LECTURE 28: SPECTRAL GEOMETRY Spectral geometry is the branch of differential geometry that studies the relations between the spectrum of the Laplace-type operator and the underline geometry. There are many many beautiful results that have been proved, and at the meantime there are also many many open problems to be studied in the future. In this last lecture, we apply Bochner formula to spectral geometry. 1. Spectral geometry ¶ Eigenvalues and eigenfunctions. In spectral geometry there are three typical spectral problems: (1) Closed setting Let (M, g) be a closed connected Riemannian manifold. We call λ an eigenvalue of ∆ if there exists smooth function u ̸= 0 so that1 ∆u + λu = 0. (2) Let (Ω, g) be a compact connected Riemannian manifold with boundary ∂Ω. (a) Dirichlet setting We call a number λ a Dirichlet eigenvalue of ∆ if there exists a smooth function u ̸= 0 so that ( ∆u + λu = 0, in Ω, u = 0, on ∂Ω. (b) Neumann setting We call a number λ a Neumann eigenvalue of ∆ if there exists a smooth function u ̸= 0 so that ( ∆u + λu = 0, in Ω, ∂νu = 0, on ∂Ω, where ∂ν represents the outer normal derivative. We have seen from PSet 1 that • All eigenvalues of ∆ are non-negative real numbers. • λ = 0 is always an eigenvalue for the closed problem and the Neumann eigenvalue problem (with eigenfunctions the constant functions), and λ = 0 is not an eigenvalue of the Dirichlet problem. • If u and v are eigenfunctions of different eigenvalues, then ⟨u, v⟩L2 = 0. According to the standard spectral theory in functional analysis, one can prove 1Here we use ∆ = div∇ = Tr(∇2 ). If one uses ∆ = −div∇ = −Tr(∇2 ) = dδ + δd, then the equation should be ∆u = λu. 1
2 LECTURE 28: SPECTRAL GEOMETRY Theorem 1.1. In all three settings above, each eigenvalue has finite multiplicity and the eigenvalues of ∆ form an increasing sequence that tends to ∞, namely 0 = λ0 < λ1 ≤ λ2 ≤ λ3 ≤ · · · ≤ λk ≤ · · · → ∞ for the closed eigenvalues and the Neumann eigenvalues, and 0 < λ1 < λ2 ≤ λ3 ≤ · · · ≤ λk ≤ · · · → ∞ for the Dirichlet eigenvalues. Moreover, one can choose an eigenbasis so that they form a complete orthonormal basis of L 2 (M) or L 2 (Ω). The simplest example being Example. For S 1 , the Laplacian eigenvalues are the squares 0, 1, 1, 4, 4, 9, 9, · · · , with eigenfunctions cos(kx) and sin(kx). Since these functions already form an orthonormal basis, there are no other eigenvalues/eigenfunctions. Similarly for T m = S 1 × · · · × S 1 , equipped with the standard flat metric, the eigenvalues are numbers of the form k 2 1 + · · · + k 2 m, with eigenfunctions cos(k · x) and sin(k · x), and again they form an orthonormal basis. [Note that in this case, the multiplicity is complicated since there may be many different ways to represent a given positive integer as the sum of m squares.] Example. One can show that the eigenvalues of the standard sphere S m are k(k + m − 1) (k = 0, 1, 2, · · ·), with multiplicity nk = m+k m − m+k−2 m . Unfortunately, other then very few examples like the sphere/the torus/the projective spaces etc (or rectangles/balls/annulus etc in the case of manifold with boundary), for most Riemannian manifolds there is no way to calculate its eigenvalues explicitly. There are two major problem in spectral geometry: • The direct problem Given information