Trends in °c Higher Education Press Partial Di®erential Equations and International Press ALM 10, pp. 241{264 Beijing-Boston Bounds of Eigenvalues on Riemannian Manifolds Jun Ling¤ Zhiqin Luy Abstract In this paper, we ¯rst give a short review of the eigenvalue estimates of Laplace operator and SchrÄodingeroperators. Then we discuss the evolution of eigenvalues along the Ricci flow, and two new bounds of the ¯rst eigenvalue using gradient estimates. 2000 Mathematics Subject Classi¯cation: 58J50, 35P15, 53C21. Keywords and Phrases: Eigenvalue, lower bounds, upper bounds, Riemannian manifolds. 1 Introduction In this paper, we discuss the eigenvalue estimates of Laplace operator and SchrÄodin- ger operators. Let (M; g) be an n-dimensional compact connected Riemannian manifold with or without boundary. Let ¢ be the Laplacian of the metric g = i n (gij)n£n. In local coordinates fx gi=1, µ ¶ 1 Xn @ p @ ¢ = p det(g) gij ; @xi @xj det(g) i;j=1 ij where the matrix (g ) is the inverse matrix of g = (gij). We consider the following three eigenvalue problems on the manifold (M; g). ¤Department of Mathematics, Utah Valley University, Orem, UT 84058, USA. Email: [email protected]. Research partially supported by the NSF under the program `The Geomet- ric Evolution Equations and Related Topics' at the MSRI at Berkeley and by the UVU School of Science and Health. yDepartment of Mathematics, University of California, Irvine, CA 92697, USA. Email: [email protected]. Research partially supported by NSF grant DMS 0347033. 242 J. Ling and Z. Lu Closed eigenvalue problem for the Laplacian. @M = ;. Find all real numbers ¸ for which there are nontrivial solutions u 2 C2(M) to the equations ¡¢u = ¸u: (1.1) Dirichlet eigenvalue problem for the Laplacian. @M 6= ;. Find all real num- bers ¸ for which there are nontrivial solutions u 2 C2(M)\C0(M) to (1.1), subject to the boundary condition u = 0 on @M: Neumann eigenvalue problem for the Laplacian. @M 6= ;. Find all real numbers ¸ for which there are nontrivial solutions u 2 C2(M) \ C1(M) to (1.1), subject to the boundary condition @ u = 0 on @M; @º where º is an outward unit normal vector ¯elds of @M. We may consider these three eigenvalue problems for other operators, for instance SchrÄodingeroperators, in similar ways. The spectrum of the Laplacian of a closed manifold consists of pure point 1 spectrum f¸igi=0 that can be arranged in the order 0 = ¸0 < ¸1 · ¸2 · ¸3 · ¢ ¢ ¢ ¡! 1: For a manifold with boundary, in the similar way, we can arrange the Dirichlet and Neumann eigenvalues like above (in Dirichlet case, the smallest eigenvalue is always positive). The eigenvalues can be characterized by the variational principle. For u 2 1 C (M), let Z Z kuk2 = juj2 + jruj2: M M 1 1 1 1 1 Let H (M), H0 (M) be the completion of C (M), C0 (M)(C functions with compact support in M) with respect to the above norm k ¢ k. 2 Let fuig be an orthonormal basis for L (M) with ¡¢ui = ¸iui, ui 2 1 1 1 C (M) \ H, where H = H (M) or H0 (M) depending on the boundary con- ditions. Then we have the following variational principal: R 2 M jruj ¸i =R inf R : uu =0; k<i 2 M k M juj It is an interesting problem to study the distribution of eigenvalues and eigen- functions since they reveal important relations between geometry of the manifold and analysis. Such a problem roots in classical analysis, physics and geomet- ric analysis. Early works in the ¯eld include Weyl's asymptotic formula [72], Courant's nodal domain theorem [28], and so on. By study the heat kernel of the Laplacian, Weyl was able to prove that n=2 n N(¸) » !n(volM)¸ =(2¼) Bounds of Eigenvalues 243 n as ¸ ! +1, where !n is the volume of the unit solid ball in R , and N(¸) is the number of eigenvalues less than or equal to ¸, multiplicity counted. Courant's nodal domain theorem states that the number of nodal domains of the k-th eigen- function is less than or equal to k + 1. These theorems are of fundamental impor- tance in understanding the Laplacian. Computing the lower order terms of the asymptotic formula, and ¯nding the limit distribution of the nodal sets are the two ongoing research projects. In this paper, we only discuss the eigenvalue estimates cotangent to the gradi- ent estimates. We refer to [65] for more details. The paper is organized as follows. In x 2, we discuss the estimates of the ¯rst non-zero eigenvalue, ¸1. In x 3, we discuss various bounds of eigenvalues, and eigenvalues on Riemann surfaces, and so on. Finally, in x 4, we give some new lower bounds of the ¯rst eigenvalue and the gap of the ¯rst two eigenvalues using the gradient estimates. 