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**Fischmann M ^{*} and Somberg P**

E. Cech Institute, Mathematical Institute of Charles University, Sokolovská 83, Praha 8 - Karln, Czech Republic

- *Corresponding Author:
- Fischmann M E. Cech Institute

Mathematical Institute of Charles University

Sokolovská 83, Praha 8 - Karln

Czech Republic

**Tel:**951 553 203

**E-mail:**[email protected]

**Received Date**: January 16, 2017; **Accepted Date:** February 17, 2017; **Published Date**: February 27, 2017

**Citation: **Fischmann M, Somberg P (2017) The Boundary Value Problem for Laplacian on Differential Forms and Conformal Einstein Infinity. J Generalized Lie Theory Appl 11: 256. doi:10.4172/1736-4337.1000256

**Copyright:** © 2017 Fischmann M, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

We completely resolve the boundary value problem for differential forms for conformal Einstein infinity in terms of the dual Hahn polynomials. Consequently, we present explicit formulas for the Branson-Gover operators on Einstein manifolds and prove their representation as a product of second order operators. This leads to an explicit description of Q-curvature and gauge companion operators on differential forms.

Boundary value problems; Einstein manifolds; Generalized hypergeometric functions; Conformal geometry; Branson- Gover operators; Q-curvature operators

The boundary value problems have ever played an important role in mathematics and physics. A preferred class of boundary value problems is given by a system of partial differential equation on manifolds with boundary or submanifold equipped with a geometrical structure. The representative examples are the Laplace and Dirac operators on Riemannian manifolds. A closely related concept is then the Poisson transform and boundary (or, submanifold) asymptotic of a solution applied to this system of PDEs, [1] for the case related to compactifications of symmetric spaces.

Fefferman and Graham [2] initiated a program allowing to view any conformal manifold as the conformal infinity of associated Poincaré-Einstein metric. Boundary value problems on the Poincaré- Einstein manifolds with prescribed boundary data are referred to as the boundary value problems for conformal infinity. It is a remarkable fact that solving such a boundary value problems leads to an algorithmical (or, recursive) construction of a series of conformally covariant differential operators on functions, spinors and differential forms [3-5]. Note that these operators were originally constructed using the ambient metric of Fefferman and Graham and tractor bundles, cf. [6-8], and were soon recognized to encode interesting geometrical quantities like Branson’s Q-curvature [9] or holographic deformations of the Yamabe and Dirac operators [10,11], and their functional determinants play a fundamental role in quantum field theories.

Assume now (M, h) to be an Einstein manifold. The main result of the present article is the complete and explicit solution of the boundary value problem for conformal Einstein infinity for the Laplace operator acting on differential forms. More presicely, we reduce the eigenvalue problem to a rank two matrix valued system of four step recurrence relations for the coefficients of asymptotic expansion of the form Laplace eigenforms. This combinatorial problem can be resolved in terms of generalized hypergeometric functions, closely related to the dual Hahn polynomials. The results analogous to ours were obtained for scalar and spinor fields [2,12]. The key property of reducing the boundary value problem for conformal Einstein infinity is the polynomial character of the 1-parameter family of metrics given by the Poincaré-Einstein metric. As an application, we produce explicit formulas for the Branson-Gover and related Q-curvature operators on differential forms on Einstein manifolds and derive their factorization as a product of second-order differential operators.

Let us briefly indicate the content of our article. The Section 2 is combinatorial in its origin with some implications to hypergeometric function theory. We introduce three series of polynomials and of degree , depending on spectral parameters. Their origin is motivated by the examples given in Subsection 3.2. We prove that satisfy a three step recurrence relation, cf. Proposition 2 and 2, while turns out to be a linear combination of for k=*m*, *m*−1,*m*−2,0, see Proposition 2. By a cascade of variable changes, we identify and as (a linear combination of) generalized hypergeometric functions, in particular are given by the dual Hahn polynomials.

In Section 3, we briefly recall the boundary value problem for conformal infinity. First of all, we determine in Proposition 3.1 its solution in terms of solution operators when the conformal infinity contains the flat metric. Then we prove in Proposition 3.2 that the polynomials and mentioned above are the organizing frame for the solution as far the conformal infinity contains an Einstein metric.

In Section 4, we discuss emergence of the Branson-Gover operators in the framework of solution operators. Furthermore, we prove in Theorem 4 that Branson-Gover operators satisfy a two step recurrence relation which immediately leads to their factorization (or, the product formula) in terms of second-order differential operators, cf. Theorem 4. Finally, we discuss explicit formulas for the gauge companion and Q-curvature operators.

