H T Chapter4

H

T

We have seen in variations of geodesics and submanifolds that a geo- metric object with the minimizing property leads to strong constraint on the global geometry. However, it is in general difficult to find a minimiz- ing object due to the hard non-linear analysis involved. For geodesics we bypass the difficulty using the completeness assumption. For minimal sur- faces this is already subtle and will only be discussed in later chapters.

There is nevertheless a situation where the analysis involved is entirely within linear elliptic PDE and a satisfactory solution is achieved. This is Hodge’s theory of harmonic forms on a compact oriented Riemannian manifold (M^m, g): every cohomology class [w] 2 ^H_dR^p (M, R) admits a unique har- monic representative which is also the absolute minimizer of the energy func- tional E(w) = R

Mw^ ⇤^wwithin the class. Indeed what Hodge proved is the famous decomposition theorem named after him

A^p(M) =Ker4^?Im4 = ^{Hp ?dAp 1?d⇤Ap+1}

for the Laplace operator4 = ^dd⇤+d^⇤d, where d^⇤ = ( 1)^mp+1⇤^d⇤^{is the} formal adjoint of d. The essential part is to show that the harmonic space H^p=Ker4is finite dimensional and consists of C^• p-forms.

Hodge developed his theory around 1930s. One has to go to Hilbert space completions (generalized functions) to start the discussion. Nowa- days a nice approach is to use Fourier transforms on L²spaces and Sobolev spaces H_s to encode the degree of differentiability. The key result is the regularity theorem called Garding’s inequality. Historically it was Hodge’s theorem which laid the foundation of general linear elliptic PDEs.

We present the details in this chapter. We also give applications of har- monic forms under the Bochner principle which describes the difference between 4 and the connection Laplacian trr² in terms of the curvature tensor. This leads to strong topological constraint when Ric 0.

127

1. Hodge⇤^operator

We need some preparation on inner product spaces in linear al- gebra. On an inner product space (V, h ^, i) ^{with dim}RV = m, we define the Hodge⇤operator on L^p(V)(the space of p-vectors)

⇤ ^{: Lp}(V) !^{Ln p}(V)

as follows: for an O.N.B. e₁, . . . , e_m 2V, we consider the “orthogonal complement”

e1^^{. . .}^^ep ⇤

7 !^ep+1^^{. . .}^^em.

By taking into account the sign, for a =e_i1^^{. . .}^^eip 2 ^Lp(V),⇤^a 2 L^{n p}(V)is the wedge of complemented e_j’s with

a^ ⇤^a =e1^^{. . .}^^em.

For general a 2 ^Lp(V),⇤is then defined by linear extensions.

For general p-vectors a, b 2 ^Lp(V), a = Â_|I|=pa_Ie_i1 ^^{. . .}^^eip, b=Â_|J|=pb_Je_j1^^{. . .}^^ejp,

a^ (⇤^b) =

Â

|I|^,|J|=p

a_Ib_J(e_i1^^{. . .}^^eip)^ ⇤(^ej₁^^{. . .}^^ejp)

=h^{a, b}i^e1^^{. . .}^^en

whereh^{a, b}iis the induced inner product on L^p(V): in the notations in Exercise3.1, for P_v = P(v1. . . v_p), P_w =P(w1. . . w_p),

h^v1^^{. . .}^^vm, w1^^{. . .}^^wmi^:=h^Pv, P_wi =^det(v^tw).

Exercise 4.1. Show that⇤ is independent of the choices of orthonor- mal basis with the same orientation and⇤² = ( 1)^{p(n p)}.

For a general oriented Riemannian manifold (M^m, g), we define Hodge ⇤ ^operator ⇤ ^{: Ap}(M) ! ^{Am p}(M) by applying the above definition on each T_p^⇤M.

Exercise 4.2. Let w = f dx¹^^{. . .}^^{dxp in a chart} (U, x). Give the explicit formula of⇤^wusing the metric tensor g_ij.

In order to make use of the exterior algebra and ⇤ operator in its full strength, we recall that vector fields are associated with their corresponding 1-forms by the “Riesz representations”:

T_pM ^{⇠ //} T_p^⇤M v ^// ˜v : w 7! h^{v, w}i^,

˜f ^oo f2 ^Tp^⇤M.

It is also customary to write ˜v as v^[ since it is a one form v_i = g_ijv^j and ˜f as f^#since it is a vector field fⁱ =g^ijf_j.

The advantage to consider forms rather than vector fields is that we may add or wedge forms in a transparent manner.

For example, in the Euclidean space M = R³, g_ij = d_ij, a vector field F =P_∂x^∂ +Q_∂y^∂ +R_∂z^∂ has ˜F= P dx+Q dy+R dz and

r^f ⌘^{grad f} =d f ,^f div F= ⇤^d⇤ ^˜F, curl F=⇤^{gd ˜F.}

Thus we can put everything we need in vector calculus in the frame- work of differential forms. Moreover, the above intrinsic (coordinate- free) expressions extend the definitions of rf , div F and curl F to arbitrary Riemannian manifolds(M, g).

