Chapter 7
Proper Fredholm Submanifolds in Hilbert Spaces
In this chapter we generalize the submanifold theory of Euclidean space to Hilbert space. In order to use results the infinite dimensional differential topo-logical we restrict ourself to the class of proper Fredholm immersions (defined below).
140 Part I Submanifold Theory
true in general. For example, the infinite dimensional differential topology developed by Smale and infinite dimensional Morse theory developed by Palais and Smale will not work for general submanifolds of Hilbert space without further restrictions. Recall also that the spectral theory of the shape operators and the Morse theory of the Euclidean distance functions of submanifolds of Rnare closely related and play essential roles in the study of the geometry and topology of submanifolds of Rn. Here again, without some restrictions important aspects of these theories will not carry over to the infinite dimensional setting. One of the main goals of this section is to describe a class of submanifolds of Hilbert space for which the techniques of infinite dimensional geometry and topology can be applied to extend some of the deeper parts of the theory of submanifold geometry.
The end point mapY :ν(M) →V for an immersed submanifoldM of a Hilbert spaceV is defined just as in Definition 4.1.7; i.e.,Y : ν(M) → V is given byY(v) =x+vforv ∈ ν(M)x.
7.1.2. Definition. An immersed finite codimension submanifoldM ofV is proper Fredholm (PF), if
(i) the end point mapY is Fredholm,
(ii) the restriction ofY to each normal disk bundle of finite radiusris proper.
Since the basic theorems of differential calculus and local submanifold geometry work for PF submanifolds just as for submanifolds of Rn, Proposition 4.1.8 is valid for PF submanifolds of Hilbert spaces. In particular, we have
dYv = (I −Av, id), (7.1.1) which implies that
7.1.3. Proposition. The end point mapY of an immersed submanifoldM of a Hilbert spaceV is Fredholm if and only ifI−Av is Fredholm for all normal vectorvofM.
7.1.4. Remarks.
(i) An immersed submanifoldM of Rn is PF if and only if the immersion is proper.
(ii) If M is a PF submanifold of V, and M is contained in the sphere of radiusrwith centerx0inV, thenv(x) =x0 −xis a normal field onM with lengthr, andY(x, v(x)) =x0. SinceY is proper on ther-disk normal bundle, M is compact. This implies thatM must be finite dimensionional. It follows that PF submanifolds of an infinite dimensional Hilbert spaceV cannot lie on a hypersphere ofV. In particular, the unit sphere ofV is not PF.
7. Proper Fredholm Submanifolds in Hilbert Spaces 141 7.1.5. Examples.
(1) A finite codimension linear subspace ofV is PF.
(2) Letϕ : V → V be a self-adjoint, injective, compact operator. Then the hypersurface
M ={x∈ V| ϕ(x), x= 1}
is PF. To see this we note that v(x) = ϕ(x)/ϕ(x) is a unit normal field to M, and Av(x)(u) = −ϕ(u)T Mx/ϕ(x) is a compact operator onT Mx, where ϕ(u)T Mx denote the orthogonal projection of ϕ(u) onto T Mx. So it follows from Proposition 7.1.3 that the end point map ofM is Fredholm. Next assume that xn ∈ M,{λnϕ(xn)}is bounded, andY(xn, λnϕ(xn)) = xn+ λnϕ(xn)→ y. Thenxnis bounded, andxn+λnϕ(xn), xn =xn2+λnis bounded, which implies thatλnis bounded. Sinceϕis compact and{λnxn}is bounded,ϕ(λnxn)has a convergent subsequence, and so{xn}has a convergent subsequence.
7.1.6. Theorem. SupposeGis an infinite dimensional Hilbert Lie group,G acts on the Hilbert spaceV isometrically, and the action is proper and Fredholm.
Then every orbitGxis an immersed PF submanifold ofV.
Proof. First we prove that the end point mapY ofM = Gxis Fred-holm. Because every isometry ofV is an affine transformation, we have
(I−Av)(ξ(x)) = (ξ(x+v))Tx,
whereξ ∈ G,v ∈ν(M)x, anduTxdenotes the tangential component ofuwith respect to the decompositionV =T Mx⊕ν(M)x. It follows from the definition of Fredholm action that the differential of the orbit map ateis Fredholm. So the two maps ξ → ξ(x) and ξ → ξ(x+v) are Fredholm maps from G to V. In particular, T(Gx)x and T(G(x+ v))x+v are of finite codimension.
So the map P : T(G(x+ v))x+v → T(Gx)x defined by P(u) = uTx is Fredholm. Hence I −Av is Fredholm, i.e., Y is Fredholm. Next we assume that xn ∈ M, vn ∈ ν(M)xn, vn ≤ r, and Y(xn, vn) → y. Then there exist linear isometry ϕn ofV andcn ∈ V such thatgn = ϕn +cn ∈ Gand xn = gn(x). Note thatdgn =ϕn,un =ϕ−n1(vn)∈ν(M)x, and
Y(gnx, vn) =ϕn(x)+cn+ϕn(un) =ϕn(x+un)+cn =gn(x+un)→ y.
