Essential Spectra of complete manifolds

Transcription

1 Essential Spectra of complete manifolds Zhiqin Lu Analysis, Complex Geometry, and Mathematical Physics: A Conference in Honor of Duong H. Phong May 7, 2013 Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 1/41

2 Let M be a compact Riemannian manifold. Then the following are true Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 2/41

3 Let M be a compact Riemannian manifold. Then the following are true 1 The spectrum on p-forms are eigenvalues 0 λ 1 λ n. 2 if f is an eigenform, and if df 0, then df is an eigenform also (similar result holds for δ). Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 2/41

4 How about these properties on a complete noncompact manifold? Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 3/41

5 How about these properties on a complete noncompact manifold? σ( ) = σ ess ( ) σ pt ( ) Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 3/41

6 How about these properties on a complete noncompact manifold? σ( ) = σ ess ( ) σ pt ( ) In this talk, we shall concentrate on the essential spectra. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 3/41

7 1 on a compact manifold, we have f + λf = 0; 2 on a non-compact complete manifold, we have f + λf L 2/ f L 2 very small. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 4/41

8 1 on a compact manifold, we have f + λf = 0; 2 on a non-compact complete manifold, we have f + λf L 2/ f L 2 very small. 3 Our result shows that we only need very small. f L f + λf L 1 f 2 L 2 Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 4/41

9 Fact On a complete non-compact manifold, the Laplacian of the distance function is locally L 1 but not L 2 in general. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 5/41

10 Fact On a complete non-compact manifold, the Laplacian of the distance function is locally L 1 but not L 2 in general. Example: let M = S 1 (, + ). Let ρ be the distance function to the point (1, 0). Then the cut locus of the function is the set I = {( 1, t) t R}, and we have ρ = δ I + a bounded function Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 5/41

11 Fact On a complete non-compact manifold, the Laplacian of the distance function is locally L 1 but not L 2 in general. Example: let M = S 1 (, + ). Let ρ be the distance function to the point (1, 0). Then the cut locus of the function is the set I = {( 1, t) t R}, and we have ρ = δ I + a bounded function Thus ρ is locally in L 1 but not in L 2. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 5/41

12 The Weyl Criterion Theorem (Classical Weyl s criterion) A point λ belongs to σ( ) if, and only if, there exists a sequence {ψ n } n N Dom( ) such that 1 n N, ψ n = 1, 2 ( + λ)ψ n 0, as n in L 2 (M). Moreover, λ belongs to σ ess ( ) of L 2 (M) if, and only if, in addition to the above properties 3 ψ n 0 weakly as n in L 2 (M). Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 6/41

13 Example The essential spectrum of R n is [0, ). Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 7/41

14 Example The essential spectrum of R n is [0, ). The proof: = 2 r + n 1 2 r r + 1 r 2 S n 1 2 r 2 for r large. For any λ > 0, sin r λ is the approximation of the eigenfunction, where r is the distance function. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 7/41

15 Example The essential spectrum of R n is [0, ). The proof: = 2 r + n 1 2 r r + 1 r 2 S n 1 2 r 2 for r large. For any λ > 0, sin r λ is the approximation of the eigenfunction, where r is the distance function. Can we at least generalize this result to asymptotically flat manifolds? The corresponding term is log det g. r Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 7/41

16 History Donnelly (1981), Escobar (1986), Escobar-Freire (1992), Jiayu Li (1989, 1994), Chen-L (1994), Zhou (1992), Donnelly (1997) generalize the result on R n to manifolds with a pole. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 8/41

17 History In 1992, K. Sturm proved the following Theorem Let M be a complete non-compact manifold whose Ricci curvature has a lower bound. If the volume of M grows uniformly sub-exponentially, then the L p essential spectra are the same for all p [1, ]. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 9/41