Exercise 4.3. Let(M, g)be a Riemannian manifold, w be its volume form.

(1) For f 2 ^C•(M), find the expression for rf in local coordi- nates(U, x).

(2) For X 2 ^C•(TM), show that (divX)w = LX(w). Deduce the expression for divX in local coordinate(U, x)from it.

Exercise 4.4. Let(M, g)be a Riemannian manifold, f 2 ^C•(M), we define the Laplace–Beltrami operator D_LB acting on C^•(M)by

4LBf =divr^f

Show that in local coordinate(U, x), 4LBf = p¹_g∂x^∂_j

✓p_ggij∂f

∂xⁱ

◆ ,

where g = det(g_ij). Deduce that the highest order term of 4LB is just the usual Laplacian and the lower order term vanishes at x =0 if we take normal coordinate.

Particularly, this shows that 4LB is the usual Laplacian when (M, g) is the standard Euclidean space. In the next section, we will generalize Laplacian to differential forms by Hodge Laplacian.

Example 4.1. The Maxwell equations describing the electric field E = (E₁, E₂, E₃)and the magnetic field B= (B₁, B₂, B₃)read as

curl E = ¹ c

∂B

∂t, (4.1)

div B =0, (4.2)

div E =4pr, (4.3)

curl B = ¹ c

∂E

∂t +^4p c ~_j, (4.4)

where r is the charge density and~j is the current density.

Consider the Minkowski space(-time) R^3,1 with x⁰ = ct and de- fine the energy–momentum tensor by the two form

F=

Â

:=

Â

E_idxⁱ^^dx0+B1dx²^^dx3+B2dx³^^dx1+B3dx¹^^dx2. Or in matrix form:

(F_ij) = 0 BB B@

0 E1 E2 E3

E₁ 0 B3 B2

E2 B3 0 B₁ E₃ B₂ B₁ 0

1 CC CA.

Then we check easily that (4.1) and (4.2) are equivalent to dF=0,

while (4.3) and (4.4) are equivalent to

⇤^d⇤^F= ^4p c ˜j.

Here j = (cr,~_j)is the density vector. The two equations can be fur- ther simplified to one equation using concepts in the next section.

Exercise 4.5. Carry out the details in Example4.1.

2. Harmonic forms

Given a closed (i.e. compact without boundary) oriented mani- fold M^m with a Riemannian metric g, we define an inner product on the space A^p(M)of smooth p-form on M by

(a, b) := Z

Mh^{a, b}i^dV = Z

Ma^ ⇤^b,

where⇤ ^{: Lp} !^{Lm p}is the Hodge⇤operator with⇤² = ( 1)^{(m p)p}. Given a cohomology class[a] 2 ^H_dR^p (M), a closed p-form a⁰ rep- resents [a] if and only if a⁰ = a+db for some b 2 ^{Ap 1}(M). An important observation made by Hodge is that we can obtain a canon- ical representation of the class[a]by requiring the normk^a+dbk^to be minimal.

Let d^⇤ be the formal adjoint¹of d : A^p(M) ! ^Ap+1(M). Namely (a, d^⇤b) := (da, b) =

Z

Mda^ ⇤^b

= Z

Md(a^ ⇤^b) ( 1)^pa^^d(⇤^b)

= ( 1)^mp+1 Z

Ma^ ⇤(⇤^d⇤^b). Since this holds for all a, we get

d^⇤ = ( 1)^mp+1⇤^d⇤ ^{: Ap+1}(M)! ^Ap(M).

1So far(A^p(M),(,))is only a pre-Hilbert space. We call d^⇤the formal adjoint of d to distinguish with the adjoint of a linear operator on Hilbert spaces

Definition 4.2. Let (M^m, g) be a Riemannian manifold, compact or not, the formal adjoint of d is defined by d^⇤ := ( 1)^mp+m+1⇤^d⇤ ^on A^p(M).

Exercise 4.6. Give the formula of d^⇤in local coordinates.

We will consider only compact manifolds in this chapter unless specified otherwise. Let a be a closed p-form which has ”minimal norm” in[a]2 ^H_dR^p (M). Then for any t 2 ^{R, b} 2 ^{Ap 1}(M),

(a+tdb, a+tdb)

= (a, a) +2(a, db)t+ (db, db)t²

 (^{a, a}) This happens if and only if that

0 = (a, db) = (d^⇤a, b), 8^b2 ^{Ap 1}(M) which is equivalent to d^⇤a =0. The discussion implies that

Lemma 4.3. A closed C^• p-form a has minimal normk^akin its de Rham cohomology class if and only if that d^⇤a =0. It is unique if it exists.

Definition 4.4. The Laplace operator on A^p(M) (or called the Hodge Laplacian) is defined by

4 = (^d+d^⇤)² =dd^⇤+d^⇤d : A^p(M) ! ^Ap(M). We call a2 ^Ap(M)is a harmonic form if4^a=0.

Lemma 4.5. For a compact Riemannian manifold(M, g), 4^a = 0 if and only if da=0 and d^⇤a =0.