Since {un} is a bounded sequence in the finite dimensional Euclidean space ν(M)x, there exists a convergent subsequenceuni → u. So we havegni(x+ uni)→ yandx+uni → x+u. It then follows from the definition of proper action thatgni has a convergent subsequence inG, which implies thatxni has a convergent subsequence inM.
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7.1.7. Proposition. LetM be an immersed PF submanifold ofV, x∈ M, v ∈ ν(M)x, and Av the shape operator of M with respect to v. Then:
(1)Av has no residual spectrum,
(2) the continuous spectrum ofAv is either{0}or empty,
(3) the eigenspace corresponding to a non-zero eigenvalue ofAv is of finite dimension,
(4)Av is compact.
Proof. SinceAv is self-adjoint, it has no residual spectrum. Note that the eigenspace ofAv with respect to a non-zero eigenvalueλis
Ker(λI −Av) = Ker(I− 1
λAv) = Ker(I −Avλ).
So (3) follows from Proposition 7.1.3. Now supposeλ= 0,Ker(Av−λI) = 0, andIm(Av−λI)is dense inT Mx. SinceAv−λIis Fredholm,Im(Av−λI) is closed and equal toT Mx, i.e.,Av −λI is invertible, which proves (2). To prove (4) it suffices to show that ifλi is a sequence of distinct real numbers in the discrete spectrum of Av andλi → λthenλ = 0. But ifλ = 0, then the self-adjoint Fredholm operator P = I −Av/λ induces an isomorphismP˜ on V /Ker(P), soP˜ is bounded. Letδ denoteP˜. Then|(1−λi/λ)−1| ≤ δ, and hence|λ−λi|/|λ| ≥1/δ >0, contradictingλi →λ.
It follows from (7.1.1) thate∈ ν(M)x is a regular point ofY if and only if I −Ae is an isomorphism. Moreover, the dimension ofKer(I −Ae) and Ker(dYe)are equal, which is finite by Proposition 7.1.3. Hence the Definition 4.2.1 of focal points and multiplicities makes sense for PF submanifolds.
7.1.8. Definition. Let e ∈ ν(M)x. The point a = Y(e) in V is called a non-focal point for a PF submanifold M of V with respect to xif dYe is an isomorphism. If m = dim(Ker(dYe)) > 0 then ais called a focal point of multiplicitymforM with respect tox.
The setΓof all the focal points ofV is called the focal set ofM inV, i.e., Γis the set of all critical values of the normal bundle mapY. So applying the Sard-Smale Transversality theorem [Sm2] for Fredholm maps to the end point mapY ofM, we have:
7.1.9. Proposition. The set of non-focal points of a PF submanifoldM of V is open and dense inV.
By the same proof as in Proposition 4.1.5, we have:
7. Proper Fredholm Submanifolds in Hilbert Spaces 143 7.1.10. Proposition. Let M be an immersed PF submanifold of V, and a ∈ V. Let fa : M → R denote the map defined by fa(x) = x− a2. Then:
(i) ∇fa(x) = 2(x − a)Tx, the projection of (x − a) onto T Mx, so in particularx0 is a critical point offaif and only if(x0−a) ∈ν(M)x0,
(ii) 12∇2fa(x0) =I −A(a−x0)at the critical pointx0 offa,
(iii)fais non-degenerate if and only ifais a non-focal point ofM inV, It follows from Propositions 7.1.9 and 7.1.10 that:
7.1.11. Corollary. If M is an immersed PF submanifold of V, thenfa is non-degenerate for allain an open dense subset ofV.
As a consequence of Proposition 7.1.7 and 7.1.10:
7.1.12. Proposition. LetM be an immersed PF submanifold ofV. Suppose x0 is a critical point offa andVλis the eigenspace ofA(a−x0)with respect to the eigenvalueλ= 0.
Then:
(i)dim(Vλ)is finite, (ii)Index(fa, x0) =
{dim(Vλ)|λ > 1}, which is finite.
Morse theory relates the homology of a smooth manifold to the critical point structure of certain smooth functions. This theory was extended to infinite dimensional Hilbert manifolds in the 1960’s by Palais and Smale ([Pa2],[Sm1]) for the class of smooth functions satisfying Condition C (see Part II, chapter 1).
7.1.13. Theorem. LetM be an immersed PF submanifold of a Hilbert space V, and a ∈ V. Then the map fa : M → R defined by fa(x) = x− a2 satisfies condition C.
Proof. We will writef forfa. Suppose
|f(xn)| ≤c, ∇f(xn) →0.
Letun be the orthogonal projection of(xn −a)ontoT Mxn, andvn the pro-jection of(xn−a)ontoν(M)xn. Sincexn−a2 ≤candun →0,{vn}is bounded (say byr). So(xn,−vn)is a sequence in ther-disk normal bundle of M, and
Y(xn,−vn) =xn−vn = (xn−a)−vn+a=un+a→ a.
Since M is a PF submanifold, (xn,−vn) has a convergent subsequence in ν(M), which implies thatxnhas a convergent subsequence inM.
144 Part I Submanifold Theory
7.1.14. Remark. LetM be an immersed submanifold ofV (not necessarily PF). Then the condition that all fa satisfy condition C is equivalent to the condition that the restriction of the end point map to the unit disk normal bundle is proper.