LECTURE28:SPECTRALGEOMETRYSpectral geometry is the branch of differential geometry that studies the relationsbetween the spectrum of the Laplace-type operator and the underline geometry.There are many many beautiful results that have been proved, and at the meantimethere are also many many open problems to be studied in the future. In this lastlecture,weapplyBochnerformulatospectral geometry.1.SPECTRALGEOMETRYI Eigenvalues and eigenfunctions.In spectral geometry there are three typical spectral problems:1)Closed settingLet (M,g)bea closed connected Riemannian manifoldWe call an eigenvalue of if there exists smooth function uO so thatl△u+入u=0.(2)Let (2,g)be a compact connected Riemannianmanifold with boundary 2.Dirichlet settingWe call a number a Dirichlet eigenvalue of if(a)there exists asmooth function u0 so that(Au+ u=0,in 2,u= 0,on a2.(b) Neumann setting We call a number ^ a Neumann eigenvalue of if thereexistsasmoothfunctionu+osothat[Au+ u=0,in 2,[O,u = 0,on a2,where O,represents the outer normal derivative.We have seen from PSet 1 that. All eigenvalues of are non-negative real numbers..\ = 0 is always an eigenvalue for the closed problem and the Neumanneigenvalue problem (with eigenfunctions the constant functions), and ^= 0is not an eigenvalue of the Dirichlet problem.. If u and y are eigenfunctions of different eigenvalues, then (u,v)r2 = 0.According to the standard spectral theory in functional analysis, one can prove1Here we use = divV = Tr(V2). If one uses = -div = -Tr(V2) = d8 + d, then theequation should beu=Au.1
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
2LECTURE28:SPECTRALGEOMETRYTheorem 1.l. In all three settings above, each eigenvalue has finite multiplicityand the eigenvalues of form an increasing sequence that tends to oo, namely0=≤≤≤≤. 00for the closed eigenvalues and the Neumann eigenvalues, and02≤≤..≤≤..00for the Dirichlet eigenvalues. Moreover, one can choose an eigenbasis so that theyform a complete orthonormal basis of L?(M) or L?(2).The simplest examplebeingErample.For S, the Laplacian eigenvalues are the squares 0,1,1,4,4, 9,9,.,witheigenfunctions cos(kc)and sin(kr).Sincethesefunctionsalreadyform an orthonor-mal basis, there are no other eigenvalues/eigenfunctions.Similarly for Tm = Si × ... × S', equipped with the standard fat metric, theeigenvalues are numbers of the form k? + ...+ km, with eigenfunctions cos(k ·r)and sin(k ·c), and again they form an orthonormal basis. [Note that in this case, themultiplicity is complicated since there may be many different ways to represent a given positiveinteger as the sum of m squares.]Erample. One can show that the eigenvalues of the standard sphere Sm arek(k+m -1) (k =0,1,2, .-), with multiplicity nk= (m+) - (m+-2).Unfortunately, other then very few examples like the sphere/the torus/the pro-jectivespaces etc(or rectangles/balls/annulus etc in the caseof manifold withboundary),for most Riemannian manifolds there is no way to calculate its eigenval-ues explicitly.There are two major problem in spectral geometry:ThedirectproblemGiveninformation of (M,g)or(2,g),whatcanwesay about these eigenvalues/eigenfunctions?The inverse problem Given the sequence of eigenvalues, what can we sayabout the geometry of (M,g) or (2,g)?I The first eigenvalue AiThe first non-zero eigenvalue Ar is very important and has received much atten-tion. Although in general one can't calculate it explicitly, we do have a variationalcharacterizationas follows.Given anysmoothfunction o,wecallR(g) = Jm/VopPdv,JMp2dVthe Rayleigh quotient of p. Then
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