In Appendices 5 and 6 we collect some standard notation, results and properties concerning generalized hypergeometric functions and Poincaré-Einstein metric.

In the present section we discuss some special polynomials characterized to satisfy certain recurrence relations. When changing the polynomial variable by a second order differential operators, they are used in Section 3.2 to determine solution operators for our boundary value problem.

Let y be an abstract variable and define the set of polynomials R_{k}(y;α),

(1)

of degree and depending on a parameter Conventionally, we set R_{0}(y;α):=1.

**Remark: **We notice that some versions of the polynomials R_{k}(y;0) have already appeared in the description of the scalar and spinor boundary value problems [2,12].

Later on, it will be important to consider in the variable y(y+1),

(2)

for all . Here we already used the notion of Pochhammer symbol, as reviewed in Appendix 5. Furthermore, we introduce the polynomials

(3)

(4)

for and two parameters .

We shall now observe basic recurrence relations satisfied by and .

Proposition: The collection of polynomials satisfies the following recurrence relation

(5)

with

**Proof:** The identity

R_{k+1}(y;0)=[y−k(k+1)]R_{k}(y;0)

for , leads to

Therefore, it remains to compare the coefficients by Rk(y;0) on both sides of (5), which is equivalent to the following set of relations among

for all such that k≤m−2, and for all . These relations can be easily verified using the identity

with l=1,2 and r=0,…,m−1. This completes the proof.

**Proposition:** The collection of polynomials satisfies the recurrence relations

with .

**Proof:** It is completely analogous to the proof of the previous proposition. The claim is now equivalent to

for such that k≤m−2, and for all . However, these identities hold due to

with l=1,2 and r=0,…,m−1. This completes the proof.

Furthermore, we introduce another set of polynomials:

(7)

of degree , and set the convention (1) .

Proposition: The polynomials and are related by

(8)

for all .

**Proof: **The left hand side is a polynomial in y of degree m, and so is the right hand side. Hence, it is sufficient to check that both sides of this polynomial identity have the same value at m+1 different points. To that aim, we choose the m-tuple i(i−1), i=1,…,m, of the roots of R_{m}(y;0). We note

and the standard combinatorial identities

allow to obtain

Hence, our claim is equivalent to

for all i=1,…,m. The identity

implies that our sum is a telescoping sum, and the only term which survives the summation process is

because i=1,…,m. Finally, for the last evaluation point of our polynomials we take :

which is exactly

, since.

This completes the proof.

**Proposition: **The set of polynomials , satisfies the following recurrence relations

(9)

with

**Proof:** The proof is straightforward. Starting from

We substitute

and shift the summation index. This yields

which completes the proof.

Finally, we introduce the set of polynomials

(10)

of degree , and we remark that (1) .

**Proposition:** The set of polynomials satisfies

(11)

Notice that .

**Proof: **By Proposition 2 and Lemma 5, we have

Comparision of the coefficients in Equation (11) by R_{k}(y;0) gives

for all such that k≤m−2. Checking these identities is straightforward and the proof is complete.

**Hypergeometric interpretation of combinatorial identities**

In this subsection we interpret our polynomials as the dual Hahn polynomials, cf. Appendix 5. The key step is to apply the variable y(y+1) to the polynomials , cf. (2). Furthermore, it follows from Proposition 2 that is a linear combination of the dual Hahn polynomials with y-independent coefficients. This linear combination can be rewritten as a sum of a hypergeometric polynomials of type (4,3) and (2,1) or, equivalently, as a linear combination with y-dependent coefficients of two dual Hahn polynomials.

Firstly, we observe that by standard Pochhammer identities our polynomials are given by generalized hypergeometric functions of type (3,2):

and hence we can express them as the dual Hahn polynomials,

When choosing for some becomes a positive integer, as required by definition of the dual Hahn polynomials.

**Remark:** One can easily realise that there is no hypergeometric series representative for . For example, the subleading coefficients in such an expansion do not factorize nicely into linear factors. Moreover, the quotients of successive coefficients are not rational functions in the summation index of the hypergeometric series (which is, in fact, the defining property of a hypergeometric series).

Secondly, as a consequence of Proposition 2, the polynomial is a linear combination of generalized hypergeometric functions of type (3,2) and (2,1),

(12)

Notice that the coefficients of previous linear combination are y-independed.

