Rational Sphere Maps

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Rational Sphere Maps

Partially supported by NSF grant DMS 13-61001

John P. D’Angelo

University of Illinois at Urbana-Champaign

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Proper maps between balls

Bn denotes unit ball in Cn.

Suppose f : Bn→ BN proper, holomorphic.

Case 1. N < n. No examples. The inverse image of a point would be a positive diml. compact complex analytic subvariety.

Case 2. N = n. Completely understood. Blaschke products if n = 1, automorphisms if n ≥ 2.

Case 3. N > n. Non-rational examples exist. If the map is smooth

and n ≥ 2, then the map is rational (Forstneriˇc) and extends

holomorphically past (Cima-Suffridge).

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Proper maps between balls

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Proper maps between balls

holomorphically past (Cima-Suffridge).

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We want to understand all rational sphere maps. It is natural to allow constant maps to the sphere. Spherical equivalence:

f = χ ◦ g ◦ ξ

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Theorem

A proper holomorphic self-map of the unit disk is a finite Blaschke product: f (z) = ei θ m Y j =1 z − aj 1 − ajz . Theorem

(almost known to Poincar´e). Proved independently by H.

Alexander and Pinchuk in the 1970’s) Assume n = N ≥ 2. Assume

f : Bn→ Bn proper, holomorphic. Then f is an automorphism,

hence a linear fractional transformation.

There should be one unified theory that explains both results.

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Note: If n = 1 and q(z) 6= 0 on closed unit disk, then there is a p

with p_q a rational sphere map, reduced to lowest terms.

Put p(z) = zdq(1_z). p is the reflection of q.

Used in the Jury stability criterion in engineering!

The analogue of reflection holds in higher dimensions, except that there is no algebraic formula for p. This point will be elaborated.

[D-Huo-Xiao] have been studying polynomial and rational maps from the circle. Ideas are similar but there are some nice

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Note: If n = 1 and q(z) 6= 0 on closed unit disk, then there is a p

with p_q a rational sphere map, reduced to lowest terms.

Put p(z) = zdq(1_z). p is the reflection of q.

Used in the Jury stability criterion in engineering!

The analogue of reflection holds in higher dimensions, except that there is no algebraic formula for p. This point will be elaborated. [D-Huo-Xiao] have been studying polynomial and rational maps from the circle. Ideas are similar but there are some nice

simplifications. For example, there is a normal form for the unitary equivalence classes of polynomial maps.

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algebraic reformulation

Write (p; q) for a polynomial map from Cn to CN+1. Thus q is

scalar-valued and not 0 on the closed unit ball. Without loss of generality p and q have no common factors.

We want ||p||2− |q|2 _{= 0 on sphere. Equivalently, there are}

polynomial maps f , g such that

||p||2− |q|2 = (||f ||2− ||g ||2)(||z||2− 1). (1)

We call ||f ||2− ||g ||2 _{the quotient form.}

Formula (1) is equivalent to saying that

||(f ⊗ z) ⊕ g ||2− ||(g ⊗ z) ⊕ f ||2

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algebraic reformulation

Write (p; q) for a polynomial map from Cn to CN+1. Thus q is

scalar-valued and not 0 on the closed unit ball. Without loss of generality p and q have no common factors.

We want ||p||2− |q|2 _{= 0 on sphere. Equivalently, there are}

polynomial maps f , g such that

||p||2− |q|2 = (||f ||2− ||g ||2)(||z||2− 1). (1)

We call ||f ||2− ||g ||2 _{the quotient form.}

Formula (1) is equivalent to saying that

||(f ⊗ z) ⊕ g ||2− ||(g ⊗ z) ⊕ f ||2

has signature pair (N, 1). In other words, as a Hermitian form, there are N positive and 1 negative eigenvalue.

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There are many rational maps. In fact,

Theorem

Let q be a (scalar-valued) polynomial with q 6= 0 on closed ball.

