For ∂M fixed normal variation of a minimal immersion M # N,

A⁰⁰(₀) = Z

k∇fk²−Ric(−→_{n ,}−→_n) + kBk²f², where−→

n =en+1.

As an application, we introduce the result which is a lemma used in the proof of positive mass theorem.

Theorem 3.38 (Schoen - Yau). If M is a compact oriented 3-fold (3-dimensional manifold) with the scalar curvature R >0, then6 ∃ M # N such that M is compact orientable surface with g(M) ≥ 1 and stable (A⁰⁰(0) ≥ 0).

P^ROOF. If M is stable, i.e. A⁰⁰(0) ≥0 and Z

M Ric(−→ n ,−→

n) + kBk² f² ≤ Z

M|∇f|², ∀ f . Let f ≡1. We apply this lemma:

Lemma 3.39. Ric(−→ n ,−→

n) = ¹₂R−K− ¹₂kBk² where K is the Gauss curvature on M.

With this lemma, we have Z

1 2R+¹

2kBk²− Z

MK ≤₀

⇒2πχ(M) =4π(1−g(M)) ≤0 a contradiction.

PROOF OF LEMMA. Pick Riemann normal coordinate at p.

R= g^ijg^k^`R_ikj`

=2(R1212+R1313+R2323)

=2(Ric(e3, e3) +R₁₂₁₂).

R_ijk` =R_ijk`+h_ikh_j`−h_i`h_jkand K =R1212= R1212+h11h22−h12h21. Also,

h₁₁ h₁₂ h21 h22

, tr(B) = h₁₁+h₂₂ =_0.

10. EXERCISES 103

kBk² = h²₁₁+h²₁₂ +h²₂₁ +h²₂₂ = 2h²₁₁ +2h²₁₂ and h11h22 −h12h21 =

−(h²₁₁+h²₁₂) = ¹₂kBk². Hence we get the formula. 10. Exercises

1. On a local coordinate chart, if ω = hdx¹∧ · · · ∧dxⁿ, show that h = ^qdet(g_ij). In general, show that v₁, . . . , v_k ⊂ _Rⁿ, the k-dim’l volume of

[v₁, . . . , v_k] = ( _k

∑

t_iv_i

0≤t_i ≤1 )

ispdet(V^tV)where

V =







| | · · · | v1 v2 · · · v_k

| | · · · |





 . 2. Check that∗² = (−1)^p⁽ⁿ⁻^p⁾.

3. Verify the original Stokes theorem inR³ Z

Ω∇ ×F·n dA = Z

∂ΩF·dr.

4. M,→_Rⁿ. Show the induced connection is torsion-free.

5. Prove the existence of Levi-Civita connection by checking this re-ally defines a covariant differentiation.

6. Prove Gauss lemma using the factΓ^α_rr =0 and then ∂rgrα =0∀α.

7. Extend the covariant derivative to all tensors. And prove the com-mutation formula:

T_jⁱ¹^···ⁱ^r

1···js;ji =T_jⁱ¹^···ⁱ^r

1···js;ij+T_j^qi²^···ⁱ^r

1···js Rⁱ_qij¹ +T_jⁱ¹^qi³^···ⁱ^r

1···js Rⁱ_qji² + · · · +Tⁱ_pj¹^···ⁱ^r

2···jsR^p_j

1ji+ · · · +T_jⁱ¹^···ⁱ^r

1···j_s−1pR^p_j

sji. 8. From these symmetries, we can get some further results:

(a) The symmetries (3.14) implies R_k`ij =R_ijk`. (b) Second Bianchi identity:

0= R_ij_[_k_`_;m_] := R_ijk`;m+R_ij`_m;k+R_ijmk;`.

9. For n ≥ 3, if Rij = λgij, then λ is constant. (hint: use Bianchi identity.)

10. Show that K(X, Y) is the Gauss curvature of surface X(u, v) = exp_p(uX+vY). (hint: may assume thatkXk = kYk =1,hX, Yi = 0)

11. Let K(X, Y) = R(X, Y, X, Y)_{. Then}

hR(X, Y)Z, Wi = K(X+W, Y+Z) −K(X+W, Y) −K(X+W, Z)

−K(X, Y+Z) −K(W, Y+Z) +K(X, Z) +K(W, Y)

−K(Y+W, X+Z) +K(Y+W, X) +K(Y+W, Z) +K(Y, X+Z) +K(W, X+Z) +K(Y, Z) −K(W, X). 12. (M, g)has constant sectional curvature at p if and only if

hR(X, Y)W, Zi = K(p)(hX, ZihY, Wi − hX, WihY, Zi)_, that is

R_ijk` =K(p)(g_ikg_j`−g_i`g_jk).

13. Complete the proofs of other parts of Synge theorem: Let M^m be compact and with positive sectional curvature K.

(a) For any homotopy class of closed curves [_γ], there exists a minimal geodesic γ0∈ [_γ]_.

(b) For m is odd, M is orientable.

14. Prove Lemma (3.29) and show by example that it fails if X is not complete.

15. Show thatHⁿ complete, K ≡ −1;Sⁿ, K ≡1.

16. M complete, simply connected with K≡1. Show that M∼=S^m. 17. For Poincare upper half plane, show that all geodesic are circles

that perpendicular to the real line. Bonus: use this to prove that the isometry group is exactly SL(2,R).

18. Determine all the geodesics on the sphereSⁿ. 19. (Do Carmo Ch.2 #8) Consider the upper half-plane

R²+ = {(x, y) ∈_R²|y>0}

with the metric given by g11 = g22 = _y¹₂, g12 = 0 (metric of Lo-batchevski’s non-euclidean geometry).

(a) Show that the Christoffel symbols of the Riemannian connec-tion are: Γ¹₁₁ =_Γ²₁₂ =_Γ¹₂₂ =0,Γ²₁₁ = ¹_y,Γ¹₁₂ =_Γ²₂₂ = −¹_y.

10. EXERCISES 105

(b) Let v0 = (0, 1) be a tangent vector at point (0, 1) of R²+ (v0

is a unit vector on the y-axis with origin at (0, 1)). Let v(t) be the parallel transport of v0 along the curve x = t, y = 1.

Show that v(t) makes an angle t with the direction of the y-axis, measured in the clockwise sense.

20. (Do Carmo Ch.3 #5) Let M be a Riemannian manifold and X ∈ X (M) (the set of all vector field of class C^∞ on M). Let p ∈ M and let U ⊂ M be a neighborhood of p. Let ϕ : (−ε, ε) ×U → _M be a differentiable mapping such that for any q ∈ U the curve t → ϕ(t, q) is a trajectory of X passing through q at t = 0 (U and ϕare given by the fundamental theorem for ordinary differential equations). X is called a Killing field (or an infinitesimal isometry) if, for each t0 ∈ (−ε, ε), the mapping ϕ(t0,·) : U ⊂ M → M is an isometry. Prove that

(a) A vector field v onRⁿ may be seen as a map v :Rⁿ →_Rⁿ; we say that the field is linear if v is a linear map. A linear field on Rⁿ, defined by a matrix A, is a Killing field if and only if A is anti-symmetric.