This follows easily from

(4^{a, a}) = (dd^⇤a+d^⇤da, a) = (d^⇤a, d^⇤a) + (da, da).

Thus, instead of looking for closed forms with minimal norm², Hodge’s observation is to look for the harmonic representative of the cohomology class[a]2 ^H_dR^p (M).

First of all, the operator4is self-adjoint, i.e.(4^{a, b}) = (a,4^b) or 4 = 4^⇤. Furthermore, the truly crucial property is that 4 ^is elliptic—a notion to be explained soon. It turns out that we may and should pose more general questions:

(1) When is the equation4^a = bsolvable?

(2) How to solve it if it is solvable?

The answer is provided by Hodge decomposition theorem whose pre- cise form is given in the next section. Let H = {^a | 4^a = 0} ^be the space of harmonic forms. Then Hodge decomposition theorem asserts that Da= bis solvable if and only if b2 ^H?, the orthogonal complement of harmonic forms.

The “only if part” is easy. For any g2 ^H,

(b, g) = (4^{a, g}) = (a,4^g) =0=) ^b2 ^H?.

For example, since M is compact, the only harmonic functions are constants. Let b 2 ^A0(M) (a function). If Da = b is solvable, then b2^H? =^R?, i.e. R

MbdV =0.

The “if part” is non-trivial even for p = 0. The proof requires a, by now standard, formalism in functional analysis. Here we give a sketch of it and leave the details to the next section.

Let L : H ! H be a continuous (i.e. with bounded norm k^Lk⁾ linear operator on a pre-Hilbert space H with adjoint operator L^⇤. Consider the equation Lw=b. For any f 2 H^,

(w, L^⇤f) = (Lw, f) = (b, f).

Thus any solution w defines a linear functional`such that (4.5) `(L^⇤f) = (b, f) 8^f2 H^.

2This amounts to prove that a minimizing sequence a_ihas a smooth limit after passing to a subsequence, known as the direct method of calculus of variations.

However in practice it relies on technical estimates on various norms which are eventually equivalent to proving the regularity theorem in the related PDE.

If we know that ` is bounded, which is the hard part and requires that b 2 <sup>H?</sup> in the case we want to apply to, then the domain of definition of` can be extended from Im L^⇤ toH, while keeping the same normk`k, by the Hahn–Banach theorem.

Definition 4.6. A bounded linear functional`onHsatisfying (4.5) is called a weak solution to Lw= b.

IfHis furthermore a Hilbert space (i.e. complete), then by Riesz representation theorem, such an`comes from some a 2 H^with

`(L^⇤f) = (a, L^⇤f) = (La, f).

Hence(La b, f) = 0 for all f 2 Hand then La= b. However, the space of C^• forms A^p(M) is never complete and one must pass to certain completion to apply the above formalism. As a result, the so- lution a is a priorily only a certain L²form and we need a “regularity theorem” to conclude that a is indeed a C^• form.

3. Elliptic operators and Hodge decomposition Consider the following typical situation:

E

✏✏

a C^• vector bundle of rank r,

M^m

f : a section

[[

a C^• manifold,

and H = ^C•(M, E)be the vector space of all C^• sections. E.g. E = L^p(T^⇤M), r=C^m_p, andH = ^Ap(M).

Definition 4.7. A R-linear map L : C^•(M, E) ! ^C•(M, E)is a linear differential operator of order d 2 ^N if under any local trivialization U ⇢ ^{Rm, E}|U ⇠=U⇥^{Rr, C•}(U, E _U) ⇠=C^•(U)^r, L is represented by a differnetial operator of order d:

L : C^•(U)^r !^C•(U)^r. That is, under the multi-index notations,

L f =

Â

|a|d

A_a(x)D^af , x2 ^U,

where f = (f¹, . . . , f^r)^t and A_a 2 ^Mr⇥r(C^•(U)). Write

L=

Â

|a|=d

A_a(x)D^a+

Â

|a|<d

A_a(x)D^a with non-trivial top order term L_d :=_Â|a|=dA_aD^a 6⌘^0.

For x = (x₁, . . . , x_m)^t 2 ^Rm, we define the polynomial in x_i’s by p_L(x, x) :=

Â

a

Aa(x)x^a

called the symbol of L over E|U. In general it depends on the trivial- ization and is not globally defined. Nevertheless we have

Exercise 4.7. Show that the top order term s_L(x, x) := p_Ld(x, x)

forms a tensor s_L 2 ^C•(Sym^d(T^⇤M)⌦^{End E}), called the principal symbol of L.

Definition 4.8. Let L be a differential operator of order d on E !^M.

(1) L is elliptic over an open subset U⇢ ^{M if} s_L(x, x)2 ^{End E}

is invertible for all x 2^{U and x} 2 ^TxM\ {⁰}^.

(2) Suppose that E ! M is equipped with a C^• bundle metric.

Then L is uniform elliptic over a subset S ⇢ ^{M if}

|^sL(x, x)v| ^C|^x|^d|^v|^, 8 ^x2 ^{S, x} 6=^0, for some C independent of x 2^S.