Then there is a (vector-valued) polynomial p such that p_q is

reduced to lowest terms and maps sphere to sphere. If q = 0 on sphere, conclusion is false, by Cima-Suffridge.

Theorem

Let p_q : Cn_{→ C}K _{be an arbitrary rational map, reduced to lowest}

terms, with ||p_q||2 _{< 1 on closed ball. Then there is a polynomial}

h : Cn→ CM _{such that} p⊕h

q maps sphere to sphere. N = M + K .

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There are many rational maps. In fact,

Theorem

Let q be a (scalar-valued) polynomial with q 6= 0 on closed ball.

Then there is a (vector-valued) polynomial p such that p_q is

reduced to lowest terms and maps sphere to sphere. If q = 0 on sphere, conclusion is false, by Cima-Suffridge.

Theorem

Let p_q : Cn_{→ C}K _{be an arbitrary rational map, reduced to lowest}

terms, with ||p_q||2 _{< 1 on closed ball. Then there is a polynomial}

h : Cn→ CM _{such that} p⊕h

q maps sphere to sphere. N = M + K .

No bounds on N or deg(h) in terms of K , n and deg(p) possible!

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Spherical equivalence seems useful only in low codimension. Classification up to homotopy ([D-Lebl]) is promising, but ... f , g are homotopy equivalent if there is a one-parameter family

of maps Ht such that H0= f and H1 = g . We assume the map

(z, t) → Ht(z) is continuous.

Spherical equivalence implies homotopy equivalence, converse is false.

If Ht(z) =P cα(t)zα, then each cα is continuous.

Since ||Ht(z)||2 =Phcα(t), cβ(t)izαzβ,

the map to the space of Hermitian forms (with appropriate topology) is also continuous.

Theorem

For N ≥ 2n, there exists a homotopy Ht of spherically inequivalent

quadratic polynomials. Thus, number of spherical equivalence classes is uncountable.

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Theorem

In fact, Lebl and I generalized this result as follows:

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Theorem

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Theorem

(D-Lebl) Let Ht be a homotopy of rational proper maps between

balls. Then either all the maps are spherically equivalent or there are uncountably many spherical equivalence classes.

Proof.

Brief sketch: First step is to show that the set of t in [0, 1] for

which Ht is spherically equivalent to H0 is closed. Second step is

to quote a result of Sierpinski: [0, 1] is not a countable union of (non-empty) disjoint closed sets.

Sierpinksi theorem: Let X be a compact, connected Hausdorff

space. If {Fn} is a closed cover of pairwise disjoint subsets, then

one of these sets is X (and the others are empty).

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Theorem

Proof.

Brief sketch: First step is to show that the set of t in [0, 1] for

which Ht is spherically equivalent to H0 is closed. Second step is

to quote a result of Sierpinski: [0, 1] is not a countable union of (non-empty) disjoint closed sets.

Sierpinksi theorem: Let X be a compact, connected Hausdorff

space. If {Fn} is a closed cover of pairwise disjoint subsets, then

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One nice thing about homotopy equivalence:

For n ≥ 2 and N fixed, there are only finitely many homotopy classes. BUT, we don’t have a good way to decide the simple question:

Suppose f , g are rational sphere maps from Cn to CN.

They are always homotopic in target dimension N + n and even in dimension N + 1 if we allow constant maps.

QUESTION. Is there some simple homotopy invariant that tells us they are NOT homotopic in target dimension N?

Degree doesn’t work!

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Surprising fact. The degree is not a homotopy invariant.

Example

f , g : B2 → B5. Each map has embedding dimension 5. These

maps are of different degree but they are homotopic in target dimension 5.

f (z, w ) = (z, zw , zw2, zw3, w4). g (z, w ) = (−w2, zw , −zw2, z2w , z2). With c denoting cosine and s denoting sine, put

Ht(z, w ) =

(cz − sw2, zw , (cz − sw2)(sz + cw2), zw (sz + cw2), (sz + cw2)2).

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Attempt to classify for a given denominator.