(b) Let X be a Killing field on M, p ∈ M, and let U be a normal neighborhood of p on M. Assume that p is a unique point of U that satisfies X(p) = 0. Then, in U, X is tangent to the geodesic spheres centered at p.

(c) Let X be a differentiable vector field on M and let f : M → N be an isometry. Let Y be a vector field on N defined by Y(f(p)) = d fp(X(p)), p ∈ M. Then Y is a Killing field if and only if X is also a Killing vector field.

(d) X is Killing⇔ h∇_YX, Zi + h∇_ZX, Yi =0 for all Y, Z ∈ X (M) (the equation above is called the Killing equation).

(e) Let X be a Killing field on M with X(q) 6=0, q∈ M. Then there exists a system of coordinates (x1, . . . , xn) in a neighborhood of q, so that the coefficients g_ij of the metric in this system coordinates do not depend on xn.

21. (Do Carmo Ch.4 #4) Let M be a Riemannian manifold with the following property: given any two points p, q ∈ M, the parallel transport from p to q does not depend on the curve that joins p to

q. Prove that the curvature of M is identically zero, that is, for all X, Y, Z ∈ X (M), R(X, Y)Z =0.

22. Define covariant derivatives on differential forms.

23. (Do Carmo Ch.4 #8)(Schur’s theorem) Let Mⁿbe a connected Rie-mannian manifold with n ≥3. Suppose that M is isotropic, that is, for each p ∈ M, the sectional curvature K(p, σ) does not depend on σ ⊂ TpM. Prove that M has constant sectional curvature, that is, K(p, σ)also does not depend on p.

24. (Do Carmo Ch.4 #10) (Einstein manifolds) A Riemannian man-ifold Mⁿ is called an Einstein manifold if, for all X, Y ∈ X (M), Ric(X, Y) = λhX, Yi, where λ : M →R is a real valued funciton.

Prove that:

(a) If Mⁿ connected and Einstein , with n ≥3, then λ is constant on M.

(b) If M³ is a connected Einstein manifold then M³has constant sectional curvature.

25. Show that the sectional curvature is the Gauss curvature.

26. (Do Carmo Ch.4 #6) Let M be a Riemannian manifold. M is a locally symmetric space if∇R = 0, where R is the curvature tensor of M.

(a) Let M be a locally symmetric space and let γ : [0,`) → M be a geodesic of M. Let X, Y, Z be parallel vector fields along γ.

Prove that R(X, Y)Z is a parallel field along γ.

(b) Prove that if M is locally symmetric, connected, and has di-mension two, then M has constant sectional curvature.

27. (Do Carmo Ch.5 #5) (Jacobi fields and conjugate points on locally symmetric spaces) Let γ : [0,∞) → M be a geodesic in a locally symmetric space M and let v = γ⁰(0) be its velocity at p = γ(0). Define a linear transformation Kv : TpM →TpM by

Kv(x) = R(v, x)v, x∈ _T_p_M.

(a) Prove that Kvis self-adjoint.

10. EXERCISES 107

(b) Choose an orthonormal basis{e₁, . . . , en}of TpM that diago-nalizes Kv, that is,

Kv(e_i) =_λ_ie_i, i=1, . . . , n.

Extend the e_i to fields along γ by parallel transport. Show that, for all t,

K_γ⁰₍_t₎(e_i(t)) =_λ_ie_i(t)_. where λi does not depend on t.

d²x_i

dt² +λ_ix_i =0, i =1, . . . , n.

(d) Show that the conjugate points of p along γ are given by γ(πk/√

λ_i), where k is a positive integer and λ_i is a positive eigenvalue of Kv.

28. (Do Carmo Ch.7 #1) If M, N are Riemannian manifolds such that the inclusion i : M ,→ N is an isometric immersion, show by an example that the strict inequality of metrics dM >dN can occur.

29. (Do Carmo Ch.7 #4) Consider the universal covering π : M→_R²\ {(0, 0)}

of the Euclidean plane minus the origin. Introduce the covering metric on M. (Note: For ˜M a covering space of M, it is possible to give the covering space a Riemannian structure such that the covering map π : ˜M → M is a local isometry.) Show that M is not complete and not extendible, and that the Hopf-Rinow theorem is not true for M (this shows that the definition of non-extendibility, though natural, is not a satisfactory one).

30. (Do Carmo Ch.7 #5) A divergent curve in a Riemannian manifold M is a differentiable mapping α : [0,∞) → M such that for any compact set K ⊂ M there exists t0 ∈ (0,∞) with α(t) ∈/ K for all t > t0 (that is, α “escapes” every compact set in M). Define the

length of a divergent curve by

tlim→∞

Z _t

|_α⁰(t)|dt.

Prove that M is complete if and only if the length of any divergent curve is unbounded.

31. (Do Carmo Ch.7 #6) A geodesic γ : [0,∞) → M in a Riemann-ian manifold M is called a ray starting from γ(₀) if it minimizes the distance between γ(₀) _{and γ}(_s), for any s ∈ (_0,_∞)_{. Assume} that M is complete, non-compact, and let p ∈ M. Show that M contains a ray starting form p.

32. (Do Carmo Ch.7 #12) A Riemannian manifold is said to be homoge-neous if given p, q ∈ M there exists an isometry of M which takes p into q. Prove that any homogeneous manifold is complete.

33. (Do Carmo Ch.8 #1) Consider, on a neighborhood inR², n>2 the metric

g_ij = ^δ^ij F²

where F6= 0 is a function of(x₁, . . . , xn) ∈_Rⁿ. Denote by F_i = ^∂F

∂x_i, F_ij = ^∂²^F

∂x_i∂x_j, etc.

(a) Show that a necessary and sufficient condition for the metric to have constant curvature K is







Fij =0, i 6= j

F(Fjj+Fii) = K+_∑ⁿ_i₌₁(Fi)².

(b) Use (a) to prove that the metric gij has constant curvature K if and only if

F =G₁(x₁) +G₂(x₂) + · · · +Gn(xn)_, where

Gi(_x_i) = _ax²_i +_b_i_x_i+_c_i and

∑

n i=1

(4cia−_b_i²) = K.

10. EXERCISES 109

gij = ^δ^ij 1+^K₄ _{∑ x}_i²²

for a metric g_ij of constant curvature K. If K <0, the metric g_ij is defined in a ball of radius of

q 4

−K.

(d) If K >0, the metric is defined on all of Rⁿ. Show that such a metric onRⁿ is not complete.

34. (Do Carmo Ch.6 #1) Let M1 and M2 be Riemannian manifolds, and consider the product M1×M2, with the product metric. Let

∇¹ be the Riemannian connection of M1 and let ∇² be the Rie-mannian connection of M2.

(a) Show that the Riemannian connection∇of M1×M2is given by∇_Y₁₊_Y₂(X₁+X2) = ∇¹_Y

1X₁+ ∇_Y²

2X2, for X₁, Y₁ ∈ X (M₁), X2, Y2 ∈ X (M2).

