Eigenvalues of Collapsing Domains and Drift Laplacian

Transcription

1 Eigenvalues of Collapsing Domains and Drift Laplacian Zhiqin Lu Dedicate to Professor Peter Li on his 60th Birthday Department of Mathematics, UC Irvine, Irvine CA January 17, 2012 Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 1/43

2 Let M be a closed manifold. By the Hodge Theorem, the spectrum of the Laplacian is made from eigenvalues of finite multiplicity. 0 = λ 0 < λ 1 λ 2 λ k. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 2/43

3 Let M be a closed manifold. By the Hodge Theorem, the spectrum of the Laplacian is made from eigenvalues of finite multiplicity. 0 = λ 0 < λ 1 λ 2 λ k. In particular, we have λ 1 > 0. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 2/43

4 The first eigenvalue λ 1 plays a very important role in differential geometry, and one of the important questions is to give a lower bound estimate of λ 1 using geometric quantities readily available. λ 1 > c(n, d, V, R, ) > 0. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 3/43

5 Review of gradient estimates Let ϕ be a smooth function on a closed manifold M. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 4/43

6 Review of gradient estimates Let ϕ be a smooth function on a closed manifold M. Let H = 1 2 ϕ 2 + F (ϕ), Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 4/43

7 Let x 0 be the maximum point. Then H(x 0 ) = 0, 0 H(x 0 ) Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 5/43

8 Let x 0 be the maximum point. Then H(x 0 ) = 0, 0 H(x 0 ) Therefore 0 2 ϕ 2 + ϕ ( ϕ) + Ric M ( ϕ, ϕ) + F (ϕ) ϕ + F (ϕ) ϕ 2. at x 0. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 5/43

9 Li s estimate Assume that the Ricci curvature is nonnegative. Let ϕ = λϕ. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 6/43

10 Li s estimate Assume that the Ricci curvature is nonnegative. Let ϕ = λϕ. Let F (x) = 1(λ + 2 ε)x2. Let H = 1 2 ϕ (λ + ε)ϕ2 2 Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 6/43

11 Li s estimate Assume that the Ricci curvature is nonnegative. Let ϕ = λϕ. Let F (x) = 1(λ + 2 ε)x2. Let H = 1 2 ϕ (λ + ε)ϕ2 2 Then ϕ j ϕ ij = (λ + ε)ϕ i 2 ϕ 2 λ(λ + ε)ϕ 2 + ε ϕ 2 0 Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 6/43

12 Li s estimate Assume that the Ricci curvature is nonnegative. Let ϕ = λϕ. Let F (x) = 1(λ + 2 ε)x2. Let H = 1 2 ϕ (λ + ε)ϕ2 2 Then ϕ j ϕ ij = (λ + ε)ϕ i 2 ϕ 2 λ(λ + ε)ϕ 2 + ε ϕ 2 0 If ϕ(x 0 ) 0, then 2 ϕ 2 ϕ i ϕ j ϕ ij 2 / ϕ 4 = (λ + ε)ϕ 2 Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 6/43

13 Li s estimate Assume that the Ricci curvature is nonnegative. Let ϕ = λϕ. Let F (x) = 1(λ + 2 ε)x2. Let H = 1 2 ϕ (λ + ε)ϕ2 2 Then ϕ j ϕ ij = (λ + ε)ϕ i 2 ϕ 2 λ(λ + ε)ϕ 2 + ε ϕ 2 0 If ϕ(x 0 ) 0, then 2 ϕ 2 ϕ i ϕ j ϕ ij 2 / ϕ 4 = (λ + ε)ϕ 2 A contradiction. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 6/43

14 Assume that max ϕ = 1. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 7/43

15 Assume that max ϕ = 1. Then ϕ 2 + λϕ 2 λ Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 7/43

16 Assume that max ϕ = 1. Then ϕ 2 + λϕ 2 λ arcsin ϕ λ Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 7/43

17 Assume that max ϕ = 1. Then ϕ 2 + λϕ 2 λ arcsin ϕ λ Let ϕ(x 1 ) = 0, ϕ(x 2 ) = 1. Integrating from x 1 to x 2, we get π 2 λd that is λ π2 4d 2. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 7/43