PLAN: Fix denominator and degree of numerator. Find all maps. I solved this problem years ago in the polynomial case.

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Definition

Let r = ||p||2− |q|2 _{and s = ||f ||}2_{− |q|}2 _{be Hermitian forms. We}

say that r is an ancestor of s or that s is a descendant of r if s = E (E (...E (r ))) where each E is a tensor product operation

E (||p||2− |q|2) = ||p||2+ (||z||2− 1) ||π(p)||2− |q|2, AND we never increase the degree.

Here π is an orthogonal projection.

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We use the term descendant both for the Hermitian form and the rational map.

Full tensor product.

(||p1||2− |q1|2) ⊗ (||p2||2− |q2|2) = ||p||2− |q|2

Leads to notion of reducibility. Analogous to factoring a Blaschke product.

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Notation, in terms of homogeneous parts.

p = pν+ ... + pm.

Assume numerator is of degree m + k, denominator of degree k. All final descendants will be of form

pm+ ... + pm+k

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Theorem

Let p be a polynomial sphere map of degree m. Then the final

descendant of p of degree m is U z⊗m. (U is unitary.) In other

words, every such p is an ancestor of z⊗m.

In terms of Hermitian forms,

||p||2− 1 7→ ... 7→ ||z||2m− 1,

which is the final descendant but is reducible.

The target CN is isomorphic to V (n, m) = Symm(Cn), the space

of homogeneous polynomials of degree m in n variables.

Thus the final descendant is pm

1 and ||pm||

2 _{is determined.}

In the general case, the target will contain an isomorphic copy of V (n, m), but many isomorphisms are possible, and there will be a non-trivial orthogonal complement.

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Theorem

Let p be a polynomial sphere map of degree m. Then the final

descendant of p of degree m is U z⊗m. (U is unitary.) In other

words, every such p is an ancestor of z⊗m.

In terms of Hermitian forms,

||p||2− 1 7→ ... 7→ ||z||2m− 1,

which is the final descendant but is reducible.

The target CN is isomorphic to V (n, m) = Symm(Cn), the space

of homogeneous polynomials of degree m in n variables.

Thus the final descendant is pm

1 and ||pm||

2 _{is determined.}

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New classification result

Given g_q, degree g is m + 1. Assume q = 1 + hz, ai. Replace z by

ei θ_{z in the identity ||g (z)||}2_{− |q(z)|}2 _{= 0 on the sphere. Expand}

in terms of homogeneous parts. Equate Fourier coefficients. Apply various operations E . The final descendant is

p

q =

pm+1+ pm

1 + hz, ai . (∗)

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The expression (*) is a rational sphere map if two things are met

hpm+1, pmi = ||z||2mhz, ai (1)

||pm+1||2+ ||z||2||pm||2 = ||z||2m+2+ ||z||2m|hz, ai|2. (2)

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We try:

pm = M(z⊗m) ⊕ 0

pm+1= hz, aiL(z⊗m) ⊕ h.

To make (1) hold we need L = (M−1)∗. Here h is a vector-valued

polynomial.

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We need to make (2) hold. After some computation we need

||h||2 = ||z||2 ||z||2m−||M(z⊗m)||2+|hz, ai|2 ||z||2m−||L(z⊗m||2. In other words, the form on the right side is non-negative definite.

In terms of eigenvalues of M, assume Mv = λv and hence ||L(v )||2₌ ||v ||2

|λ|2 .

After some computation:

||h||2 = ||z||2m(1 − |λ|2) ||z||2−|hz, ai|

2

|λ|2 .

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We need to make (2) hold. After some computation we need

||h||2 = ||z||2 ||z||2m−||M(z⊗m)||2+|hz, ai|2 ||z||2m−||L(z⊗m||2. In other words, the form on the right side is non-negative definite. In terms of eigenvalues of M, assume Mv = λv and hence

||L(v )||2₌ ||v ||2 |λ|2 .