In particular, ellipticity on U implies uniform ellipticity over any compact subsets S ⇢^U.

Remark 4.9. In a similar manner we define differential operators L : C^•(E) ! ^C•(F) of order d between vector bundles over M. The principal symbol is a section s_L 2 ^C•(Sym^d(T^⇤M)⌦^Hom(E, F)) and L is elliptic if s_L(x, x)is invertible for all x 2 ^{M, x} 2 ^T⇤M\ {⁰}^. In particular rk E = rk F. But E and F need not be isomorphic. For our current purpose it is enough to consider the case E =F.

Exercise 4.8. Show that s₄(x, x) = |^x|²Id for the Hodge Laplacian 4^{on Ap}(M), hence it is uniformly elliptic on M.

Now we state two fundamental results in elliptic PDE.

Let Lw = bbe a linear elliptic PDE on E ! M over a compact manifold M. In order to apply the Hilbert space formalism we as- sume that(M, g)is Riemannian and E has a bundle metric h. If E is a complex vector bundle we require that the metric h is hermitian.

Denote by L : H ! H ^with H = ^C•(E). His a pre-Hilbert space under(f , g):=R

Mh(f , g)dV. Denotek^fk = (^{f , f})^1/2. Then

Theorem 4.10 (Regularity theorem). Any weak solution of Lw = bis automatically smooth.

Theorem 4.11 (Compactness theorem). For a sequence a_n 2 H^{, if} k^ank  ^{C and} k^Lank  C are both bounded, then {^an} has a Cauchy subsequence.

Remark 4.12. We will show later that both theorems are consequences of the G˚arding inequality: for f 2 H^,

k^fks+d ^C(k^{L f}ks+k^fks),

where d is the order of L, and the norms are Sobolev norms to be defined and studied in section 5.

Now we return to the case L=D,H = ^Ap = A^p(M). By assum- ing the above two PDE theorems, we prove

Theorem 4.13 (Hodge decomposition theorem). Let H = ^Hp := {^a 2 ^Ap(M)| 4^a =0}be the space of harmonic p-forms.

(1) dim H <•, and

(2) A^p =^{H ?}4^Ap. That is, Im4 =^H?. PROOF. (1) If dim H =•, we select

u1, u2, . . .2 ^H, k^uik =^{1, u}i ?^uj 8ⁱ6= ^j

satisfying Du_i = 0. It is clear that{^ui} has no Cauchy subsequence hence contradicts to the compactness theorem (Theorem4.11). Thus

l :=dim H <• and H ⇢ His a closed subspace. In particular H^? is also a closed subspace and A^p =H H^?. Indeed, pick an O.N.B.

{^w1, w2, . . . , w_l}of H. For any a2 ^Ap, a =b+

Â

(a, w_i)w_i =: b+H(a) where b 2^{H?and H}(a)is the harmonic projection of a.

(2) We need to show H^? =4^Ap. The direction “ ” is clear:

(4^{a, g}) = (a,4^g) =0 8^g 2^H. For the other direction we need the following

Claim 4.14. There exists c >0 such thatk^bk  ^ck4^bk^{for all b}2 ^H?. Let a2 ^H?and we define a linear functional`on Im4^by

`(4^f) := (a, f).

It is well defined: if4^f1 =4^{fthen f}1 f 2 ^{Hand then} (a, f₁) = (a, f). Also ` is a bounded linear functional on Im4^{: let b} = f H(f)2 ^{H?, then}

k`(4<sup>f</sup>)k = k`(4^b)k = k(^{a, b})k  k^ak · k^bk

^ck^ak · k4^bk = (^ck^ak) · k4^fk^.

By Hahn–Banach theorem, ` can be extended to a bounded linear function onH = <sup>Ap. That is,</sup>`is a weak solution to4^w =a.

By the regularity theorem (Theorem4.10), there exists a smooth w 2 ^{Ap such that}4^w=a. So H^? ⇢ 4^Apfollows. ⇤ PROOF OF CLAIM 4.14. Suppose the contrary, then there exists a sequence b_j 2^H?withk^bjk = ^{1 and}k4^bjk !^0.

By the compactness theorem (Theorem 4.11), we may assume that{^bj}is itself a Cauchy sequence. For y2 ^{Ap, define}

`(y) := lim

j!•(b_j, y).

The linear functional`is clearly bounded: k`k ^{1, and}

`(4^y) = lim

j!•(b_j,4^y) = lim

j!•(4^bj, y) =0

by Cauchy’s inequality, i.e.`is a weak solution of4^b =0.

By the regularity theorem (Theorem4.10), there exists a b 2 ^Ap such that`(y) = (b, y)and4^b=0, i.e. b2 ^H.

Then (b_j, y) ! (^{b, y}) and in particular 0 = (b_j, b) ! k^bk^{2 and} thus b =0. On the other hand, the Cauchy sequence{^bj} ^{has limit} bimplies thatk^bk = ^limj!•k^bjk =1, which is a contradiction. ⇤

The proof actually applies to more general cases:

Exercise 4.9. Extend the Hodge decomposition for any elliptic op- erator L : C^•(E) ! ^C•(F) between two vector bundles E, F over a compact M. (Assuming regalarity and compactness theorems.)