(b) For every p ∈ M₁, the set (M₂)_p = {(p, q) ∈ M₁×M₂; q ∈ M₂} is a submanifold of M₁×M₂, naturally diffeomorphic to M2. Prove that(M₂)_pis a totally geodesic submanifold of M1×_M₂_.

(c) Let σ(x, y) ⊂T(_p,q)(M1×M2)be a plane such that x ∈ TpM1

and y ∈ TqM2. Show that K(σ) = 0.

35. (Do Carmo Ch.6 #2) Show that x :R² →_R⁴given by x(θ, ϕ) = √¹

2(cos θ, sin θ, cos ϕ, sin ϕ), (θ, ϕ) ∈_R²

is an immersion of R² into the unit sphere S³(1) ⊂ _R⁴, whose image x(_R²)is a torusT²with sectional curvature zero in the in-duced metric.

36. (Do Carmo Ch.6 #4) Let N₁ ⊂ M₁, N₂ ⊂ M₂ be totally geodesic submanifolds of the Riemannian manifolds M1 and M2 respec-tively. Prove that N1×_N₂is a totally geodesic submanifold of the product M1×M2with the product metric.

37. (Do Carmo Ch.6 #5) Prove that the sectional curvature of the Rie-mannian manifold S²×_S² with the product metric, where S² is

the unit sphere inR³, is non-negative. Find a totally geodesic, flat torus,T², embedded inS²×_S².

38. (Do Carmo Ch.6 #8) (The Clifford torus) Consider the immersion x :R²→_R⁴given in the above exercise35.

(a) Show that the vectors

e₁= (−sin θ, cos θ, 0, 0), e₂ = (0, 0,−sin ϕ, cos ϕ)

form an orthonormal basis of the tangent space, and that the vectors

n1 = √¹

2(cos θ, sin θ, cos ϕ, sin ϕ), n₂ = √¹

2(−_{cos θ,}−sin θ, cos ϕ, sin ϕ) form an orthonormal basis of the normal space.

(b) Use the fact that

hSn_k(e_i)_{, e}_ji = −h∇_e_in_k, eji = h∇_e_ie_j, nki_,

where ∇is the covariant derivative (that is, the usual deriv-ative) ofR⁴, and i, j, k = 1, 2, to establish that the matrices of Sn1 and Sn2 with respect to the basis{e1, e2} are

Sn₁ =

−1 0 0 −1

, Sn2 =

1 0

0 −1

(c) From the above exercise 35, x is an immersion of the torus T² into S³(1) (the Clifford torus). Show that x is a minimal immersion.

39. (Do Carmo Ch.6 #11) Let f : Mⁿ⁺¹ → R be a differentiable func-tion. Define the Hessian, Hess f of f at p ∈ M as the linear operator

Hess f : TpM→TpM,(Hess f)Y= ∇_Y grad f , Y ∈ TpM, where ∇ is the Riemannian connection of M. Let a be a regular value of f and let Mⁿ ⊂ Mⁿ⁺¹be the hypersuperface in M defined by M = {p∈ M| f(p) = a}. Prove that:

(a) The Laplacian∆ f is given by

∆ f =trace(Hess f).

10. EXERCISES 111

(b) If X, Y ∈ X (M), then

h(Hess f)Y, Xi = hY,(Hess f)Xi.

Conclude that Hess f is self-adjoint, hence determines a sym-metric bilinear form on TpM, p∈ M, given by(Hess f)(X, Y) = h(Hess f)X, Yi, X, Y ∈ TpM.

|grad f|

(d) Observe that every embedded hypersurface Mⁿ ⊂ Mⁿ⁺¹ is locally the inverse image of a regular value. Conclude from (c) that the mean curvature H of such a hypersuperface is given by

H = −¹ ndivH,

where N is an appropriate local extension of the unit normal vector field on Mⁿ ⊂ Mⁿ⁺¹.

40. (Do Carmo Ch.8 #8) (Riemannian submersions) A differentiable mapping f : Mⁿ⁺^k → Mⁿ is called a submersion if f is surjective, and for all ¯p ∈ M, d f¯p : T¯pM → T_f₍_¯p₎M has rank n. In this case, for all p ∈ M, the fiber f⁻¹(p) = Fp is a submanifold of M and a tangent vector of M, tangent to some Fp, p ∈ M, is called a vertical vector of the submersion. If, in addition, M and M have Riemannian metrics, the submersion f is said to be Riemannian if, for all p ∈ M, d fp : TpM → T_f₍_p₎M preserves lengths of vectors orthogonal to Fp. Show that:

(a) If M1×M2 is the Riemannian product, then the natural pro-jections πi : M1×M2 → Mi, i =1, 2 are Riemannian submer-sions.

(b) Let the tangent bundle TM be given the Riemannian metric as:

hV, Wi₍_p,v₎ = hdπ(V), dπ(W)i_p+ h^Dv

dt (0), Dw ds (0)i_p

for (p, v) ∈ TM, V, W tangent vectors at (p, v) in TM where V = α⁰(0), W = β⁰(0) for curves α, β chosen such that α(t) =

(p(t), v(t)), β(t) = (q(s), w(s)), p(0) = q(0) = 0, v(0) = w(0) = v (cf. Do Carmo Ch.3 #2). Show that the projection π : TM→ M is a Riemannian submersion.

41. (Do Carmo Ch.8 #9) (Conneciton of a Riemannian submersion) Let f : M → M be a Riemannian submersion. A vector ¯x ∈ T¯pM is horizontal if it is orthogonal to the fiber. The tangent space T¯pM then admits a decomposition T¯pM = (T¯pM)^h⊕ (T¯pM)^v, where (T¯pM)^h and (T¯pM)^v denote the subspaces of horizontal and ver-tical vectors, respectively. If X ∈ X (M), the horizontal lift X of X is the horizontal field defined by d f¯p(X(¯p)) = X(f(p)).

(a) Show that X is differentiable.

(b) Let ∇ and ∇ be the Riemannian connections of M and M respectively. Show that

∇_XY = ∇_XY+¹

2[X, Y]^v, X, Y ∈ X (M), where Z^vis the vertical component of Z.

42. (Do Carmo Ch.8 #10) (Curvature of a Riemannian submersion) Let f : M → M be a Riemannian submersion. Let X, Y, Z, W ∈ X (_M), X, Y, Z, W be their horizontal lifts, and let R and R be the curvature tensors of M and M respectively. Prove that:

(a)

R(X, Y)Z, W

= hR(X, Y)Z, Wi −¹ 4

[X, Z]^v,[Y, W]^v +¹

[Y, Z]^v,[X, W]^v−¹ 2

[Z, W]^v,[X, Y]^v (b) K(_σ) = K(_σ) +³₄[X, Y]^v

2 ≥K(_σ), where σ is the plane gen-erated by the orthonormal vectors X, Y ∈ X (M)and σ is the plane generated by X, Y.