of (M, g) or (Ω, g), what can we say about these eigenvalues/eigenfunctions? • The inverse problem Given the sequence of eigenvalues, what can we say about the geometry of (M, g) or (Ω, g)? ¶ The first eigenvalue λ1. The first non-zero eigenvalue λ1 is very important and has received much attention. Although in general one can’t calculate it explicitly, we do have a variational characterization as follows. Given any smooth function φ ̸= 0, we call R(φ) = R M |∇φ| 2dVg R M φ2dVg the Rayleigh quotient of φ. Then
LECTURE 28: SPECTRAL GEOMETRY 3 Theorem 1.2 (Variational characterization of λ1). For the closed or the Neumann eigenvalue problem, λ1 = inf{R(φ) | 0 ̸= φ ∈ H 1 (M), Z M φ = 0.}, while for the Dirichlet eigenvalue problem, λ1 = inf{R(φ) | 0 ̸= φ ∈ H 1 0 (M).} Proof. For any φ in the given space, we may expand φ = P∞ k=1 ckuk. We may assume ∥φ∥L2 = 1, i.e. Pc 2 k = 1. Then R(φ) = X k≥1 λkc 2 k ≥ X k≥1 λ1c 2 k = λ1. On the other hand, if we take φ = u1, then R(φ) = R(u1) = λ1. □ Remark. For example, given bounded domain Ω, the Poincar´e inequality states that there exists constant C so that Z Ω |u| 2 dVg ≤ C Z Ω |∇u| 2 dVg, ∀u ∈ H 1 0 (Ω). Now in view of the above theorem, the smallest (=the best) constant C for the Poincar´e inequality to be true is the reciprocal of the first Dirichlet eigenvalue of Ω. Remark. One also has variational characterization of higher eigenvalues λk for all k. 2. Some results on the first eigenvalue λ1 Now suppose (M, g) is closed and we focus on the first nonzero eigenvalue. ¶ Lichnerowitz estimate for λ1. Now we apply Bochner formula to prove a lower bound estimate for λ1. Theorem 2.1 (Lichnerowitz). Let (M, g) be a closed Riemannian manifold with Ric ≥ (m − 1)k for some k > 0. Then the first eigenvalue λ1 ≥ mk. Proof. According to Corollary 1.3 in Lecture 28, for any u ∈ C ∞(M), 1 2 ∆(|∇u| 2 ) ≥ 1 m (∆u) 2 + ⟨∇u, ∇(∆u)⟩ + Rc(∇u, ∇u). So if we take u be an eigenfunction, i.e. ∆u + λu = 0, then we get (1) 1 2 ∆|∇u| 2 ≥ − λ m u∆u − λ⟨∇u, ∇u⟩ + Rc(∇u, ∇u). Integrate over M and apply the Green’s formula − Z M u∆udx = Z M |∇u| 2 dx
4 LECTURE 28: SPECTRAL GEOMETRY we get 0 ≥ Z M λ m − λ + (m − 1)k |∇u| 2 dx. This implies λ m − λ + (m − 1)k ≤ 0, i.e. λ ≥ mk. □ ¶ Obata’s λ1 rigidity theorem. One can prove that the first eigenvalue of the standard sphere S m is m. In fact, this is the only case where λ1 = m if (M, g) satisfies the conditions in Theorem 2.1. Theorem 2.2 (Obata). Let (M, g) be a closed Riemannian manifold with Ric ≥ (m − 1)k for some k > 0. If λ1 = mk, then (M, g) is isometric to the sphere S m k . Proof. Without loss of generality we may assume k = 1. If λ1 = m, then from the proof above we see Rc(∇u, ∇u) = (m − 1)|∇u| 2 . Since ∆(u 2 ) = 2u∆u + 2|∇u| 2 (see PSet 1 for −∆), from (1) we get 1 2 ∆ |∇u| 2 + u 2 ≥ −u∆u − m|∇u| 2 + (m − 1)|∇u| 2 + u∆u + |∇u| 2 = 0. It follows ∆ (|∇u| 2 + u 2 ) ≡ 0 since its integral over M is 0. In other words, |∇u| 2 + u 2 = constant. We normalize u so that maxM u 2 = 1. Since ∇u = 0 at the maximum/minimum points of u, we get |∇u| 2 + u 2 = 1 and max M u = − min M u = 1. Now let p, q ∈ M be points such that u(p) = −1, u(q) = 1. Let l = d(p, q) and let γ : [0, l] → M be a normal geodesic from p to q. Let f(t) = u(γ(t)). Then f ′ (t) p 1 − f 2 (t) ≤ |∇u(γ(t))| p 1 − u(γ(t))2 = 1. Integrating both sides we get π = Z l 0 f ′ (t) p 1 − f 2 (t) dt ≤ Z l 0 dt = l = d(p, q). So diam(M, g) ≥ π. But by Bonnet-Meyer, diam(M, g) ≤ π. So diam(M, g) = π. Finally by Cheng’s maximal diameter theorem, (M, g) is isomorphic to S m. □