For the Hodge Laplacian4, a lot more can be said.

Definition 4.15. Since A^p =^H (4^Ap), we define the Green operator

G := 8<

:

0 on H,

D ¹ on H^?. i.e. I =H+DG,

where H : A^p !^His the harmonic projection. Hence Ga is the unique solution in H^?for Dw =a H(a).

Recall that a bounded linear operator is compact if the image of any bounded subset has compact closure.

Exercise 4.10. Show that

(1) G commutes with any operator T with T4 = 4^T.

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

Notice that from4 = ^dd⇤+d^⇤d we see immediately that d4 = dd^⇤d =4^{d and d⇤}4 =^d⇤dd⇤ =4^{d⇤. Hence}

[d, G] = [d^⇤, G] = [D, G] =0.

The Hodge decomposition can be refined to A^p =H Im4 =^{H Im d Im d⇤,}

which is still an orthogonal decomposition. A precise formula could be given in terms of the Green operator G. Indeed we can rewrite

the identity I = H+DG as

a = H(a) + (dd^⇤+d^⇤d)Ga

= H(a) +d(d^⇤Ga) +d^⇤(dGa). In particular we have:

Proposition 4.16. If da=0 then a= H(a) +d(d^⇤Ga).

Therefore, the harmonic representation of a cohomology class is unique: if a₁ a₂ =db then H(a₁) H(a₂) = H(db) =0.

Here is another simple yet important application of the Hodge decomposition theorem:

Theorem 4.17 (Poincar´e duality). Let M be a C^•compact oriented man- ifold of dimension m. Then the natural pairing

H_dR^p (M)⌦^H_dR^{m p}(M) ! ^Hm_dR(M) ⇠=^R : (w, h)7!

Z

Mw^^h, is a perfect pairing. In particular,

H_dR^p (M) ⇠= H_dR^{m p}(M)^⇤.

PROOF. Let g be a metric on M. It suffices to prove the theorem using harmonic representatives of de Rham cohomology.

Since⇤4 = 4⇤, we see that⇤ ^{: Hp} ! ^{Hm p} which maps har- monic forms to harmonic forms. From⇤²= ( 1)^{p(n p)} we conclude that⇤ ^{: Hp} ⇠=^{Hn p}. This isomorphism depends on g and is not the natural one stated in the theorem.

Nevertheless it shows that Hⁿ is spanned by⇤¹=dV_gand hence H_dR^m (M) ⇠= ^R under[W] 7! R

MW (the trace map). Moreover, it also implies that the natural pairing is perfect by noticing that

(w,⇤^w)7!

Z

Mw^ ⇤^w =k^wk² 6=⁰

if[w]6=0. This completes the proof. ⇤

Exercise 4.11. (1) Let M = R^m/L where L ⇢ ^Rm is a lattice gen- erated by m linearly independent vectors. Let g be the flat metric induced from R^m. Determine H^p(M) and show that the wedge of harmonic forms is still harmonic.

(2) Give an example(M, g)and two harmonic forms w, h so that w^^h is not harmonic.

4. Bochner Principle

Given any C^• manifold M, letrbe an affine connection on TM.

We denote the induced connection on T^⇤M and^VpT^⇤M again byr^. For w 2 ^Ap(M),r^w(X0, X1, . . . , X_p) 2 ^Ap(M)⌦^A1(M)is given by

(r^w)(X0, . . . , X_p) = (rX₀w)(X1, . . . , X_p). We define the anti-symmetrization(r^w)^alt 2 ^Ap+1(M)by

(r^w)^alt(X₀, . . . , X_p) =

Â

( 1)^j(rXjw)(X₀, . . . , cX_j, . . . , X_p).

Exercise 4.12. Show that ris torsion-free if and only if (r^w)^alt = dw.

Now, we assumeris the Levi-Civita connection on(M, g). Definition 4.18. For X, Y 2 ^C•(TM), we define r²_X,Y acting on any tensor fields T by

r²_X,Y^T =rXrYT rrXYT.

The connection Laplace acting on any tensor fields T is defined by the trace tr(r^2T)with respect to g.

Exercise 4.13. For f 2 ^C•(M), show that 4LBf = tr(r^2f), where 4LBis the Laplace–Beltrami operator. Also, we have4 = 4LB.

In general,4^{and tr}r²are related by

Proposition 4.19 (Bochner formula). Let M be compact.

4 = ^trr²

Â

i,j

hⁱ^ⁱe_jR(e_i, e_j),

where{^ej}is an orthonormal local frame of TM,{^hi}is its dual frame.

In the literature this is also known as the Bochner–Lichnerwicz–

Witezenb¨ock formula.³.

Exercise 4.14. With the notations in proposition4.19, for w 2 ^Ap(M), prove that

(4.6) dw =

Â

hⁱ^ reiw; d^⇤w =

Â

i_eireiw.