18 History In 1997, J-P. Wang proved that, if the Ricci curvature of a manfiold M satisfies Ric (M) δ/r 2, where r is the distance to a fixed point, and δ is a positive number depending only on the dimension, then the L p essential spectrum of M is [0, + ) for any p [1, + ]. In particular, for a complete non-compact manifold with non-negative Ricci curvature, all L p spectra are [0, + ). Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 10/41

19 History In 2011, L-Zhou proved the following Theorem Let M be a complete non-compact Riemannian manifold. Assume that lim Ric M (x) = 0. x Then the L p essential spectrum of M is [0, + ) for any p [1, + ]. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 11/41

20 This is the joint work with N. Charalambous. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 12/41

21 Weak Weyl criterion Theorem (Weyl s criterion for quadratic forms) A point λ belongs to σ( ) if, and only if, there exists a sequence {ψ n } n N Dom( 1/2 ) such that 1 n N, ψ n = 1, 2 ( λ)ψ n 0 in n L2 (M). Moreover, λ belongs to σ ess ( ) if, and only if, in addition to the above properties w 3 ψ n 0 in n L2 (M). Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 13/41

22 Theorem (Charalambous-L) Let f be a bounded positive continuous function over [0, ). Let H be a densely defined self-adjoint nonnegative operator on some Hilbert space H. A positive real number λ belongs to the spectrum σ(h) if, and only if, there exists a sequence {ψ n } n N Dom(H) such that 1 n N, ψ n = 1, 2 (f(h)(h λ)ψ n, (H λ)ψ n ) 0, as n and 3 ((H + 1) 1 ψ n, (H λ)ψ n ) 0, as n. Moreover, λ belongs to the essential spectrum σ ess (H) of H if, and only if, in addition to the above properties 4 ψ n 0, weakly as n in H. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 14/41

23 Since H is a densely defined self-adjoint operator, the spectral measure E exists and we can write H = We pick ε > 0 such that λ > ε, and E(λ + ε) E(λ ε) = 0. We write 0 ψ n = ψ 1 n + ψ 2 n, λ de. (1) where and ψ 2 n = ψ n ψ 1 n. ψ 1 n = λ ε 0 de(t)ψ n, Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 15/41

24 Then we have (f(h)(h λ)ψ n, (H λ)ψ n ) = (f(h)(h λ)ψ 1 n, (H λ)ψ 1 n) + (f(h)(h λ)ψ 2 n, (H λ)ψ 2 n) c 1 ψ 1 n 2 + (f(h)(h λ)ψ 2 n, (H λ)ψ 2 n) c 1 ψ 1 n 2, where the positive number c 1 is the infimum of the function f(t)(t λ) 2 on [0, λ ε]. Therefore ψ 1 n 0. On the other hand, we similarly get ((H + 1) 1 ψ n, (H λ)ψ n ) c 2 ψ 2 n 2 c 3 ψ 1 n 2, If criteria (2), (3) are satisfied, then, by the two inequalities above, we conclude that both ψ 1 n, ψ 2 n go to zero. This is a contradiction to ψ n = 1, and the theorem is proved. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 16/41

25 new criterium Corollary A nonnegative real number λ belongs to the essential spectrum σ ess ( ), if there exists a sequence {ψ n } n N of smooth functions such that 1 n N, ψ n L 2 = 1, 2 ψ n L ( λ)ψ n L 1 0, as n. 3 ψ n 0, weakly as n in L 2. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 17/41

26 new criterium Corollary A nonnegative real number λ belongs to the essential spectrum σ ess ( ), if there exists a sequence {ψ n } n N of smooth functions such that 1 n N, ψ n L 2 = 1, 2 ψ n L ( λ)ψ n L 1 0, as n. 3 ψ n 0, weakly as n in L 2. Proof: ( 1) 1 is bounded in L. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 17/41

27 Theorem (Charalambous-L) Let M be a complete noncompact manifold. Assume that with respect to a fixed point lim inf r Ric ( r, ) = 0 r and assume that vol (M) =. Then the essential spectrum of M is [0, ). Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 18/41

