# Cheng's eigenvalue comparison theorem

Cheng's eigenvalue comparison theorem 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 "Tamanho" of the domain must also account for its curvature.[1] The theorem is due to Cheng (1975b) by Shiu-Yuen Cheng. Using geodesic balls, it can be generalized to certain tubular domains (Lee 1990).

Conteúdo 1 Teorema 2 Veja também 3 Referências 3.1 Citações 3.2 Bibliography Theorem Let M be a Riemannian manifold with dimension n, and let BM(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(BM(p, r)) of the Dirichlet problem in BM(p, r) with the first eigenvalue in BN(k)(r) for suitable values of k. There are two parts to the theorem: Suppose that KM, the sectional curvature of M, satisfies {estilo de exibição K_{M}leq k.} Então {lambda de estilo de exibição _{1}deixei(B_{N(k)}(r)certo)leq lambda _{1}deixei(B_{M}(p,r)certo).} 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, {nome do operador de estilo de exibição {Ric} (X,X)geq k(n-1)|X|^{2}.} Então, with the same notation as above, {lambda de estilo de exibição _{1}deixei(B_{N(k)}(r)certo)geq lambda _{1}deixei(B_{M}(p,r)certo).} 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 ^ Chavel 1984, p. 77 ^ Chavel 1984, p. 70 Bibliography Bessa, G.P.; Montenegro, J.F. (2008), "On Cheng's eigenvalue comparison theorem", Anais de Matemática da Sociedade Filosófica de Cambridge, 144 (3): 673–682, doi:10.1017/s0305004107000965, ISSN 0305-0041. Chavel, Isaque (1984), Autovalores na geometria Riemanniana, Pure Appl. Matemática., volume. 115, Imprensa Acadêmica. Cheng, Shiu Yuen (1975uma), "Eigenfunctions and eigenvalues of Laplacian", Geometria diferencial (Proc. Simpósios. Matemática pura., Volume. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 2, Providência, R.I.: Sociedade Americana de Matemática, pp. 185–193, MR 0378003 Cheng, Shiu Yuen (1975b), "Eigenvalue Comparison Theorems and its Geometric Applications", Matemática. Z., 143: 289-297, doi:10.1007/BF01214381. Lee, Jeffrey M. (1990), "Eigenvalue Comparison for Tubular Domains", Anais da American Mathematical Society, Sociedade Americana de Matemática, 109 (3): 843–848, doi:10.2307/2048228, JSTOR 2048228. McKean, Henry (1970), "An upper bound for the spectrum of △ on a manifold of negative curvature", Jornal de Geometria Diferencial, 4: 359–366. Lee, Jeffrey M.; Richardson, Ken (1998), "Riemannian foliations and eigenvalue comparison", Ana. Global Anal. Geom., 16: 497–525, doi:10.1023/UMA:1006573301591/ Categorias: Theorems in Riemannian geometryChinese mathematical discoveries

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