2 Bounds on the ¯rst eigenvalue 2.1 Lichnerowicz bound and the gradient estimates We ¯rst discuss the lower bounds for the ¯rst eigenvalue ¸1 of the Laplacian on an n-dimensional compact Riemannian manifold (M; g) with positive Ricci curvature Ric ¸ (n ¡ 1)Kg with constant K > 0: (2.2) If @M = ;, the classical Lichnerowicz theorem [42] for the ¯rst eigenvalue of the Laplacian gives ¸1 ¸ nK: (2.3) We sketch the proof here. Take a nontrivial solution u to (1.1). By the Schwarz inequality, we have 1 ¸ jr2uj2 ¸ (¢u)2 = ¡ u¢u: (2.4) n n The Bochner-Lichnerowicz formula gives 1 ¢(jruj2) = jr2uj2 + rur¢u + Ric(ru; ru): 2 Using the Ricci lower bound (2.2) and (2.4), we have 1 ¸ ¢(jruj2) ¸ ¡ u¢u ¡ ¸jruj2 + (n ¡ 1)Kjruj2: 2 n Integrating the above inequality over M and using the divergence theorem, we have Z · ¸ ¸ 0 ¸ ¡ ¸ + (n ¡ 1)K jruj2 n M Z n ¡ 1 = (¡¸ + nK) jruj2; n M 244 J. Ling and Z. Lu which gives Inequality (2.3). In 1962, M. Obata [58] proved that the equality in (2.3) holds if and only if the manifold M is isometric to the n-sphere of constant sectional curvature K. This can be proved easily as follows by S.Y. Cheng's generalized Toponogov theorem [25], which states that if an n-dimensional compact Riemannian manifoldp has positive Ricci lower bound (n ¡ 1)K > 0, and maximal diameter d(M) = ¼= K, then it is isometric to the n-sphere. In fact, if ¸1 = nK, then using the Bochner- Lichnerowicz formula for one more time, we will get that jruj2 +Ku2 is a constant. Let max u2 = 1, then we must have jruj p p = K: 1 ¡ u2 Moreover, we must have max u = 1 and min u = ¡1. Let p; q be two points on M with u(p) = 1 and u(q) = ¡1. Integrating the above equality along the geodesic connecting p; q, we get p d(p; q) ¸ ¼= K: Bonnet-Myersp Therorem says that the opposite inequality holds. Thus d(M) = ¼= K and by the theorem of Cheng, we proved that M has to be an n-sphere. The results of Lichnerowicz and Obata for closed manifold were generalized to compact manifold with boundary. To state those results, we ¯rst make the following de¯nition: De¯nition 2.1. A manifold with boundary is called weakly convex if the mean curvature of the boundary is nonnegative with respect to the outward normal of the boundary, convex if the second fundamental form of the boundary is nonnegative with respect to the outward normal of the boundary, strongly convex if the second fundamental form of the boundary is positive with respect to the outward normal of the boundary. R. Reilly [64] proved that the same Lichnerowicz-type lower bound holds for the ¯rst Dirichlet eigenvalue of a manifold with weakly convex boundary and that the equality holds if and only if M isp isometric to a closed hemisphere of the Euclidean sphere of Sn(K) of radius 1= K. J.F. Escobar [30] proved the similar result for the ¯rst Neumann eigenvalue of a manifold with convex boundary. For a closed Riemannian manifold with nonnegative Ricci curvature, Li-Yau [40] introduced the method of gradient estimate and derived a lower bound for the ¯rst eigenvalue of the Laplacian in terms of the diameter d of the manifold. They proved ¼2 ¸ ¸ : 1 2d2 Zhong and Yang [80] re¯ned Li-Yau's gradient estimate and obtained the bound ¼2 ¸ ¸ : (2.5) 1 d2 Recently, based on the strong maximum principle, Hang and Wang [32] proved that actually strict inequality holds for dimension ¸ 2. This is a very interesting result. Bounds of Eigenvalues 245 Peter Li conjectured (cf. [73]) that, when the Ricci curvature of the manifold is positive, then Zhong-Yang's estimate can be further sharpened. In fact, the ¯rst ¼2 eigenvalue of the sphere of dimension n is n d2 , which is n-times the Zhong-Yang's estimate. In view of the case on spheres, one version of Li's conjecture is as follows: Conjecture 2.2 (P. Li). For a compact manifold with Ric ¸ (n ¡ 1)K > 0, the ¯rst eigenvalues ¸1, with respect to the closed, the Neumann, or the Dirichlet Laplacian satis¯es ¼2 ¸ ¸ + (n ¡ 1)K: 1 d2 ¼2 Note that by the theorem of Myers, we always have d2 ¸ K. Thus the conjecture, if true, will give a common generalization of the result of Lichnerowicz's and the one obtained by gradient estimate. In this direction, D.G. Yang [73] proved that the ¯rst Dirichlet eigenvalue of the Laplacian satis¯es ¼2 1 ¸1 ¸ + (n ¡ 1)K; d~2 4 if the manifold has weakly convex boundary, where d~ is the interior diameter of the manifold (the diameter of the largest inscribed ball in the manifold).