After some computation:

||h||2 _{= ||z||}2m_{(1 − |λ|}2_{) ||z||}2₋|hz, ai|2

|λ|2 .

Using Cauchy-Schwarz, works if ||a|| ≤ |λ| ≤ 1.

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To find all solutions: there is an invertible linear M such that

pm+1= q1(M−1)∗(z⊗m) ⊕ h.

pm = M(z⊗m) ⊕ 0

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Thus: assume degree g is m + 1 and degree q is 1. Assume p_q is a

final descendant. Then the target space for p_q contains an

isomorphic copy of V (n, m). The orthogonal complement contains excess; it is determined up to a unitary.

CN = V ⊕ W .

V is isomorphic to V (n, m) and πV(pm) = pm.

πV(pm+1) is divisible by q1.

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Theorem

Let f be a proper rational mapping between balls with linear

denominator 1 + q1 and of degree m + 1. The following hold:

I There is a finite number of tensor operations such that

Es ◦ ... ◦ E1(f ) =

pm+1+ pm

1 + q1

. (∗)

I There is an invertible linear M such that

pm+1= q1(M−1)∗(z⊗m) ⊕ h.

pm = M(z⊗m) ⊕ 0.

I The Hermitian form defined by

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Similar ideas hold in general. Given q with q(z) 6= 0 on the sphere, how do we construct p? What are all the possible p?

Write q = 1 + q1+ ... + qk.

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On the sphere we have || k X j =0 pm+j||2= | k X j =0 qj|2 X j ,l hpm+j, pm+li = X j ,l qjql.

Replace z by ei θz and use homogeneity to get:

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On the sphere we have || k X j =0 pm+j||2= | k X j =0 qj|2 X j ,l hpm+j, pm+li = X j ,l qjql.

Replace z by ei θz and use homogeneity to get:

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Equate Fourier coefficients to get k + 1 equations, which hold on

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For each a with 0 ≤ a ≤ k we get identities that hold everywhere: For a = k,

hp_m+k, pmi = qk||z||2m = hqkz⊗m, z⊗mi

hp_m+k, pm+1i + hpm+k−1, pmi||z||2= qkq1||z||2m+ qk−1||z||2m+2

and finally (when a = 0) X

l

||p_m+l||2 _||z||2k−2l ₌X

|q_l|2 _||z||2m+2k−2l_.

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These identities have a nice interpretation as upper trace identities.

Consider a matrix whose column vectors are A0, ...Ad.

These identities say:

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In domain dimension 1, things simplify.

One can identify a homogeneous polynomial with a constant! D-Huo-Xiao have used the upper trace identities to give a normal form for polynomial sphere maps from the unit disk, under unitary equivalence in the target.

I’ll just write it out for degree 2 and then for degree 4, the first case hard enough to illustrate the idea.

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Write f (z) = A0+ A1z + A2z2.

The normal form becomes   |A0| −α|A2| 0 0 a11 0 0 α|A0| |A2|   where (|A0|2+ |A1|2)(1 + |α|2) + |a11|2 = 1.

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Now for degree four. Write

f (z) = A0+ A1z + A2z2+ A3z3+ A4z4.

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The condition hA0, A4i = 0 enables us to choose coordinates       |A0| ∗ ∗ ∗ 0 0 ∗ ∗ ∗ 0 0 ∗ ∗ ∗ 0 0 ∗ ∗ ∗ 0 0 ∗ ∗ ∗ |A4|      

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      |A₀| ∗∗ ∗∗ −α|A₄| 0 0 ∗ ∗ ∗ 0 0 ∗ ∗ ∗ 0 0 ∗ ∗ ∗ 0 0 α|A0| ∗∗ ∗∗ |A4|      

Regard the ** as complex parameters. The rest of the identities become upper-trace identities on the three-by-three matrix in the center. Since everything depends only on the inner products, things are unitarily invariant. We can then use a unitary to put the middle matrix in upper triangular form.