28 Theorem (Charalambous-L) Let M be a complete noncompact manifold. Assume that with respect to a fixed point lim inf r Ric ( r, ) = 0 r and assume that vol (M) =. Then the essential spectrum of M is [0, ). If vol (M) <, and if we assume that the volume doesn t decay exponentially, then the same conclusion is true. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 18/41

29 Theorem (Charalambous-L) The essential spectrum of a complete shrinking Ricci soliton is [0, ). Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 19/41

30 Lemma For all ε > 0 there exists an R 1 > 0 such that for r > R 1 one of the following holds (a) If vol (M) is infinite ρ 2ε V (r + 1) B p(r)\b p(r 1 ) (b) If vol (M) is finite ρ 2ε (vol (M) vol (B(r))) + 2vol ( B p (r)) M\B p(r) Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 20/41

31 Let x, y, R be large positive numbers such that x > 2R > 2µ + 4 and y > x + 2R. We take the cut-off function ψ : R + R, smooth with support on [x/r 1, y/r + 1] and such that ψ = 1 on [x/r, y/r] and ψ, ψ bounded. We let ϕ( r) = ψ( ρ R ) ei λ ρ (2) ϕ + λϕ C R + C ρ Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 21/41

32 When the volume of M is infinite, if we choose R, x large enough, then ϕ + λϕ ε V (y + R) M we can prove In general, V (y + R) V (y) is not correct. but by the sub exponential volume growth, we can do that. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 22/41

33 By the curvature assumption, the volume of the manifold grows subexponentially, which implies that there exists a sequence of y k such that V (y k + R) 2 V (y k ). Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 23/41

34 By the curvature assumption, the volume of the manifold grows subexponentially, which implies that there exists a sequence of y k such that V (y k + R) 2 V (y k ). If not, then for any k, Thus V (R + k) V (R + k 1) Ce ε(r+k) V (R + k) 2 k V (R) We get a contradiction when k. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 23/41

35 Now we turn to the essential spectrum of forms. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 24/41

36 Theorem The essential spectrum of the Laplacian on functions is contained in that of 1-forms. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 25/41

37 Theorem The essential spectrum of the Laplacian on functions is contained in that of 1-forms. Corollary Assume the Ricci curvature of M is asymptotically nonnegative, then the essential spectrum on 1-forms is [0, ). Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 25/41

38 Moreover, we have Theorem Suppose that λ belongs to the essential spectrum of the Laplacian on k-forms, σ ess (k, ). Then one of the following holds: (a) λ σ ess (k 1, ), or (b) λ σ ess (k + 1, ). Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 26/41

39 Let λ > 0 and λ σ ess (k, ). For any ε > 0, have a k-form ω ε such that ω ε L 2 = 1, ( λ)ω ε L 2 ε ω ε L 2 (3) and ω ε 0 weakly as ε 0. If we choose ε < 1 λ, then we obtain 2 ω ε L 2 λ 2 ω ε L 2. (4) Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 27/41

40 WLOG, we assume that ω ε are smooth with compact support. We have ω, ω 1 2λ ω 2 L λ (λ2 ε 2 ) ω 2 L 2 c ω 2 L 2 M where c > 0 is a constant. Integration by parts yields ω, ω = dω 2 L + 2 δω 2 L 2 c ω 2 L2. (5) M Since d and δ commute with the Laplacian, then they also commute with ( + 1) m for any integer m. For m = 1, 2, we compute (( + 1) m dω,( λ)dω) + (( + 1) m δω, ( λ)δω) after integration by parts. = (( + 1) m ω, ( λ)ω) Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 28/41

41 Furthermore, (( + 1) m ω, ( λ)ω) = (( + 1) m+1 ω, ( λ)ω) (λ + 1)(( + 1) m ω, ( λ)ω). Since ( + 1) m is a bounded operator for m = 0, 1, 2, we obtain (( +1) m dω, ( λ)dω)+(( +1) m δω, ( λ)δω) Cε ω 2 L 2 for some constant C. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 29/41