We were informed by Christian Krattenthaler that our can be organized by the following two expressions based on various generalized hypergeometric functions:

**Proposition:** The set of polynomials , for has the following descriptions:

(13)

where . Additionally it holds

(14)

where the coefficients in the linear combination (14) are y-depended.

**Proof:** The proof of Equation (13) is based on the elementary identity

The standard Pochhammer identity allows to decompose our generalized hypergeometric function _{4}F_{3} into two summands, which lead to Equation (14). The proof is complete.

We start with a brief reminder about the boundary value problem for conformal infinity and the Laplace operator acting on differential forms, [5]. Then we proceed to its complete solution in the case when the conformal infinity contains the flat or an Einstein metric.

Let (M,h) be a Riemannian oriented manifold of dimension n ≥3. Note that all statements given below extend to the semi-Riemannian setting by careful checking the number of appearances of minuses induced by the signature. The differential has a formal adjoint given by the codifferential when acting on p+1-forms. Here we denoted by the Hodge operator on (M,h). The form Laplacian

is formally self-adjoint differential operator of second order.

Consider the Poincaré-Einstein space (X,g_{+}) associated to (M,h), see 6 for the description of its construction. A differential p-form ω on X uniquely decomposes (note a different convention compared to ref. [5]) off the boundary as

(15)

for differential forms and characterized by trivial contraction with the normal vector field ∂r. It is straightforward to verify that the form Laplacian on X acting in the splitting (15) is

(16)

Here and denote the Hodge operator and codifferential with respect to the 1-parameter family of metrics hr on M. In the case when n is odd, the form Laplacian can be expanded as a power series around r=0, while in even dimensions n there appear additional log(r)- terms coming from the Poincaré-Einstein metric, cf. [5].

For we consider the eigenequation

(17)

with ω a p-form on X. The boundary value problem for conformal infinity consists of finding an asymptotic solution ω∈Ω^{p}(X) of Equation (17) with prescribed boundary value ?∈Ω^{p}(M). The construction of a solution for this boundary value problem is algorithmically described in ref. [5]. For a manifold with general conformal structure (M,h), this algorithm is quite complicated due to the complexity in the construction of the Poincaré-Einstein metric. As we shall see in next subsections, there is rather explicit solution when the conformal infinity is metrizable by the flat or an Einstein metric.

**Conformally flat metric**

Let be the euclidean space. Then the associated Poincaré-Einstein metric can be realized as the hyperbolic metric

on the upper half space . Consider the asymptotic expansion of a p-form on , given by

(18)

for and . Formally, one can solve Equation (17) for a given initial data in terms of the solution operators

(19)

which are h-natural differential operators with rational polynomial coefficients in λ determining and uniquely for all . Notice that the solution operators turn out to be well-defined for and and by construction for .

**Remark:** Due to the absence of curvature, the solution operators are given in terms of δ^{h}(dδ^{h})^{j−1}, (δ^{h}d)^{j} and (dδ^{h})^{j}.

**Proposition:** Let (M,h) be the euclidean space . Then

for all and

Proof: For and see Equation (??), reduces to

where all operators are considered with respect to . The ansatz (18) solves Equation (17) iff the following system is satisfied:

for and arbitrary, while . It is now straightforward to check that the solution operators satisfy the recurrence relations and the proof is complete.

Let (M,h) be an Einstein manifold normalized by Ric(h)=2λ(n-1) h for some constant λ∈R. This implies that the (normalized) scalar curvature and the Schouten tensor are given by J=nλ and , respectively. In this case the Poincaré-Einstein metric is of the form

(20)

for . The polynomial type of J(r) implies that one can explicitly compute the form Laplacian, especially the term P′ in Equation (??). From now on we use the abbreviation β:=n2p.

Lemma: Let (M.h) be an Einstein manifold, normalized by Ric(h)=2λ(n−h)h. Then in the splitting (15), it holds

where

**Proof:** The explict formula for hr, cf. (20), leads to the explicit form of Equation (??). In more detail, we note

for .Hence, we get on p-forms

which implies . Furthermore, for p and (p−1)-forms ω^{(+)},ω^{()} as introduced in (15), we have

The result then follows from (??).

The eigenequation (17), acting in the splitting , is equivalent to the system

(21)

(22)

Using the polynomial type of , we multiply Equation (21) by J(r)^{3} and Equation (22) by J(r)^{2}. As a result, the coefficients in both equations are polynomials of degree 3 and 2 in r^{2}, respectively. This is the key step to formulate

**Proposition:** Let (M,h) be an Einstein manifold with the normalization given by for a constant . The eigenequation (17) acting on

for some (unknown) differential forms and , is equivalent to the following recurrence relations:

(23)

(24)

with coefficients

depending on and the initial data and . Furthermore, it holds

Proof: Due to the polynomial type of the coefficients in Equations (21) and (22) after multiplication with appropriate powers of J(r), and the ansatz for ω, we obtain the recurrence relations by comparing the coefficients by Based on and eveness of involved coefficients in r, we get for all . This completes the proof.