LECTURE 28: SPECTRAL GEOMETRY 5 ¶ Reilly’s formula. Let Ω be a compact smooth manifold with smooth boundary M = ∂Ω. Then one can define the second fundamental form of M (as a Riemannian submanifold of Ω) as follows: For any p ∈ M, the ✿✿✿✿✿✿✿✿✿✿✿✿✿✿ vector-valued✿✿✿✿✿✿✿✿ second ✿✿✿✿✿✿✿✿✿✿✿✿✿ fundamental✿✿✿✿✿✿ form II at p is a symmetric bilinear map II : TpM × TpM → NpM, (X, Y ) 7→ (∇Ω X Y ) ⊥, where X, Y are smooth vector fields whose value at p are X and Y respectively [According to PSet 2, II(X, Y ) is well-defined and is symmetric]. Since in the hypersurface case there is only one normal dimension, we may study the (scalar-valued) second fundamental form h : TpM × TpM → R, (X, Y ) 7→ h(X, Y ) := −⟨II(X, Y ), ν⟩. If we pick a local orthonormal coordinate system {ei} near p ∈ M, where em+1 is the out normal direction, then for any X = Xi ei , Y = Y j ej ∈ TpM, one has h(X, Y ) = Xm i,j=1 hijX iY j , where hij = −⟨∇ei ej , em+1⟩ = ⟨∇ei em+1, ej ⟩. The trace of h, H := Tr(h) = X i hii, is known as the mean curvature of M at p. By integrating Bochner formula, one can prove the following useful formula obtained by R. Reilly in 1977: Theorem 2.3 (Reilly’s formula). Let Ω be a compact Riemannian manifold of dimension m + 1, with smooth boundary M = ∂Ω. Then for any f ∈ C ∞(Ω), m m + 1 Z Ω (∆Ω f) 2≥ Z M Hf 2 ν +2 Z M fν∆ Mf + Z M h(∇Mf, ∇Mf)+Z Ω RcΩ (∇f, ∇f). Moreover, the equality holds if and only if fij = ∆Ωf m+1 δij , i.e. ∇2 f = ∆Ωf m+1 Id. Proof. For simplicity we write ∆Ωf = g, and write f|∂Ω = u. So in what follows we may abbreviate ∆Ωf = ∆f, ∇Ωf = ∇f and ∆Mu = ∆u, ∇Mu = ∇u. By Bochner formula we have 1 2 ∆(|∇f| 2 ) ≥ 1 m + 1 g 2 + ⟨∇f, ∇g⟩ + RcΩ (∇f, ∇f), with equality if and only if ∇2 f = ∆f m+1 Id. Integrate and in view of Green’s formula Z Ω ⟨∇f, ∇g⟩ = − Z Ω g∆f + Z ∂Ω gfν
6 LECTURE 28: SPECTRAL GEOMETRY we get 1 2 Z Ω ∆(|∇f| 2 ) ≥ 1 m + 1 Z Ω g 2 + Z Ω ⟨∇f, ∇g⟩ + Z Ω RcΩ (∇f, ∇f) = −m m + 1 Z Ω g 2 + Z M gfν + Z Ω RcΩ (∇f, ∇f). In what follows we will prove (⋆) 1 2 Z Ω ∆(|∇f| 2 ) = − Z M Hf 2 ν + Z M gfν − 2 Z M fν∆u − Z M h(∇u, ∇u) from which the theorem follows. We choose orthonormal frame near p so that em+1(p) = ν(p) and ∇em+1 em+1 = 0. Cover M by such coordinate neighborhoods and let ρα be a partition of unity subordinate to this covering (together with the open set Ω \ M). As we have seen in Lecture 4, P α ραdiv(X) = P α div(ραX). So 1 2 Z Ω ∆(|∇f| 2 ) = 1 2 X α Z Ω ρα∆(|∇f| 2 ) = 1 2 X α Z Ω div(ρα∇(|∇f| 2 )). So by divergence theorem, 1 2 Z Ω ∆(|∇f| 2 ) = 1 2 X α Z M ⟨ρα∇(|∇f| 2 ), ν⟩ = 1 2 X α Z M ρα∂ν(|∇f| 2 ). Thus we may compute in the above local coordinates. Since |∇f| 2 = P i (eif) 2 , 1 2 Z M ρα∂ν(|∇f| 2 ) = Z M ρα mX +1 i=1 (eif)(em+1eif) = Z M ρα (em+1f)(em+1em+1f) +Xm i=1 (eif)(em+1eif) . Note that ∇em+1 em+1 = 0 in the neighborhood. So we get, at all x ∈ M, em+1em+1f = mX +1 i=1 eieif − (∇ei ei)f − Xm i=1 eieif − (∇ei ei)f = mX +1 i=1 eieif −(∇ei ei)f − Xm i=1 eieif −(∇ei ei) T f + Xm i=1 (∇ei ei) ⊥f = ∆f − ∆u + Xm i=1 ⟨∇ei ei , em+1⟩fν = g − ∆u − Hfν.