42 Estimate (5) gives (( + 1) m dω, ( λ)dω) + (( + 1) m δω, ( λ)δω) C ε( dω 2 L 2 + δω 2 L 2) for a possibly larger constant C. Thus there exists a subsequence of ω εn with ε n 0 (which we also denote as ω ε ) such that either (( + 1) m dω ε, ( λ)dω ε ) 1 2 C ε dω ε 2 L2, or is valid. (( + 1) m δω ε, ( λ)δω ε ) 1 2 C ε δω ε 2 L 2 Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 30/41

43 Now we consider the essential spectrum on p-forms. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 31/41

44 Now we consider the essential spectrum on p-forms. Theorem Let M be a complete noncompact manifold. Assume that the curvature of M goes to zero at infinity. Then the essential spectra of M on k-forms are [0, ). Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 31/41

45 Definition A nilpotent structure on a manifold M is a sheaf N of vector fields such that for each p there exists a neighborhood U p and an action of group G p on a Galois covering Ũp of U p with the following properties 1 The connected component N p of G p is nilpotent; 2 The deck transformation group Γ p of the covering: Ũ p U p is contained in G p ; 3 G p is generated by N p and Γ p ; 4 [Γ p : Γ p N p ] = [G p : N p ] is finite; 5 The lift of N to Ũp is generated by the action of N p. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 32/41

46 Theorem (Cheeger-Fukaya-Gromov) There exists ε n > 0 and k n with the property that, for each M M n there exists a nilpotent structure N such that the following holds in addition 1 U p contains B εn (p, M); 2 The injectivity radius of Ũp is larger than ε n ; 3 [G p : N p ] < k n. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 33/41

47 Definition A Riemannian metric g on M is said to be (ε, k)-round, if there exists a nilpotent structure N on M such that it satisfies (1)-(3) in addition and that the section of N is a killing vector field of the metric g. Theorem (Cheeger-Fukaya-Gromov) For each δ there exist ε = ε(δ, n) and k = k(δ, n) such that for each Riemannian manifold (M, g) in the class M n we can find a metric g δ on M with the following properties 1 (M, g δ ) is (ε, k)-round; 2 g g δ < δ. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 34/41

48 Let M be a complete noncompact manifold whose curvature goes to zero at infinity. for any σ > 0 small, we can rescale the metric of M so that at infinity, the curvature still tends to zero. By the above theorems, there exists a neighbor U p such that 1 The injectivity radius of Ũp is greater than ε n ; 2 U p = A N, where A is a Eucliean ball and N is an infra nilpotent manifold. 3 The metric on U p can be approximated by (ε, k)-round metric. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 35/41

49 Rescaling back, we obtain the following result Theorem There exists a nonnegative integer 0 k n 1 such that for any R > 0, there is any embedding B(R) N M where B(R) is the ball of radius R in the Euclidean space and N is an infranilmanifold. That is, N is the quotient of a Nilpotent Lie group by a discrete group. Moreover, the embedding is almost isometric in the sense that 1 2 g 0 f (g) 2g 0, where g 0, g are the Riemannian metrics of B(R) N and M, respectively. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 36/41

50 A Prposition essentially due to Lott. Proposition Let N be a compact nilpotent manifold and let B(σ) be the ball of radius δ. Then there exists a content depending only on σ such that the first eigenvalue with respect to the Dirichlet Laplacian is bounded. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 37/41

51 Essential spectrum of the Laplacian on p forms on hyperbolic manifolds. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 38/41

52 Theorem The L p essential spectrum of the Laplacian on k-forms, is exactly the set of points to the right of the parabola Q p = { ( N p + is k)(n p + is + k N) s R }. Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 39/41

53 Thank you! Zhiqin Lu, Dept. Math, UCI Essential Spectra of complete manifolds 40/41