Exercise 4.15. (1) Prove the Bochner formula4.19using (4.6).

(2) Denote byr^⇤ the formal adjoint ofr, show that

r^⇤r = ^trr^2.

Hence, another common form of Bochner formula is given by 4 = r^⇤r

Â

i,j

hⁱ^ⁱe_jR(e_i, e_j).

Now, following Bochner, we use Proposition 4.19 to derive the topological constraints given by the Ricci curvature conditions.

Corollary 4.20 (Bochner). Let(M, g)be a closed Riemannian manifold.

(1) If Ric>0 then b1 =h¹(M) =0. More precisely,

(2) If Ric 0 then h¹(M)  ^m=dim(M). If furthermore 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/G where G⇠=^Zn is a lattice.

PROOF. Before proving the statements, we first prove an identity also due to Bochner.

3In general, an identity expressing difference between two second–order el- liptic operator with the same principle symbols in terms of curvatures is known as Weitzenb¨ock formula or Bochner formula. Such formula was first indicated by Witzenb¨ock in 1925, yet it was Bochner who first used the formula to relate topol- ogy and curvature estimates on compact manifolds (cf. Corollary 4.20) in 1948.

There are many variants for formula of such type in different contexts. For in- stance, in 1963, Lichnerwicz developed an analogous formula for Dirac operators on spin bundles, which we will discuss in later chapter.

Claim 4.21. For any q 2 ^A1(M),

h4^{q, q}i = ¹₂4|^q|²+|r^q|²+Ric(˜q, ˜q), where ˜q2 ^C•(TM)is the metric dual of q (cf. section 1).

Given p 2 M, assume that {^hi}^m_i=1 is the dual frame of a local orthonormal frame{^ei}^m_i=1of TM around p. We write q =Â^m_i=1a_ihⁱ.

Since{^ei}is an orthonormal frame, g(e_i, e_j) = d_ijand hencereie_i = 0. The connection Laplace of q is then given by

trr^2q =

Â

i=1r²e_i,e_iq =

Â

i=1re_ire_iq.

Therefore, from Proposition4.19we deduce h4^{q, q}i =

Â

i hre_ire_iq, qi

| {z }

(I)

h

Â

i,j

hⁱ^ⁱe_jR(e_i, e_j)q, qi

| {z }

(II)

.

By direct calculation, (I) =

Â

i

e_ihreiq, qi |reiq|² = ¹

24|^q|² |r^q|^2, and

(II) =

Â

i,j

(R(e_i, e_j)q)(e_j)h^{hi, q}i

Notice that for X, Y, Z 2 ^C•(M), w 2 ^A1(M), a direct computa- tion shows(R(X, Y)w)(Z) = w(R(X, Y)Z). Therefore,

(II) =

Â

i,j

q(R(e_i, e_j)e_j)h^{hi, q}i

=

Â

i,j,k,l

a_ka_ih^Rl_jij^el, e_ki =

Â

i,j,k

a_ka_iR_kjij

= Ric(

Â

k

a_ke_k,

Â

i

a_ie_i) = Ric(˜q, ˜q). As a result, we have proved:

h4^{q, q}i = ¹₂4|^q|²+|r^q|²+Ric(˜q, ˜q).

We now prove (1) and (2) by contradiction. If h¹(M) 6= 0, we take a non-zero cohomology class[q] 2 ^H_dR¹ (M). By Hodge decomposition

(cf. 4.13), we choose q 6= 0 to be the harmonic 1-form representing [q]6= ^0.

For (1), if Ric > 0, choose p 2 M such that |^q(p)| > ^{0 achieves} maximal. Then(4|^q|²)(p) 0 by second derivative test. However, this contradicts to Ric(˜q, ˜q) > 0.

For (2), if Ric 0 , by taking integration on both sides of4.21, 0= ¹

2 Z

M4|^q|²+ Z

M|r^q|²+ Z

MRic(˜q, ˜q). By Stokes’ theorem, since ∂M =∆, we have

0= Z

M|r^q|²+ Z

MRic(˜q, ˜q).

Since Ric 0, we must conclude that r^q ⌘ 0, i.e. q is parallel and is determined by q_q 2 ^Tq^⇤M,8^q 2 ^{M. Hence h1}  ^{dim T}q^⇤(M) = m.

Also,r^q ⌘^{0 and Ric} 0 in turn implies Ric(˜q, ˜q) ⌘0. Now, if Ric>

0 at one point p and q6= 0, we have Ric(˜q, ˜q) >0, a contradiction.

On the other hand, if the equality holds, h¹ = dim M = m, then the universal cover ˜M ! M has m parallel 1-forms. Therefore it has m parallel vector fields. We conclude that ˜M ⇠=^Rm. ⇤

5. Fourier Transform and Sobolev Spaces

Let us first introduce some standard notations.

(1) x= (x1, . . . , x_m) 2^Rm.

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

(3) a= (a1, . . . , a_m)multi-index, a_i 2^N[ {⁰}^, |^a| = Â^m_i=1a_i. (4) x^a =x^a₁¹x₂^a2· · ·^xam^m.

