r)\right).}$

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,

${\displaystyle \operatorname {Ric} (X,X)\geq k(n-1)|X|^{2}.}$

Then, with the same notation as above,

${\displaystyle \lambda _{1}\left(B_{N(k)}(r)\right)\geq \lambda _{1}\left(B_{M}(p,r)\right).}$

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.^[2]

See also

• Comparison theorem
• Eigenvalue comparison theorem

References

Citations

Bibliography

• Bessa, G.P.; Montenegro, J.F. (2008), "On Cheng's eigenvalue comparison theorem",
  ISSN 0305-0041
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• Chavel, Isaac (1984), Eigenvalues in Riemannian geometry, Pure Appl. Math., vol. 115, Academic Press.
• MR 0378003
• Cheng, Shiu Yuen (1975b), "Eigenvalue Comparison Theorems and its Geometric Applications", Math. Z., 143: 289–297,
  doi:10.1007/BF01214381
  .
• Lee, Jeffrey M. (1990), "Eigenvalue Comparison for Tubular Domains", Proceedings of the American Mathematical Society, 109 (3), American Mathematical Society: 843–848,
  JSTOR 2048228
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• McKean, Henry (1970), "An upper bound for the spectrum of △ on a manifold of negative curvature", Journal of Differential Geometry, 4: 359–366.
• Lee, Jeffrey M.; Richardson, Ken (1998), "Riemannian foliations and eigenvalue comparison", Ann. Global Anal. Geom., 16: 497–525,
  doi:10.1023/A:1006573301591
  /