18 Remark The choice of F is highly technical. Zhong and Yang chose F as F (x) = 1 x 2 + a( 4 π (arcsin x + x 1 x 2 2x)), and proved λ 1 π2 d 2. (S-Y Cheng, D. Yang, J. Ling, Hang-Wang, etc) Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 8/43

19 Similar estimates can be obtained for compact manifolds with (convex) boundary. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 9/43

20 Let M be a compact manifold with boundary. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 10/43

21 Let M be a compact manifold with boundary. Let λ j be the Dirichlet eigenvalues of M. is called the fundamental gap. λ 2 λ 1 Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 10/43

22 Let M be a compact manifold with boundary. Let λ j be the Dirichlet eigenvalues of M. λ 2 λ 1 is called the fundamental gap. We have λ 2 λ 1 > 0 Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 10/43

23 This is joint with Julie Rowlett. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 11/43

24 A special case Theorem Let φ be an eigenfunction of the first Dirichlet eigenvalue. M ε := {(x, y) x M, 0 y εφ 2 (x)} M R +. Let µ 1 (ε) be the first Neumann eigenvalue of M ε with respect to := g + 2 y, Then µ 1 (ε) λ 2 λ 1 as ε 0. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 12/43

25 We conclude that the gap problem is a sub-problem of estimating the first Neumann eigenvalue. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 13/43

26 Proof: variational principle. Let ϕ 1, ϕ 2 be the eigenfunctions with respect to the eigenvalues λ 1, λ 2. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 14/43

27 Proof: variational principle. Let ϕ 1, ϕ 2 be the eigenfunctions with respect to the eigenvalues λ 1, λ 2. Using ϕ 2 /ϕ 1 to be the testing function for M ε, we obtain µ 1 (ε) M ε (ϕ 2 /ϕ 1 ) 2 M ε (ϕ 2 /ϕ 1 ) 2 = λ 2 λ 1 Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 14/43

28 Let ψ ε be an eigenfunction of M ε with respect to µ 1 (ε). Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 15/43

29 Let ψ ε be an eigenfunction of M ε with respect to µ 1 (ε). To obtain the other side of the inequality, for any 0 r ε, we take ψ(x, rφ(x)) as the testing functions and average over r. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 15/43

30 M ε is never a convex domain. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 16/43

31 M ε is never a convex domain. Neumann first eigenvalue estimate. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 16/43

32 M ε is never a convex domain. Neumann first eigenvalue estimate. Example. A domain of two big disks connecting by a narrow channel: very small first eigenvalue. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 16/43

33 The van de Berg and Yau Conjecture Let Ω be a compact domain of R n with convex boundary. Then λ 2 λ 1 > 3π2 d 2, where d is the diameter of the domain Ω. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 17/43

34 History 1 In 1985, Singer-Wong-Yau-Yau proved that λ 2 λ 1 π2 4d 2 Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 18/43

35 History 1 In 1985, Singer-Wong-Yau-Yau proved that λ 2 λ 1 π2 4d 2 2 The conjecture was proved by B. Andrews and J. Cutterbuck in Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 18/43

36 The first Neumann eigenvalue of M ε is not so small: µ 1 (ε) 3π2 d 2 O(ε). Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 19/43

37 Bakry-Émery Geometry A Bakry-Émery manifold is a triple (M, g, e φ dv g ), where (M, g) is a Riemannian manifold and φ is a function. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 20/43

38 Bakry-Émery Geometry A Bakry-Émery manifold is a triple (M, g, e φ dv g ), where (M, g) is a Riemannian manifold and φ is a function.the Bakry-Émery Ricci curvature is defined to be Ric = Ric + Hess(φ), Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 20/43