(5) ∂^a = _∂xa1^∂|a|

1 ···∂x^amm , D^a = ( i)^|a|∂^a.

To prove the regularity theorem and compactness theorem (cf.

theorem4.10,4.11), we need to define the concept of Sobolev space. To achieve this, we first need the some rudiments in Fourier analysis.

Definition 4.22 (Fourier transdorm). For x 2 ^{Rm, x} 2 ^{Rm, let f} 2 C₀^•(^Rm, C)be a smooth function with compact support. The Fourier

transform of f is defined as ˆf(x) = (2p) ^m/2

Z

R^me ^ix·xf(x)dx.

Also, recall the convolution of two functions f ⇤^{g :}=

Z

R^m f(x y)g(y) dy= Z

R^m f(z)g(x z)dz.

It is a standard trick to use convolution to construct smooth ap- proximation of functions with inferior smothness. Let f 2 ^C₀^•(^Rm) withR

R^m f =1, f 0, f(0) 6=0. Such a function is sometimes called a molifier. The existence of molifiers is evident from our construc- tion on bump functions in chapter 1. For u 2 ^{R+, define f}u(x) :=

u1^mf⇣

xu

⌘.We can see that R

R^m f_u = 1. Such a sequence f_u is called a

“d-function”, and we can write it as d0= lim

u!0f_u.

Proposition 4.23. For g2 ^{C0, f}u⇤^g(x)! ^g(x)as u!⁰+is a smooth approximation of identity⁴.

Exercise 4.16.

(1) Prove the above proposition.

(2) Show that C^•₀ (R^m) is dense in L²(R^m) (with respect to L²- norm).

Thus, we may extend the definition Fourier transform ˆf to f 2 L²(^Rm) by continuity. That is, we define ˆf = lim_k! bf_k, where f_k 2 C₀^•(^Rm) and f_k ! ^{f in L2}(^Rm). Recall that we also have inverse Fourier transform.

Definition 4.24 (Inverse Fourier Transform). For g(x) 2 ^C₀^•(^Rm), the inverse Fourier transform ˇg(x)is defined as⁵

ˇg(x):= (2p) ^m/2 Z

R^me^ix·xg(x)dx.

4That is, d₀=lim_u!0fuis the identity element for⇤

5There are several conventions for Fourier transform and its inversion. An- other common convention is that ˆf(x) = R

R^m f(x)e ^ix·xdx while the inversion is given by ˇg(x):= (2p) ^mR

R^mg(x)e^ix·xdx.

Similarly, we can define ˇg for g 2 ^L2(^Rm) by continuity argu- ments. We list the basic properties and correspondence with con- volutions in the following theorem and refer the proof to [Gil95], Chapter 1.1.

Theorem 4.25 (Basic Properties of Fourier Transform). For f 2 ^C₀^•(^Rm), we have

(1) ˇˆf= f .

(2) D_x^a ˆf(x) = \x^af(x),_D\^a_xf(x) =x^a ˆf(x). (3) [_f ⇤^g= ˆfˆg, ˆf⇤ ˆg = f^d·^g.

Moreover, Fourier Transform on L²(^Rm)is an isometry onto itself⁶, i.e.

L²(R^m) ^{ˆ· //} L²(R^m)

ˇ·

oo .

From the formula dD_x^af =x^a ˆf, we notice that derivatives of f will corresponds multiplications of ˆf in the frequency (phase) space. This formula enlightens the idea of weak derivatives. First, for k 2 ^N, f 2 ^C•0 (R^m, C), any multi-index a with |^a| = k, we consider the norm to evaluate the L²-norms of its derivatives:

Z

R^m

Â

|a|k

|^Daf(x)|^2dx.

Thus, from dD^a_xf =x^a ˆf, we have

|^f|²_k ^:=

Â

|a|k

Z

R^m|^Daf(x)|^2dx

=

Â

|a|k

Z

R^m|^Ddaf(x)|^2dx =

Â

|a|k

Z

R^m|^xa|²|^bf(x)|^2dx.

Observe that there exists constant C₁, C2 >0 such that C1(1+|^x|²)^k 

Â

|a|=k|^xa|²^C2(1+|^x|²)^k.

6This is called Plancherel theorem

Hence, the norm is equivalent to the norm⁷ Z

R^m(1+|^x|²)^k|^bf(x)|^2dx,

which is free of any differentiation. More generally, we have the following definition.

Definition 4.26.

(1) For s 2 ^R, which is regarded as order of L²-derivatives , we define

Sobolev s-norm of a L²-function f by

|^f|²s := Z

R^m(1+|^x|²)^s| ^ˆf(x)|^2dx, and the inner product(·^,·)s by

(f , g)_s := Z

R^m(1+|^x|²)^s ˆf(x)ˆg(x) dx.

2) The Sobolev s-space H_s(^Rm)is defined to⁸be the completion of C₀^•(^Rm)in L²(^Rm)(w.r.t the s-norm).

7Another common choice of weight is(1+|^x|)^2k.