43. (Do Carmo Ch.8 #11) (The complex projective space) Let Cⁿ⁺¹\ {0} = {(z0, . . . , zn) = Z 6=0|z_j =x_j+iy_j, j =0, . . . , n} be the set of all non-zero (_n+₁)-tuples of complex numbers zj. Define equivalence relation on Cⁿ⁺¹\ {0}: (z0, . . . , zn) ∼ W = (w0, . . . , wn) if zj = λwj, λ ∈ _{C, λ} 6= 0. The equivalence class

10. EXERCISES 113

of Z will be denoted by[Z](the complex line passing through the origin and through Z). The set of such classes is called, by analogy with the real case, the complex projective space Pⁿ(_C) of complex dimension n.

(a) Show thatPⁿ(_C) has a differentiable structure of a manifold of real dimension 2n and thatP¹(_C)is diffeomorphic toS². (b) Let(Z, W) = z0w0+ · · · +znwn be the hermitian product on

Cⁿ⁺¹, where the bar denotes complex conjugation. Identify Cⁿ⁺¹ ≈_R²ⁿ⁺²by putting z_j =x_j+iy_j = (x_j, y_j). Show that

S²ⁿ⁺¹ = {N ∈ _Cⁿ⁺¹ ≈_R²ⁿ⁺²| (N, N) = 1} is the unit sphere inR²ⁿ⁺².

(c) Show that the equivalence relation ∼ induces on S²ⁿ⁺¹ the following equivalence relation: Z ∼W if e^iθZ =W. Establish that there exists a differentiable map (the Hopf fibering) f : S²ⁿ⁺¹→_Pⁿ(_C)such that

f⁻¹([Z])

={e^iθN ∈ _S²ⁿ⁺¹| N ∈ [Z] ∩_S²ⁿ⁺¹, 0 ≤θ ≤2π}

=[Z] ∩_S²ⁿ⁺¹.

(d) Show that f is a submersion.

44. (Do Carmo Ch.8 #12) (Curvature of the complex projective space) Define a Riemannian metric on Cⁿ⁺¹\ {0} in the following way:

If Z∈ _Cⁿ⁺¹\ {0}and V, W ∈ TZ(_Cⁿ⁺¹\ {0}), h_{V, W}i_Z = ^Real(V, W)

(Z, Z) ^.

Observe that the metrich , i restricted toS²ⁿ⁺¹ ⊂ _Cⁿ⁺¹\ {0} co-incides with the metric induced fromR²ⁿ⁺².

(a) Show that, for all 0 ≤_θ ≤_{2π, e}^iθ _: _S²ⁿ⁺¹ →_S²ⁿ⁺¹is an isom-etry, and that, therefore, it is possible to define a Riemannian metric onPⁿ(_C) in such a way that the submersion f is Rie-mannian.

(b) Show that, in this metric, the sectional curvature ofPⁿ(_C)is given by

K(_σ) =1+3 cos²ϕ,

where σ is generated by the orthonormal pair X, Y, cos ϕ = hX, iYi, and X, Y are the horizontal lifts of X and Y, respec-tively. In particular, 1≤K(σ) ≤4.

Chapter 4

H

ODGE

T

HEOREM

1. Harmonic Forms

Let A^p(M)be differential p-forms on a compact closed manifold Mⁿ with a given Riemannian metric g and an inner product

(α, β):= Z

Mhα, βidV = Z

Mα∧ ∗β,

where∗ : Λ^p →_Λⁿ⁻^p,∗²= (−1)⁽ⁿ⁻^p⁾^p. We get a pre-Hilbert space.

Now the question is that: for[_α] ∈ H_dR^p (M), can we pick a (or the best) representative of[α]? e.g. infβk_α+dβk_.

Denote d^∗ as the adjoint of d : A^p(M) → A^p⁺¹(M). (α, d^∗β) = (dα, β) =

Mdα∧ ∗_β

= Z

Md(α∧ ∗β) − (−1)^pα∧d(∗β)

= (−₁)^np⁺¹ Z

Mα∧ ∗(∗d∗_β) So d^∗ = (−1)^np⁺¹∗d∗.

Let dα=0.

(α+tdβ, α+tdβ) = (α, α) +2(α, dβ)t+ (dβ, dβ)t²≥ (α, α)

⇐⇒ 0 = (α, dβ) = (d^∗α, β), ∀ _β ⇐⇒ d^∗α =0.

Hence we have dα=0, d^∗α=0⇔ k_αkis minimal.

Here we introduce the Laplace operator

∆ = (d+d^∗)²=dd^∗+d^∗d.

We can see that

(_{∆α, α}) = (dd^∗α+d^∗dα, α) = (d^∗α, d^∗α) + (dα, dα) =0,

115

and then∆α=0⇔dα =0, d^∗α =0.

Clearly, this operator∆ is self-adjoint. And the truly crucial prop-erty we need is that∆ is elliptic.

So the general question becomes: how to solve∆α = _{β? when is} it solvable? The answer goes:

LetH= {α |_∆α=0}. ∆α= βis solvable if and only if β∈ _H^⊥. The “if part” is easier. For any γ∈ _H,

(β, γ) = (_{∆α, γ}) = (α,∆γ) =0 ⇒ β∈ _H^⊥.

For example, let β ∈ A⁰(M) (a function). If∆α = β is solvable, then β ∈_H^⊥ =_R^⊥, i.e. R

Mβ dV =0.

Conversely, the “only if part” comes from a standard formulation in fundamental analysis.

Let L :H → Hbe a bounded linear operator on some pre-Hilbert spaceH. Consider the equation Lw=β. For any φ ∈ H,

(w, L^∗φ) = (Lw, φ) = (β, φ).

We can see that any solution w defines a linear functional`such that

`(L^∗φ) = (β, φ) ∀φ∈ H. Such a`is called a weak solution for Lw =β.

Remark 4.1. If H is a Hilbert space (i.e. complete), then by Riesz representation theorem, such`will come from some α∈ H_.

2. Hodge Decomposition Theorem

Here we recall two theorems in standard analysis.

Theorem 4.2 (Regularity theorem). For Lw = β elliptic linear PDE, any weak solution is automatically smooth.

Theorem 4.3(Compactness theorem). For a sequence αn ∈ H, ifkαnk ≤ C, kLαnk ≤ C are bounded, then {αn} has a Cauchy subsequence (w.r.t.

L²norm).

2. HODGE DECOMPOSITION THEOREM 117

Both of the two theorems are implied by the Garding inequality.

For f ∈ H,

k_fk_s₊_d ≤_C(kL fk_s+ kfk_s),

where d is the order of L, and the norms are Sobolev norms.

Now back to the case L=_∆,H = A^p(M) = A^p. LetH =_H^p _:= {_α ∈ A^p(M) |_∆α=₀}be the set of harmonic p-forms.

By assuming the two theorems, we have the theorem.

Theorem 4.4(Hodge decomposition theorem).

(1) dimH <_{∞, and}

(2) A^p =_H^p⊕_∆A^p_{, i.e. Im∆} =_H^⊥.