3LECTURE28:SPECTRALGEOMETRYTheorem 1.2 (Variational characterization of Ai).For the closed or the Neumanneigenvalue problem,= inf[R() I0EH'(M),§ = 0.]while forthe Dirichlet eigenvalue problem,A = inf(R() [ 0 H(M),)Proof. For any in the given space, we may expand =E=, Ckuk We mayassume llll2 =1, i.e. c=1. ThenR(0) =≥ 1= >1k≥1k≥1口On the other hand, if we take = ui, then R()=R(ui)=1Remark.For example,given bounded domain 2,the Poincare inequality states thatthere exists constant C so that[apPdVg≤c / ivudVg,Vue Hi(2).Now in view of the above theorem,the smallest (=thebest)constant Cfor thePoincareinequalitytobetrueis thereciprocal of thefirstDirichleteigenvalue of 2Remark. One also has variational characterization of higher eigenvalues A, for all k.2.SOME RESULTS ON THEFIRST EIGENVALUE入1Now suppose (M,g)is closed and wefocus on the firstnonzeroeigenvalueI Lichnerowitz estimate for Ar.Now we apply Bochner formula to prove a lower bound estimate for AiTheorem 2.1 (Lichnerowitz).Let (M,g) be a closed Riemannian manifold withRic≥ (m-1)k for somek>0.Then the first eigenvalueAi≥mk.Proof. According to Corollary 1.3 in Lecture 28, for any u E C(M),(/Vul2) ≥ =(Au)? + (Vu, V(Au)) + Rc(Vu, Vu).2So if we take u be an eigenfunction, i.e. Au + Au = O, then we getA≥-uu-(Vu, Vu)+ Re(Vu, Vu),(1)Integrate over M and apply the Green's formulauud=Vupda
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
4LECTURE28:SPECTRALGEOMETRYwe get>+(m-1)k)/Vul2dcmThis impliesT-入+ (m- 1)k≤ 0,m口i.e. ^≥mk.IObata'si rigiditytheorem.One can prove that the first eigenvalue of the standard sphere sm is m. In fact,this is the only case where >i =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 somek>o.If i=mk,then (M,g)is isometric tothe sphere Sm.Proof. Without loss of generality we may assume k = 1. If Ai = m, then from theproof above we seeRc(Vu, Vu) = (m - 1)/Vul2.Since△(u2)=2uu+2/Vu|2 (see PSet 1for-△), from (1)weget(/V?+)≥--m/V+(m-1)/V?++/V?=02It follows (/Vu|2+ u?)= 0 since its integral over M is 0. In other words,/Vuj?+u?=constant.We normalize u so that maxm u? = 1. Since Vu = 0 at the maximum/minimumpoints of u, we getIVu? +?=1and-minu= 1.maxu=MMNow let p, q E M be points such that u(p) = -1, u(q) = 1. Let I = d(p, q) andlet :[o,] →M be a normal geodesic from p to q. Let f(t) = u((t). Thenf'(t)[Vu((t)|= 1.V1-u((t)2V1- f2(t)Integratingbothsideswegetf'(t)dt = l= d(p,q)T/1-f2(t)So diam(M,g) ≥ π. But by Bonnet-Meyer, diam(M,g) ≤ π. So diam(M,g) = π口Finally by Cheng's maximal diameter theorem, (M, g) is isomorphic to Sm
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. □
LECTURE28:SPECTRALGEOMETRY5Reilly's formula.Let2beacompactsmoothmanifoldwithsmoothboundaryM=2.Thenonecan definethesecond fundamentalformof M(asaRiemanniansubmanifold of2)asfollows:ForanypEM,theyector-valued secondfundamental formII atpisa symmetricbilinear mapII: T,M ×T,M-→N,M,(X,Y)-(V)+,where X,Y are smooth vector fields whose value at p are X and Y respectively[According to PSet 2, I(X,Y) is well-defined and is symmetric]. Since in the hypersurfacecase there is only one normal dimension, we may study the (scalar-valued) secondfundamentalformh:T,M ×T,M -→R, (X,Y) -→h(X,Y) := -(II(X,Y),v)If we pick a local orthonormal coordinate system e,) near p E M,where em+1 isthe out normal direction, then for any X = X'ei,Y -yje, e T,M, one hash(X,Y) = h,X'yi,ij=1where hij = -(Ve,ej,em+1) = (Ve,em+1,ej). The trace of h,H := Tr(h) = hiis known as the mean curvature of M at p.By integrating Bochner formula, one can prove the following useful formulaobtained byR.Reilly in1977:Theorem 2.3 (Reilly's formula). Let be a compact Riemannian manifold of di-mension m+1, with smooth boundaryM =02. Then for any f eC(2),m(A"f)?≥ / Hf2+2 / fuAMf+ / h(VMf,VMf)+ / Rc(Vf,f)m+1Moreover, the equality holds if and only if fi, =oig, i.e.2f =fId.m+1Proof. For simplicity we write °f = g, and write flan = u. So in what follows wemay abbreviatef=f,f=-f andMu=Au,Mu=Vu.ByBochnerformula wehave11A(IVfI)≥g?+(Vf,Vg) +Rc(Vf,f),2m+1withequality ifandonlyf=dIntegrateandivewofGrn'sfomula((Vf, Vg) = - / gaf + / gfu
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