39 Bakry-Émery Geometry A Bakry-Émery manifold is a triple (M, g, e φ dv g ), where (M, g) is a Riemannian manifold and φ is a function.the Bakry-Émery Ricci curvature is defined to be Ric = Ric + Hess(φ), and the Bakry-Émery Laplacian is φ = φ. The operator can be extended as a self-adjoint operator with respect to the weighted measure e φ dv g ; it is also known as a drifting or drift Laplacian. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 20/43

40 Relation to Bakry-Émery geometry Let u 1 = λ 1 u 1, u 2 = λ 2 u 2. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 21/43

41 Relation to Bakry-Émery geometry Let u 1 = λ 1 u 1, u 2 = λ 2 u 2. Then u 2 = (λ 2 λ 1 ) u log u 1 u 2 u 1 u 1 u 1 Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 21/43

42 Relation to Bakry-Émery geometry Let u 1 = λ 1 u 1, u 2 = λ 2 u 2. Then u 2 = (λ 2 λ 1 ) u log u 1 u 2 u 1 u 1 u 1 Or in other word, φ u 2 u 1 = (λ 2 λ 1 ) u 2 u 1 Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 21/43

43 Relation to Bakry-Émery geometry Let u 1 = λ 1 u 1, u 2 = λ 2 u 2. Then u 2 = (λ 2 λ 1 ) u log u 1 u 2 u 1 u 1 u 1 Or in other word, Variational Principle φ u 2 u 1 = (λ 2 λ 1 ) u 2 u 1 λ 2 λ 1 = inf f 2 e φ f 2 e φ, where f are functions such that fe φ = 0. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 21/43

44 Theorem Let (M, g, φ) be a compact Bakry-Émery manifold. Let M ε := {(x, y) x M, 0 y εe φ(x) } M R +. Let {µ k } k=0 be the eigenvalues of the Bakry-Émery Laplacian on M. If M, assume the Neumann boundary condition. Let µ k (ε) be the eigenvalues of M ε for := g + 2 y, where g is the Laplacian with respect to the Riemannian metric g on M. Then µ k (ε) = µ k + O(ε 2 ) for k 0. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 22/43

45 Let M be a convex manifold. Let ϕ be a Neumann eigenfunction of M ε. Let ψ = ϕ(x, 0). Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 23/43

46 Let M be a convex manifold. Let ϕ be a Neumann eigenfunction of M ε. Let ψ = ϕ(x, 0). Theorem (New Maximum Principle) With the above notations, we have at (x 0, 0) o(1) 2 ψ 2 + ψ ( ϕ) + Ric ( ψ, ψ) + F (ψ) ϕ + F (ψ) ψ 2. (Recall that = + 2 y) Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 23/43

47 Key Lemmas Let H = 1 2 ϕ 2 + F (ϕ). Let (x 0, 0) be the maximum point of H on y = 0. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 24/43

48 Lemma 2 H y 2 = 2 log f( ψ, ψ) + ( ) 2 2 ϕ + o(1). y 2 Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 25/43

49 Lemma As ε 0, 2 ϕ log f(x) ψ = o(1), y2 ( ψ, 2 ϕ y ) 2 ( 2 log f)( ψ, ψ) 2 ψ( ψ, log f) = o(1). Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 26/43

50 Neumann boundary conditions We have ϕ (x, 0) = 0 y ϕ (x, εf(x)) ε f(x) ϕ(x, εf(x)) = 0 y Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 27/43

51 Neumann boundary conditions We have ϕ (x, 0) = 0 y ϕ (x, εf(x)) ε f(x) ϕ(x, εf(x)) = 0 y By the mean-value theorem, we have 2 ϕ (x, ξ(x)) = O(ε ϕ ) y2 Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 27/43

52 Neumann boundary conditions We have ϕ (x, 0) = 0 y ϕ (x, εf(x)) ε f(x) ϕ(x, εf(x)) = 0 y By the mean-value theorem, we have 2 ϕ (x, ξ(x)) = O(ε ϕ ) y2 C α estimate for ϕ yy in order to get ϕ yy (x, 0). Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 27/43