PROOF. Firstly, we can infer that dimH < ∞ by compactness. If dimH =∞, we may select

u1, u2, . . . ∈_H,k_u_ik =_{1, u}_i ⊥_u_j∀_i 6= _j

satisfying ∆ui = 0. But it’s clear that {ui} has no Cauchy subse-quence.

Also by compactness,H ⊂ His a closed subspace. We have the decomposition A^p=_H⊕_H^⊥.

Hence we can pick{w₁, w2, . . . , w_l}, an O.N.B. ofH such that α =β+

∑

l i=1

(α, wi)wi = β+H(α) where β ∈_H^⊥and H(α)is the harmonic projection of α.

We still need to showH^⊥ =_∆A^p. It’s clear thatH^⊥ ⊃_∆A^psince

(_{∆α, γ}) = (α,∆γ) =0, ∀γ∈ _H.

The other side comes from the following claim:

Claim 4.5. There exists c>0 such thatk_βk ≤ ck_∆βk_,∀ _β∈ _H^⊥_. If the claim holds, let α ∈_H^⊥. Define a linear functional`by

`(_∆^∗φ) := (α, φ)_{, on∆A}^p ⊂ A^p.

Check that`is bounded: let φ₁ =φ−H(φ).

k`(_∆φ)k = k`(_∆φ₁)k = k(α, φ₁)k ≤ kαk · kφ₁k (claim) ≤ck_αk · k_∆φ₁k = (ck_αk) · k_∆φk.

By Hahn-Banach theorem, ` can be extended to a bounded linear function on A^p(M)_{, i.e.} `is a weak solution to∆w=_α.

Then by regularity theorem, there exists a smooth w ∈ A^p(M) such that∆w=α. So finallyH^⊥ ⊂_∆A^p.

Now we prove the claim.

PROOF OF CLAIM. Suppose the contrary, there exists a sequence of β_j ∈_H^⊥,k_β_jk = 1,k_∆β_jk → 0.

By compactness theorem, we may assume β_jitself is a Cauchy se-quence. Define`(_ψ) =_lim_j_→_∞(_β_j_{, ψ}) _{for ψ} ∈ A^p which is bounded.

And

`(_∆ψ) = lim

j→∞(βj,∆ψ) = lim

j→∞(_∆β_j, ψ) =0, i.e. `is a weak solution of∆β =0.

By regularity theorem, there exists a β ∈ A^p(M)such that`(ψ) = (β, ψ)and∆β=0, i.e. β∈ _H.

But kβ_jk = 1 and β_j ∈ _H^⊥ imply that kβk = 1 6= 0, β ∈ _H^⊥,

which leads a contradiction.

So far, these works for any elliptic L. In particular for∆, we can go further. Since A^p =H⊕ (_∆A^p), we define the Green operator

G :=







0 onH

∆⁻¹ onH^⊥ , i.e. I = H+_∆G

where H : A^p → H is the harmonic projection. Hence Gα is the unique solution inH^⊥ for∆w=α−H(α).

Exercise 4.1.

(1) G will commute with any operator T with T∆=_∆T.

(2) G is a bounded, self-adjoint, compact operator.

3. BOCHNER PRINCIPLE 119

Notice that ∆ = dd^∗ +d^∗d, d∆ = _{∆d, d}^∗_∆ = _∆d^∗, and then [d, G] = [d^∗, G] = [_{∆, G}] = 0. So

A^p=_H⊕_Im∆=_H⊕Imd⊕Imd^∗. We can rewrite the identity I =H+_{∆G as}

α = H(α) + (dd^∗+d^∗d)Gα

= H(_α) +d(d^∗Gα) +d^∗(dGα). So firstly we have the following result.

Proposition 4.6. If dα=0, α= H(_α) +d(d^∗Gα).

Notice that the harmonic representation is unique. (e.g. α1−_α₂= dβ, H(α1) −H(α2) = H(dβ) =0)

There are some applications.

Proposition 4.7(Poincar´e duality).

H_dR^p (M) ∼= H_dRⁿ⁻¹(M)^∗.

PROOF. Since ∗_∆ = _∆∗, ∗² = (−1)^p⁽ⁿ⁻^p⁾, by using harmonic forms,

H^p ∼=^∗ Hⁿ⁻^p.

3. Bochner Principle

Proposition 4.8(Bochner formula). Let M be compact.

∆= −tr∇²−

∑

i,j

ηⁱ∧ι(e_j)R(e_i, e_j),

for{e_j}an O.N.B.,{_ηⁱ}a dual basis. And∇²_x,y = ∇_x∇_y− ∇_∇_x_y. Exercise 4.2. Prove the Bochner formula4.8.

Corollary 4.9.

(1) If Ric > 0 (In fact, Ric ≥ 0, and > 0 at one point), b1 = h¹(M) = 0.

(2) If Ric≥0, h¹(M) ≤n =dim(M)and if Ric>0 at some point, then h¹(M) = 0. The equality holds if and only if M is a flat torus, i.e. M ∼=Rⁿ/Γ, Γ ∼=Zⁿ.

PROOF. For (i), let θ ∈ A¹(M). Say θ = _∑_ia_iηⁱwhere {_ηⁱ}is the dual frame to the local frame{e_i}_{at some p.}

At p, we have

∇²_e

i,e_i = ∇_e_i∇_e_i− ∇_∇

eie_i. So

0= h_{∆θ, θ}i −

∑

h∇_e_i∇_e_i_{θ, θ}i

| {z }

(1)

− h

∑

^ηⁱ^∧^ι⁽^e^j^)(∇ⁱ^∇^j^{− ∇}^j^∇ⁱ⁾^η^k^{, η}^`ⁱ^a^k^a^`

| {z }

(2)

By direct calculation, (1) =

∑

eih∇_e_i_{θ, θ}i − |∇_e_i_θ|²= −¹

2∆|_θ|²− |∇_θ|²_. (2) =

∑

^h^R^k^jij^ηⁱ^{, η}^`ⁱ^a^k^a^` ⁼^Ric⁽^{θ, θ}⁾^.

Therefore

0= h_{∆θ, θ}i = ¹

2∆|_θ|²+ |∇_θ|²+Ric(θ, θ).

• _{If θ} 6= _{0, let} |_θ(p)| > 0 be maximal. Then ∆|_θ|² ≥ _{0 by} second derivative test. But this contradicts to Ric(_{θ, θ}) >_0.

• In case of Ric≥0 and>0 at one point.

0 = −¹ 2

M∆|_θ|²+ Z

|∇_θ|²+ Z

MRic(_{θ, θ})_. By Stokes’ theorem, since ∂M =_{∅, we have}

0= Z

|∇_θ|²+ Z

MRic(_{θ, θ}) ⇒ ∇_θ ≡_0.

i.e. θ is parallel and then θ is constant. And this contradicts to Ric >0 at one point.

For (ii), again Ric ≥ 0 implies θ 6= 0 is parallel, determined by some θp∈ T_p^∗(M).

Hence h¹ ≤dim T_p^∗(M) = dim M.

