# 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 "Taille" 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).

Contenu 1 Théorème 2 Voir également 3 Références 3.1 Citations 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, satisfait {style d'affichage K_{M}leq k.} Alors {style d'affichage lambda _{1}la gauche(B_{N(k)}(r)droit)leq lambda _{1}la gauche(B_{M}(p,r)droit).} 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, {nom de l'opérateur de style d'affichage {Ric} (X,X)geq k(n-1)|X|^{2}.} Alors, with the same notation as above, {style d'affichage lambda _{1}la gauche(B_{N(k)}(r)droit)geq lambda _{1}la gauche(B_{M}(p,r)droit).} 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 ^ Chaval 1984, p. 70 Bibliography Bessa, G.P.; Montenegro, J.F. (2008), "On Cheng's eigenvalue comparison theorem", Actes mathématiques de la Cambridge Philosophical Society, 144 (3): 673–682, est ce que je:10.1017/s0305004107000965, ISSN 0305-0041. Chavel, Isaac (1984), Valeurs propres en géométrie riemannienne, Pure Appl. Math., volume. 115, Presse académique. Cheng, Shiu Yuen (1975un), "Eigenfunctions and eigenvalues of Laplacian", Géométrie différentielle (Proc. Colloques. Mathématiques pures., Volume. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 2, Providence, R.I.: Société mathématique américaine, pp. 185–193, M 0378003 Cheng, Shiu Yuen (1975b), "Eigenvalue Comparison Theorems and its Geometric Applications", Math. Z., 143: 289–297, est ce que je:10.1007/BF01214381. Lee, Jeffrey M. (1990), "Eigenvalue Comparison for Tubular Domains", Actes de l'American Mathematical Society, Société mathématique américaine, 109 (3): 843–848, est ce que je:10.2307/2048228, JSTOR 2048228. McKean, Henry (1970), "An upper bound for the spectrum of △ on a manifold of negative curvature", Journal de géométrie différentielle, 4: 359–366. Lee, Jeffrey M.; Richardson, Ken (1998), "Riemannian foliations and eigenvalue comparison", Anne. Global Anal. Geom., 16: 497–525, est ce que je:10.1023/UN:1006573301591/ Catégories: Theorems in Riemannian geometryChinese mathematical discoveries

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