53 It is not hard to write down the eigenfunctions formally. Let ϕ be a Neumann eigenfunction of M ε with eigenvalue λ. Let ϕ = y k ϕ k, k=0 where ϕ k are functions on M. Then we have for all k 0. ϕ k + λϕ k + (k + 1)(k + 2)ϕ k+2 = 0 Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 28/43

54 Since ϕ/ y = 0, we have ϕ 1 = 0 and hence ϕ 2k+1 = 0 for all k. Let Aϕ = ϕ λϕ. Then ϕ 2k+2 = Aϕ 2k (2k + 1)(2k + 2) = Ak+1 ϕ 0 (2k + 2)!. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 29/43

55 Since ϕ/ y = 0, we have ϕ 1 = 0 and hence ϕ 2k+1 = 0 for all k. Let Aϕ = ϕ λϕ. Then ϕ 2k+2 = Formally, we have Aϕ 2k (2k + 1)(2k + 2) = Ak+1 ϕ 0 (2k + 2)!. ϕ = k=0 y k A k (2k)! = cosh(y A)ϕ 0. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 29/43

56 The differential equation for ϕ 0 follows from the Neumann boundary condition A sinh(εf(x) A)ϕ0 ε f (cosh(y A)ϕ 0 ) = 0. y=εf(x) We are not able to prove the full regularity of the above equation at this moment. But a partial solution, namely, a good approximation to the eigenfunctions, is enough for our application. Very Roughly speaking, we proved ϕ = ϕ 0 + y 2 ϕ 2 + O(ε 3 ). Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 30/43

57 The estimates 1 Let Ω be a bounded domain in R n with smooth boundary; Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 31/43

58 The estimates 1 Let Ω be a bounded domain in R n with smooth boundary; 2 Let f be a positive smooth function on Ω which is constant near the boundary Ω; Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 31/43

59 The estimates 1 Let Ω be a bounded domain in R n with smooth boundary; 2 Let f be a positive smooth function on Ω which is constant near the boundary Ω; 3 Define Ω ε = {(x, y) R n+1 x Ω, 0 y εf(x)} for any ε; Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 31/43

60 The estimates 1 Let Ω be a bounded domain in R n with smooth boundary; 2 Let f be a positive smooth function on Ω which is constant near the boundary Ω; 3 Define Ω ε = {(x, y) R n+1 x Ω, 0 y εf(x)} for any ε; 4 Let ϕ be a Neumann eigenfunction of Ω ε such that ϕ 2 = ε. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 31/43

61 The estimates 1 Let Ω be a bounded domain in R n with smooth boundary; 2 Let f be a positive smooth function on Ω which is constant near the boundary Ω; 3 Define Ω ε = {(x, y) R n+1 x Ω, 0 y εf(x)} for any ε; 4 Let ϕ be a Neumann eigenfunction of Ω ε such that ϕ 2 = ε. 5 Then 3 ϕ x j y 2 = O(1), etc Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 31/43

62 Schauder estimate 1 Consider the estimate 3 ϕ x j y 2 = O(1); Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 32/43

63 Schauder estimate 1 Consider the estimate 3 ϕ x j y 2 2 By the normalization ϕ 2 = ε; = O(1); Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 32/43

64 Schauder estimate 1 Consider the estimate 3 ϕ x j y 2 2 By the normalization ϕ 2 = ε; 3 ϕ should be bounded; = O(1); Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 32/43

65 Schauder estimate 1 Consider the estimate 3 ϕ x j y 2 2 By the normalization ϕ 2 = ε; 3 ϕ should be bounded; = O(1); 4 Meanvalue theorem suggests that ϕ yy 1/ε 2. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 32/43

66 Let ψ be a Bakry-Émery Neumann eigenfunction. Define the function η := φ ψ, and on M ε, let U := ψ y2 η. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 33/43

67 Let ψ be a Bakry-Émery Neumann eigenfunction. Define the function η := φ ψ, and on M ε, let U := ψ y2 η. ψ + η = µψ. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 33/43

68 Since ψ and η are independent of y, U = µψ y2 η = µu + O(y 2 ), Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 34/43