4. FOURIER TRANSFORM 121

Moreover, if the equality holds, h¹ = dim M = m, then the uni-versal cover ˜M −→ M has m parallel 1-forms. Therefore it has m parallel vector fields.

Hence ˜M∼=R^m.

4. Fourier Transform

Here we define some notions in this section:

(1) x= (x₁, . . . , xm) ∈_R^m_.

(2) x·y= x1y1+x2y2+ · · · +xmym,|x| = (x·x)^1/2.

(3) α= (α1, . . . , αm)multi-index, αi ∈_N∪ {0}, |α| = _∑^m_i₌₁αi. (4) x^α =x^α₁¹x^α₂²· · ·x^αm^m.

(5) d^α = ^∂^|α|

∂x^α1₁ ···∂x_m^αm, D^α = (−i)^|α|d^α.

To prove the regularity theorem and compactness theorem, we need to investigate the concepts of weak derivatives, which we in-troduce here is defined through Fourier transformations.

Definition 4.10 (Fourier transformation). For ξ ∈ _R^m, x ∈ _R^m, let f ∈ C₀^∞(_R^m,C) be a smooth function with compact support. The Fourier transformation of f is defined as

ˆf(ξ) = (2π)⁻^m/2 Z

R^me^ix^·^ξf(x)dx.

Also recall the convolution of two functions f ∗g :=

R^m f(x−y)g(y) dy= Z

R^m f(z)g(x−z)dz.

On the purpose of doing operations on weak functions, in many times we can use convolutions to construct smooth approximations.

Let f ∈ C₀^∞(_R^m) with R

R^m f = 1, f ≥ 0, f(0) 6= 0. For u ∈ R+, define fu(x) := _u¹m f

x u

. We can see that R

R^m fu = 1. Such a sequence fu is called a “δ-function”, and we can write it as δ0 =

ulim→0fu.

Proposition 4.11. For g∈ C⁰, fu∗g(x) → g(x)as u→0+is a smooth approximation, i.e., δ0 =limu→0 fu is the identity element for∗.

We also have inverse Fourier transforms defined as in the follow-ing.

Definition 4.12 (Inverse Fourier Transform).

ˇg(x) := Z

R^me^iξ^·^xg(_ξ) dξ.

Now we list the basic properties and correspondence with con-volutions in the following theorem.

Theorem 4.13(Basic Properties of Fourier Transform).

(1) ˇˆf= f .

(2) D^α_ξ ˆf(ξ) = \x^αf(x),_D\^α_x_f(x) =ξ^α ˆf(ξ). (3) [_f ∗g= ˆf ˆg, ˆf∗ ˆg = fd·g.

(4) Fourier transform is an isometry, i.e., L²(_R^m) ^ˆ^· ^// L²(_R^m)

ˇ·

oo .

5. Sobolev spaces

By observing the formula dD^α_xf = ξ^αˆf, we notice that derivatives of f will corresponds multiplications of ˆf in the frequency space.

This formula enlightens the idea of weak derivatives.

Let s ∈ _R^m, which is regarded as the order of L²-derivative. De-fine the s-norm of a L²-function f by

|f|²_s := Z

R^m(1+ |ξ|²)^s| ˆf(ξ)|²dξ, and the inner product(·,·)_s by

(f , g)_s := Z

R^m(1+ |ξ|²)^s ˆf(ξ)ˆg(ξ) dξ.

We can easily check that C₀^∞(_R^m) ,→ L²(_R^m).

Let the Sobolev space Hs(_R^m) be the completion of C^∞₀ (_R^m) in L²(_R^m)(w.r.t the s-norm). Then again by the formula dD_x^αf =_ξ^α _ˆf,

D^α : Hs∩C₀^∞ −→ H_s_−|_α_|,

|D^α_xf|²_s_−|_α_| = Z

R^m(1+ |ξ|²)^s^−|^α^||ξ^α ˆf(ξ)|²dξ.

5. SOBOLEV SPACES 123

The core concept here is, while we define the (weak) derivative by multiplication in frequency space, how does f ∈ C^k actually if ˆf lies in some Sobolev space Hs, i.e. ξ^α ˆf(_ξ)lies in L²?

Theorem 4.14(key lemmas).

(1) (Sobolev lemma) If s>k+m/2 and f ∈ Hs, then f ∈C^kand

|f|_∞,k ≤C|f|_s.

(2) (Rellich lemma) If s>t, Hs ,→ Ht is a compact imbedding.

(3) (interpolation) Let s>t >u.∀e >0,∃C(e)such that

|f|_t ≤_e|f|_s+C(_e)|f|_u for all f ∈ C₀^∞.

P^ROOF.

(1) Let k =0. First we consider f ∈ C₀^∞(_R^m).

|_f(x)| = Z

R^me^ixξ ˆf(ξ)dξ

= Z

R^m

he^ixξ ˆf(_ξ)(1+ |_ξ|²)^s/2ⁱ(1+ |_ξ|²)⁻^s/2dξ

≤ |f|_s Z

R^m

(1+ |_ξ|²)^s ^dξ (H ¨older’s inequality)

≤C|f|_s

for some C depending on s >m/2.

Now for any f ∈ Hs, we can choose a sequence f_j −→^H^s f . By above result,

|f_i− f_j|_∞ ≤C|f_i− f_j|_s

infers that fi forms a Cauchy sequence in C⁰ and then f ∈ C⁰. So

|f|_∞ ≤ |f − f_i|_∞+ |f_i|_∞ ≤C(|f − f_i|_s+ |f_i|_s) and then|f|_∞ ≤C|f|_s by taking i→ _∞.

For k>0, apply the same argument to D^αf ,|α| =k.

|D^αf|_∞ ≤C|D^αf|_s ≤C|f|_s₊_k,

⇒ |f|_∞,k ≤C|f|_s₊_k. Hence Hs ⊂C^k.

(2) Now consider K ⊂_cpt _R^m, fn ∈ Hs with supp(fn) ⊂ K,

|fn|_s ≤C.

Let g ∈ C₀^∞(_R^m) with g ≡ 1 on K. Then g· fn = fn, ˆfn = ˆg∗ ˆfn.

∂_jˆf_n =∂_j(ˆg∗ ˆf_n) = (∂_jˆg) ∗ ˆf_n.

|∂_j ˆfn(ξ)| ≤ Z

|∂_jˆg(ξ−ζ) ˆfn(ζ)|dζ

≤ |fn|_s Z

|∂jˆg(ξ−ζ)|² (1+ |_ξ|²)^s ^dζ

≤Ch(ξ) ≤ C₁

where h(ξ)is a continuous function in ξ.

Again apply the same process to ˆfn(ξ), and we also have

| ˆf_n(ξ)| ≤C₁.

Hence ˆfn is a uniformly bounded and equicontinuous sequence on any compact subset. By Arzela-Ascoli theo-rem and a diagonal argument, there exists convergent sub-sequences of ˆfn (still denoted as ˆfn) uniformly on each com-pact subset.