LECTURE 28: SPECTRAL GEOMETRY 7 For the second term we use em+1eif = eiem+1f + (∇em+1 ei)f − (∇ei em+1)f = eiem+1f + mX +1 j=1 ⟨∇em+1 ei , ej ⟩fj − mX +1 j=1 ⟨∇ei em+1, ej ⟩fj = eiem+1f + Xm j=1 ⟨∇em+1ei , ej ⟩fj − Xm j=1 hijfj , where in the last step we used the facts ⟨∇em+1 ei , em+1⟩ = −⟨ei , ∇em+1em+1⟩ = 0, ⟨∇ei em+1, em+1⟩ = 1 2 ei |em+1| 2 = 0 and hij = ⟨∇ei em+1, ej ⟩. So we get three terms. For the first one we have X α Z M ρα Xm i=1 (eif)(eiem+1f) = X α Z M ρα⟨∇Mf, ∇Mfν⟩ = − Z M (∆u)fν. For the second term, we have Pm i,j=1(eif)⟨∇em+1 ei , ej ⟩fj = 0 since S :=Xm i,j=1 (eif)⟨∇em+1 ei , ej ⟩fj = X i,j ⟨∇em+1 ei , ej ⟩fifj =− Xm i,j=1 ⟨ei , ∇em+1 ej ⟩fifj =−S. Finally for the last term, Xm i,j=1 (eif)⟨∇ei em+1, ej ⟩fj = Xm i,j=1 hijfifj = h(∇u, ∇u). So we get the desired equality (⋆). □ Remark. If we don’t apply Cauchy-Schwartz inequality at the first step, then Z Ω ((∆f) 2 − |∇2 f| 2 ) = Z M Hf 2 ν + 2fν∆ Mf + h(∇Mf, ∇Mf) + Z Ω Rc(∇f, ∇f). ¶ Yau’s conjecture. A Riemannian submanifold Mm of N is called minimal if it has mean curvature ✿✿✿✿✿✿✿✿ H = 0. Minimal submanifolds are very important objects in Riemannian geometry, especially the branch “submanifold geometry”. As an application we prove Theorem 2.4 (Choi-Wang, 1984). Let Mm be a compact connected embedded oriented minimal hypersurface in a compact oriented Riemannian manifold N m+1. Suppose N has Ricci curvature RicN ≥ mk > 0, then λ1(M) ≥ mk 2 . Proof. Since RicN > 0, by Bochner theorem (c.f. Corollary 1.7 in Lecture 28), b1(N) = 0. Let Ω be a tubular neighborhood of M. Then the Mayer-Vietories sequence of de Rham cohomologies, 0 → H 0 (N) → H 0 (N \ M) ⊕ H 0 (Ω) → H 0 ((N \ M) ∩ Ω) → H 1 (N)
8 LECTURE 28: SPECTRAL GEOMETRY becomes 0 → R → H 0 (N \ M) ⊕ R → R ⊕ R → 0. It follows that H0 (N \ M) ∼= R ⊕ R, i.e. N \ M contains exactly two connected components. We denote N \ M = Ω1 ∪ Ω2, ∂Ω1 = ∂Ω2 = M. Now let u ∈ C ∞(M) be an eigenfunction associated to λ1 = λ1(M), i.e. ∆Mu + λ1u = 0. Without loss of generality, we assume Z M h(∇u, ∇u)dVg ≥ 0, where we regard M as ∂Ω1. [If this inequality is not true, then the analogue inequality for Ω2 holds and we proceed with Ω2 instead of Ω1.] Let f be a solution to ( ∆N f = 0, in Ω1 f = u, on M = ∂Ω1. By Reilly’s formula, 0 ≥ −2λ Z M ufν + Z M h(∇u, ∇u) + mk Z Ω1 |∇f| 2 ≥ −2λ Z M ufν + mk Z Ω1 |∇f| 2 . Since ∆f = 0, by Green’s formula we get Z M ufν = Z ∂Ω1 ffν = Z Ω1 |∇f| 2 , thus 0 ≥ (−2λ1 + mk) Z Ω1 |∇f| 2 . It follows λ1 ≥ mk/2. □ In particular, if we take N m+1 = S m+1 we get λ1(M) ≥ m 2 . This lower bound is half of the conjectured bound by Yau in 1982: Conjecture (Yau). For any embedded minimal hypersurface M of S m+1, one has λ1(M) ≥ m. Dedicated to Yau for his 75th Birthday