69 Since ψ and η are independent of y, U = µψ y2 η = µu + O(y 2 ), We compute directly, U 0 on B I B II ; n = Mε ε3 f 2 f η 2(1 + ε 2 f 2 ) 1/2 on B III. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 34/43

70 Define k 1 w = αϕ + U + α j ϕ j, j=0 for suitable α, α j, and for eigenfunctions ϕ j. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 35/43

71 We have 1 w L 2 (M ε) = O(ε 5/2 ), w L 2 (M ε) = O(ε 5/2 ) (By Poincaŕe inequalities); Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 36/43

72 We have 1 w L 2 (M ε) = O(ε 5/2 ), w L 2 (M ε) = O(ε 5/2 ) (By Poincaŕe inequalities); 2 w = O(ε 2 ) (By Moser iteration) ; Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 36/43

73 We have 1 w L 2 (M ε) = O(ε 5/2 ), w L 2 (M ε) = O(ε 5/2 ) (By Poincaŕe inequalities); 2 w = O(ε 2 ) (By Moser iteration) ; 3 w/ y = O(ε 2 ) (Usual maximum principle); Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 36/43

74 We have 1 w L 2 (M ε) = O(ε 5/2 ), w L 2 (M ε) = O(ε 5/2 ) (By Poincaŕe inequalities); 2 w = O(ε 2 ) (By Moser iteration) ; 3 w/ y = O(ε 2 ) (Usual maximum principle); 4 w/ y C 2,α ε M); = O(ε 2 ) (Schauder estimates on M ε and Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 36/43

75 We have 1 w L 2 (M ε) = O(ε 5/2 ), w L 2 (M ε) = O(ε 5/2 ) (By Poincaŕe inequalities); 2 w = O(ε 2 ) (By Moser iteration) ; 3 w/ y = O(ε 2 ) (Usual maximum principle); 4 w/ y C 2,α ε M); = O(ε 2 ) (Schauder estimates on M ε and 5 w y = O(ε2 ) (rerun (3) in this more complicated case); Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 36/43

76 We have 1 w L 2 (M ε) = O(ε 5/2 ), w L 2 (M ε) = O(ε 5/2 ) (By Poincaŕe inequalities); 2 w = O(ε 2 ) (By Moser iteration) ; 3 w/ y = O(ε 2 ) (Usual maximum principle); 4 w/ y C 2,α ε M); = O(ε 2 ) (Schauder estimates on M ε and 5 w y = O(ε2 ) (rerun (3) in this more complicated case); 6 Re-run (4), we get the conclusion. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 36/43

77 Applications/future projects Hearing the shape of a triangle? Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 37/43

78 Conclusions Conclusion 1 We don t have to redo gradient estimates for Bakry-Émery geometry in eigenvalue estimates. Because such kinds of estimates are always true if in the Riemannian cases they are true. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 38/43

79 Conclusions Conclusion 2 We obtained a new Maximum principle which is valid for nonconvex domiains M ε. That means, gradient estimates developed by Li and Li-Yau can be extended to some non-convex cases. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 39/43

80 Known results 1 We hear the shape of a triangle if we hear all of its eigenvalues. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 40/43

81 Known results 1 We hear the shape of a triangle if we hear all of its eigenvalues. 2 (Chang-DeTurck) There exists a number N = N(λ 1, λ 2 ) such that if we hear the first N eigenvalues, we hear the shape of the triangle. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 40/43

82 We make the following Conjecture There exists an absolute number N, such that the first N (Dirichlet or Neumann) eigenvalues are enough to determine the shape of a triangle. Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 41/43

83 The following result is useful and stimulating to our conjecture. Peter Li, Andrejs Treibergs, Shing-tung Yau How to hear the volume of convex domains. Geometry and nonlinear partial differential equations (Fayetteville, AR, 1990), , Contemp. Math., 127, Amer. Math. Soc., Providence, RI, Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 42/43

84 Happy Birthday, Peter! Zhiqin Lu, Dept. Math, UCI Eigenvalues of Collapsing Domains and Drift Laplacian 43/43