Now we want to show that fn converges in Ht for s>t.

|f_j−f_k|²_t = Z

R^m|ˆf_j− ˆf_k|²(₁+ |_ξ|²)^t dξ

We decompose the domain into two parts and have the fol-lowing estimates.

6. ELLIPTIC OPERATORS AND GARDING’S INEQUALITY 125

For|ξ| ≥r,(1+ |ξ|²)^t ≤ (1+r²)^t⁻^s(1+ |ξ|²)^s and Z

|ξ|≥r|ˆf_j− ˆf_k|²(1+ |ξ|²)^t dξ

≤ (₁+r²)^t⁻^s Z

R^m|ˆf_j− ˆf_k|(₁+ |_ξ|²)^s dξ

≤2C(1+r²)^t⁻^s.

For any e>0, we can choose r large such that 2C(1+r²)^t⁻^s <

e/2.

On the other hand, {|ξ| ≤ r} is a compact set. We can pick j, k large such that

|ξ|≤r

| ˆf_j− ˆf_k|²(1+ |_ξ|²)^t dξ <e/2.

Hence|f_j− f_k|²_t <_efor j, k large enough.

(3) For s>t >u, there exists C_e such that

(1+ |ξ|²)^t ≤e(1+ |ξ|²)^s+Ce(1+ |ξ|²)^u.

Then we get the interpolation inequality immediately.

6. Elliptic Operators and Garding’s Inequality

Now we consider:

(E, k)

a vector bundle of rank k, e.g. E=_Λ^p(T^∗M), k= (^m_k).

f : a section

compact manifold,

and C^∞(M, E)is the vector space of all C^∞ sections.

Definition 4.15. We say L : C^∞(M, E) → C^∞(M, E) is a linear differ-ential operator of order d ∈ N, if locally we can regard L as a map between locally bundles:

C^∞(U, E

U) −→^L C^∞(U, E U)_, L f =

∑

Aα(x)D^αf , x∈U,

where U ⊂_R^m_{, E}

U ∼=U×_R^k_{, f} = (f¹,· · · _{, f}^k)_{and A}_α _{being k}×k matrices.

In particular, we usually write L=

∑

|α|=d

Aα(x)D^α+

∑

|α|<d

Aα(x)D^α where∑|α|=dA_α 6=_0.

• L is elliptic over U if p(x, ξ) =

∑

|α|=d

Aα(x)ξ^α(x ∈ U, ξ 6=0) is an invertible matrix. p is called the symbol of L.

• L is strictly elliptic over S if

∑

Aα(_x)_ξ^α·_v

≥_C|_ξ|^|_α||_v|_, ∀ _x ∈_{S, ξ} 6=₀ for some C independent of ξ.

In particular, ellipticity implies uniform ellipticity over any compact subsets.

Now we want to reduce to the Euclidean case.

Pick a local trivialization(U_i, φ_i, ψ_i)where U_iare charts on M, φ_i is a P.O.U. and ψ is the trivialization of E

U_i.

∼ U_i×_R^k ψ

Ui M

For any f ∈ C^∞(M, E), f =_∑_i fⁱσi, i.e. φ_i(f) = (f¹,· · · , f^k). And for s∈ _R,

|_f|_s =

∑

|_ψ_i(φi· _f)|s.

6. ELLIPTIC OPERATORS AND GARDING’S INEQUALITY 127

Then we can define Hs(M, E)to be the completion of C^∞(M, E)with respect to the norm| · |_s.

Denote H₀(M, E) = L²(M, E). Remark 4.16.

[

s∈R

Hs =: H−∞ ⊃ · · · ⊃ Hs ⊃· · ·⊃^s^<^t Hs ⊃ · · · ⊃ H∞ := ^\

s∈R

Hs =C^∞. Note that∪_sHs =C^∞ follows from the Sobolev lemma.

Exercise 4.3.

(1) Show that all| · |_sare independent of choice of (U_i, φ_i, ψ_i)as long as φ_i, ψ_i (and their inverse) have bounded derivation (i.e. norm equivalence).

(2) All the key lemmas still hold.

To ensure the regularity, we need to show that the weak solu-tion belongs to some Hs for higher order s actually and use the key lemmas. This estimate comes from the following theorem.

Theorem 4.17(Garding inequality). P is an elliptic operator of order d on(M, E). There exists C >0 such that

|f|_s₊_d ≤C(|P f|_s+ |f|_s), for all f ∈ H_s+d.

PROOF. Under some local chart (U_i, φ_i, ψ_i), we rewrite P = P₀+ P₁+P₂where

P0 =

∑

|α|=d

Aα(0)D^α, P1 =

∑

|α|<d

Aα(x)D^α, P2 =

∑

|α|=d

(Aα(x) −Aα(0))D^α for some p∈ U_i, p ↔x =0.

Locally, by uniform ellipticity at p,

|P₀f|²_s = Z

(1+ |_ξ|²)^s|dP₀f(_ξ)|²dξ

= Z

(1+ |ξ|²)^s

∑

α=d

Aα(0)ξ^α ˆf(ξ)

dξ

≥C⁰ Z

(₁+ |_ξ|²)^s|_ξ|^2d| ˆf(_ξ)|²dξ.

(|P0f|_s+ |f|_s)² ≥ Z

(1+C⁰⁰|_ξ|^d)²(1+ |ξ|²)^s|ˆf(ξ)|²dξ

≥ Z

C(1+ |ξ|²)^d(1+ |ξ|²)^s|ˆf(ξ)|²dξ ≥C₁²|f|²_s₊_d. We can easily get the estimates

|P1f|_s ≤C2|f|_s₊_d₋₁

|P2f|_s ≤ ^C¹

2 |f|_s+d+C2|f|_s+d−1

by choosing U small enough a priori.

Exercise 4.4. Show that in the proof of this inequality4.17, there ex-ists open neighborhood U 3 p independent of f such that the above estimate for P2f holds. Notice that P2f(0) = 0.

Hence

|P f|_s+ |f|_s ≥ |P₀f|_s− |P₁f|_s− |P₂f|_s+ |f|_s

≥ ^C¹

2 |f|_s₊_d−2C₂|f|_s₊_d₋₁.

⇒ |f|_s₊_d ≤C(|P f|_s+ |f|_s+ |f|_s₊_d₋₁). By interpolation inequality, (e=_1/2C)

|f|_s₊_d₋₁ ≤ ¹

2C|f|_s₊_d+C(_e)|f|_s_. Finally we have

2|_f|_s₊_d ≤_C|_{P f}|_s+_C^˜|_f|_s_.

7. Proof of Compactness and Regularity Theorem

Theorem 4.18 (Compactness theorem). For a sequence αn ∈ H_{, if} kαnk ≤C,kLαnk ≤C are bounded, then{αn}has a Cauchy subsequence (w.r.t. L²norm).

PROOF. Since un ∈ C^∞on compact M, un, Lunis uniformly bounded in L²-norm. By Garding inequality,

|un|₂ ≤C(|Lun|₀+ |un|₀) ≤ C.^˜

7. PROOF OF COMPACTNESS AND REGULARITY THEOREM 129

By Rellich lemma, un ∈ H2 and un has a Cauchy subsequence in

H₀.