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Return to Cn and general denominator. Because

X

||p_m+j||2 ||z||2k−2j =X|q_j|2 ||z||2m+2k−2j, there is (a large dimensional!) unitary matrix U with

      pm+k ... pm+j⊗ z⊗(k−j) ... pm⊗ z⊗k       = U       qk z⊗m ... qj z⊗(m+k−j) ... q0 z⊗(m+k)       .

Thus we could put pm+j = qjz⊗m⊗ vj ⊕ fm+j.

For simplicity in the next slide we choose U in a simple way. The method does not construct all possible numerators. One must take additional subspaces into account.

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Theorem

Let f be a proper rational mapping between balls with denominator of degree k and of degree m + k. The following hold:

I There is a finite number of tensor operations such that

Es ◦ ... ◦ E1(f ) = Pk j =0pm+j Pk j =0qj . (∗)

I Given q we can construct p as follows.

I Choose vj with hvi, vji = 1 for i 6= j and

pm+j = qjz⊗m⊗ vj⊕ fm+j.

I The Hermitian form defined by

||z||2m X

j

|qj|2(1 − |vj|2)||z||2k−2j

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Automorphisms are homotopic to the identity. But, as the target dimension rises, we get uncountably many spherical equivalence classes, whereas

Theorem

(D-Lebl) For n ≥ 2 and any N, the number of homotopy classes

for rational proper maps from Bn→ BN is finite.

Sketch of Proof.

Step 1 is a degree bound: If n ≥ 2, and f is a proper rational map

from Bn to BN, then

d ≤ N(N − 1)

2(2n − 3).

(This bound is not sharp. Long ago I conjectured certain bounds. Still open.)

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Automorphisms are homotopic to the identity. But, as the target dimension rises, we get uncountably many spherical equivalence classes, whereas

Theorem

(D-Lebl) For n ≥ 2 and any N, the number of homotopy classes

for rational proper maps from Bn→ BN is finite.

Sketch of Proof.

Step 1 is a degree bound: If n ≥ 2, and f is a proper rational map

from Bn to BN, then

d ≤ N(N − 1)

2(2n − 3).

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Step 2. Let f =p_q be proper. Put R = ||p||2− |q|2_.

R is of degree at most d in z, of total degree at most 2d , and is

divisible by ||z||2_{− 1. R determines a Hermitian form on the}

finite-dimensional vector space of polynomials of degree at most d . This form has one negative eigenvalue.

The (vector) coefficients of p and coefficients of q are bounded. (Assume q(0) = 1.)

R must be divisible by ||z||2− 1.

Equate Fourier coefficients in n variables after replacing z by ei θz.

Obtain various linear conditions on the inner products of the vector

coefficients. These generalize pν ⊥ pm from before.

Putting it together gives finitely many polynomial inequalities on the coefficients.

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Step 2. Let f =p_q be proper. Put R = ||p||2− |q|2_.

R is of degree at most d in z, of total degree at most 2d , and is

divisible by ||z||2_{− 1. R determines a Hermitian form on the}

finite-dimensional vector space of polynomials of degree at most d . This form has one negative eigenvalue.

The (vector) coefficients of p and coefficients of q are bounded. (Assume q(0) = 1.)

R must be divisible by ||z||2− 1.

Equate Fourier coefficients in n variables after replacing z by ei θz.

Obtain various linear conditions on the inner products of the vector

coefficients. These generalize pν ⊥ pm from before.

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Step 3. Conclusion: A proper rational map corresponds to the intersection of the unit sphere in a finite-dimensional real vector space with a set described by finitely many polynomial inequalities. Such a set is semi-algebraic and can have at most a finite number of components.

Each component corresponds to a collection of homotopic rational proper mappings with target dimension at most N.

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Theorem

The previous result generalizes the following.

Theorem

If f , g are spherically equivalent proper polynomial maps preserving 0, then they are unitarily equivalent. There exists a one-parameter family of spherically inequivalent quadratic polynomial mappings

from Bn→ B2n.