8Here is an alternative way to define Sobolev space. The notion of weak derivative can be defined more directly by ”integration by part”. That is, we say v 2 L²(R^m, C) is a a-th L²-weak derivative for a locally integrable function

f 2L¹_loc(R^m)if it satisfies Z

R^mvjdx= ( 1)^|a|

Z

R^m f(∂^aj)dx, 8^j2C₀^•(R^m, C).

For k2N, if we assume that f 2L²(R^m)having L²-derivative d^af for any|^a| k, we then define Sobolev k-norm by

(4.7) |f|²_k ^:=

Z R^m

Â

|a|k

|d^af(x)|²dx,

and define the Sobolev space H^k(R^m)by

(4.8) H^k(R^m):={u2 L¹_loc(R^m):9^∂af 2L²(R^m), 8|^a| k}^.

The definition is in fact equivalent to our definition. More generally, for 1 p <

•, we can consider W^k,p consisting of locally integrable function f having L^p- weak derivative up to order k. We can also define the Sobolev norm on W^k,pby

|f|k,p=^⇣Â_|a|k|d^af|^p_L^p⌘1/p.

Then again by the formula dD^a_xf =x^a ˆf, D^a : H_s\^C0^• ! ^H_{s |}a|,

|^Daxf|²_{s |a|} = Z

R^m(1+|^x|²)^{s |a|}|^{xa ˆf}(x)|^2dx.

The key question here is that: although we define the weak de- rivative by multiplication in frequency space, when is f is actually C^k-differentiable if ˆf lies in some Sobolev space H_s, i.e. x^a ˆf(x)lies in L²-space? This is answered by the following

Theorem 4.27 (key lemmas in Euclidean spaces).

(1) (Sobolev lemma) If s> k+m/2 and f 2 ^Hs, then f 2 ^{Ck and} there exists a universal constant C >0 depending only on s such that

|^f|_C^k ^C|^f|s.

where |^f|_C^k = Â_|a|ksup_x2Rm|^∂af(x)| is the supremum norm of C^k-functions.

(2) (Rellich lemma) If s>t, H_s ,! ^Ht is a compact imbedding.

(3 (interpolation inequality) Let s>t >u.8 ^e>0,9 ^C(e)such that

|^f|t ^e|^f|s+C(e)|^f|u

for all f 2 ^C0^•. PROOF.

(1) Let k =0. First, we consider f 2 ^C0⁰(R^m):

|^f(x)| = Z

R^me^ixx ˆf(x) dx

= Z

R^m

he^ixx ˆf(x)(1+|^x|²)^s/2i(1+|^x|²) ^s/2dx

 |^f|s

✓_Z

R^m

1

(1+|^x|²)^{s dx}

◆1/2

(Cauchy–Schwarz inequality)

^C|^f|s,

for some C depending on s >m/2. In other words, |^f|_C⁰  C|^f|s, for some constant C>0 depending only on s.

Now, for any f 2 ^Hs(^Rm), we can choose a sequence f_j ^H!^{s f with f}j 2 ^C₀⁰(^Rm). By the above result,

|^fi f_j|_C⁰ ^C|^fi f_j|s

infers that f_i forms a Cauchy sequence in C⁰. Thus, f_i ! ^f uniformly implies that f 2 ^{C0. So}

|^f|_C⁰  |^{f f}i|_C⁰+|^fi|_C⁰ ^C(|^{f f}i|s+|^fi|s) and then|^f|_C⁰ ^C|^f|sby taking i !^•.

For k >0 and s > k+m/2, apply the same argument to D^af , for any multi-index a with|^a| =^k:

|^Daf|_C⁰ ^C|^Daf|s k ^C|^f|s.

Hence, for any a with |^a| = ^{k, Daf} 2 ^C0(^Rm), and thus H^s ⇢^Ck. Also, we obtain|^f|_C^k ^C|^f|s.

(2) Consider K ⇢cpt R^m, f_n 2 ^C•\^Hs with supp(f_n) ⇢ ^{K and}

|^fn|s ^C.

Let g 2 ^C0^•(^Rm) with g ⌘ 1 on K. Then g· ^fn = f_n and thus ˆf_n = ˆg⇤ ^ˆfn. We then have

∂_jˆfn =∂_j(ˆg⇤ ^ˆfn) = (∂_jˆg)⇤ ^ˆfn.

|^∂jˆf_n(x)|  Z

R^m

∂ ˆg

∂x_j(x h)ˆf_n(h) dh

 |^fn|s

Z

R^m

|(^∂x_jˆg)(x h)|² (1+|^h|²)^{s dh}

!¹₂

(Cauchy–Schwarz).

We denote h_j(x) := ^⇣R

R^m(1+|^h|²) ^s|(^∂xjˆg)(x h)|^2dh⌘

12

. From theorem4.25, for any multi-indices a, b,

x^bD^a_xˆg = \D^b_x(x^bg),

and D^xb(x^bg) 2 ^C•0 (^Rm) since g 2 ^C•0 (^Rm). We conclude from Plancherel’s theorem that |^xbD_x^aˆg|0 = |^Dxb(x^bg)|0 <