Theorem 4.19(Regularity theorem). L is an elliptic operator of d ∈ _N.

If Lu=v, v ∈ Ht, u∈ H−_∞ := ∪_s_∈_RHs, then u∈ H_t+d.

P^ROOF. Suppose u ∈ Hs for some s ∈ R. It’s enough to show that if

Lu ∈ H_s−d+1 ⇒u ∈ H_s+1. The key here is to show that the difference quotient

u^h(x) := ^u(x+h) −u(x)

|h| ∈ Hs

for any vector h.

u\(· +h)(ξ) = Z

e⁻^iξxu(x+h)dx

=e^ihxi Z

e⁻ⁱ⁽^x⁺^h⁾^ξu(x+h)d(x+h)

=e^ihξuˆ(ξ).

⇒u^b^h(ξ) = ^e

ihξ −₁

|h| ^ˆh(ξ) =

ih·_ξ

|h| +o(|h|)

ˆ u.

So if u^h(x) ∈ Hs, then for h 6=0 small,

⇒ Z

(1+ |ξ|²)^s|ξ|²|uˆ(ξ)|²dξ <_∞ since u ∈ Hs.

So we get

u∈ Hs,|u^h|_s ≤C ∀ h6=0⇐⇒ u ∈ Hs+1. Now we need to get estimate for|u^h|_s.

Lemma 4.20.

L(u^h) = (Lu)^h−L^h(T_hu)

where T_hu(x) = u(x+h) denotes the translation operator, and L^h =

∑α A^h_α(x)D^α.

PROOF OF LEMMA.

Aα(x)D^α(u(x+h) −u(x))

= A_α(x+h)D^αu(x+h) −A_α(x)D^αu(x)

− (Aα(x+h) −Aα(x))D^αu(x+h).

Back to the proof. Notice that Lu∈ H_s−d+1infers that|(Lu)^h|_s₋_d is uniformly bounded for any small h. And then

|u^h|_s ≤C(|L(u^h)|_s₋_d+ |u^h|_s₋_d)

≤C(|(Lu)^h|_s₋_d+ |L^h(T_hu)|_s₋_d+ |u^h|_s₋_d)

≤C(|Lu|_s₋_d₊₁+ |u|_s₋_d₊₁).

So|u^h|_s ≤C for any small h and hence|u|_s+1≤C.

8. Exercises

1. (a) G will commute with any operator T with T∆=_∆T.

(b) G is a bounded, self-adjoint, compact operator.

2. Prove the Bochner formula4.8.

3. (a) Show that all| · |_s are independent of choice of (U_i, φ_i, ψ_i)as long as φi, ψi(and their inverse) have bounded derivation (i.e.

norm equivalence).

(b) All the key lemmas still hold.

4. Show that in the proof of this inequality 4.17, there exists open neighborhood U 3 p independent of f such that the above esti-mate for P2f holds. Notice that P2f(0) =0.

5. (Warner Ch.6 #6) Derive explicit formulas for d,∗, δ and∆ in Eu-clidean space. In particular, show that if

α =

∑

i₁<···<ip

α_Idx_i₁∧ · · · ∧dx_i_p, then

∆α= (−1)

∑

i₁<···<i_p

∑

n i=1

∂²αI

∂x²_i

dxi₁∧ · · · ∧dxi_p.

8. EXERCISES 131

6. (Warner Ch.6 #7) Let ϕ belong to the C^∞ periodic functionsP on the plane. Prove that

∂²ϕ

∂x∂y

≤ ¹ 2k_∆ϕk

7. (Warner Ch.6 #8) The Rellich lemma 4.14 says that the natural injection i : Ht → Hs for s < t is a compact operator; that is, it takes bounded sequences into sequences with convergent subse-quences. An analogous example of this phenomenon is the fol-lowing. Let C denote the Banach space of periodic continuous functions on the real line, say with period 2π, and with norm the sup-normk · k_∞. Let C¹be the subset of C consisting of functions with continuous first derivative. As a norm for C¹we take

kfk = kfk_∞+

d f dx

_∞.

Use Arzela-Ascoli theorem to prove that the natural injection i : C¹ →C is a compact operator.

8. (Warner Ch.6 #9) We shall consider a number of elliptic equations of the form Lu= f on the real line. In each case, f will be smooth and periodic of period 1, and we look for solution u also periodic of period 1. This restriction to periodic functions makes this in essence a problem on a compact space, the unit circle. We let u⁰ = du/dx, etc.

(a) u⁰ = f . This is the simplest example of an elliptic operator which exhibits all of the essential ingredients of the theory.

What is the formal adjoint of this differential operator? Show that there is a solution u (periodic) if and only if f is orthogo-nal to the kernel of this adjoint.

(b) u⁰−u = f . What is the kernel (in the periodic functions) in the case? What are the necessary and sufficient conditions on

f for there to exist a periodic solution?

(c) u⁰⁰ = f . Show that this operator is formally self-adjoint. Show that there is a periodic solution if and only if f is orthogonal

to the kernel; and using the fact that Z _x

_Z _t

0 f(s)ds

dt =

Z _x

0 f(s)

_Z _x

0 dt

ds,

show that the unique solution orthogonal to the kernel is u(x) =

Z _x

0 t(x−1)f(t)dt+ Z ₁

x x(t−1)f(t)dt

−¹ 2

Z ₁

0 t(_t−₁)_f(_t)_dt.

This explicitly exhibits the Green’s operator for this case.

(d) u⁰⁰ +4π²u = f . Show that this operator is formally self-adjoint. What is the kernel? Derive an explicit formula for the solution u, and show that u is periodic if and only if f is orthogonal to the kernel.

9. (Warner Ch.6 #12) Let α and β be n-forms on a compact oriented manifold Mⁿ such thatR

Mα = R

Mβ. Prove that α and β differ by an exact form.

10. (Warner Ch.6 #13) Show that the compactness theorem cannot be strengthened to the assertion of the existence of a subsequence which is convergent in A^p(M).

11. (Warner Ch.6 #16) (The Eigenvalues of the Laplacian) This is an extended exercise in which the fundamental properties of the eigen-functions and eigenvalues of the Laplacian are developed. Proofs for the more difficult parts are outlined, and in some cases are given nearly in full.

Consider the Laplace-Beltrami operator∆ acting on the p-forms A^p(M)for some fixed p. A real number λ corresponding to which there exists a not identically zero p-form u such that ∆u = λu is called an eigenvalue of∆. If λ is an eigenvalue, then any p-form u such that∆u=_λuis called an eigenfunction of∆ corresponding to the eigenvalue λ. The eigenfunctions corresponding to a fixed λ form a subspace of A^p(M)called the eigenspace of the eigenvalue λ.

(a) Prove that the eigenvalues of∆ are non-negative.

(b) Prove that the eigenspaces of∆ are finite dimensional.

在文檔中 Differential Geometry Chin-Lung Wang (頁 102-140)