In order to get an insight into the solution structure of the recurrence relations (23) and (24), we present several low-order approximations:

Second-order approximation: The relation (23) for j=2 gives

and (24) for j=2 implies

for

Hence are well-defined for. As we will see later, for the right hand side of Equation (26) is proportional to the second-order Branson-Gover operator.

**Fourth-order approximation: **The relation (23) for j=4 gives

in terms of the operator

The relation (24) for j=4 reduces to

(27)

where

Hence are well-defined for the right hand side of Equation (27) is proportional to the fourth-order Branson-Gover operator.

**Sixth-order approximation:** The relation (23) for j=6 gives

in terms of

The relation (24) for j=6 yields after some computations

(28)

expressed in terms of operators

Hence are well-defined for. For the right hand side of Equation ( 28) is proportional to the sixth-order Branson-Gover operator.

The previous approximations indicate the following definition of the solution operators:

for Here we have taken the evaluation of the polynomials and at

Remark: The inspiration for the definition of comes from the scalar case, cf. [2]. The reason is that for 0-forms, Equation (23) becomes trivial and , while Equation (24) is solved by with vanishing term .

The proof of the next theorem is mainly based on the combinatorial identities discussed in Section 2.

Theorem: Let and for all . The solution of the recurrence relation (23) and (24) is given by

for all and boundary data .

**Proof:** In order to shorten the notation, we introduce

First of all, we verify the recurrence (24). In terms of solution operators, it reads

This can be decomposed, due to and dealing with dδ^{h} and δ^{h}d as independent commuting variables, into two independent claims. The first is

while the second is given by

Note that

We first notice that Equation (30) was proved in Proposition 2. Now we proceed to Equation (??). By (δ^{h})^{2}=0 and Lemma 5, we have

Furthermore,

for all k∈?. Applying Proposition 2 to , for k=m,m−1,m−2, allows to rewrite Equation (??) just in terms of for appropriate collection of values of k. It turns out that Equation (??) is equivalent to the three-times repeated application of the recurrence relation in Proposition 2 to

which finally proves Equation (??).

Now we proceed to prove

Using two ingredients: due to (δh)^{2}=0, we have

and due to Lemma 5, we get

we see that Equation (32) is equivalent to

(33)

We replace the terms and , using Proposition 2, byfor appropriate collection of values of k. Then it turns out that Equation (??) is equivalent to the two-times application of the recurrence relation in Proposition 2 to

This proves the theorem.

This section is focused on the origin and properties of the Branson- Gover operators and their derived quantities on Einstein manifolds. In addition, we present another proof of a result in ref. [13] on the decomposition of Branson-Gover operators as a product of secondorder differential operators.

Let (M,h) be a Riemannian manifold of dimension n. For p=0,…,n the Branson-Gover operators [8] are differential operators

of order 2N, for for even n), of the form

Where LOT is the shorthand notation for the lower order (curvature correction) terms. They generalize the GJMS operator [6]

in the sense that . The key property of Branson- Gover operators is that they are conformally covariant,

Here denotes the evaluation with respect to the conformally related metric . In the case of even dimensions n and the critical Branson-Gover operators factorize

by two additional differential operators

called the gauge companion and the Q-curvature operator, respectively. Similarly to , these relatives are quite complicated operators in general, but in the case when the underlying metric is flat or Einstein we shall present closed formulas for them.

Now let (M,h) be an n-dimensional Einstein manifold with normalization given by for a constant By ref. [5], it follows that one can recover Branson-Gover operators as residues of solution operators, see Equation (34). More precisely, we have

The right hand side on the previous display is exactly the Branson- Gover operator of order , when acting on differential p-forms:

(35)

Note that there is no obstruction to the existence in even dimensions n. Introducing the following normalization for Branson- Gover operators,

(36)

has the effect that the factors appearing in Theorem 4 are differential operators with polynomial coefficients.

**Remark: **The normalization factor can vanish only in even dimensions n, due to β=n−2p. It vanishing is characterized by: there exists an l∈{1,…,N} such that , or there exists an l∈{1,…,N−1} such that

Proposition: Let and p=0,…,n when n is odd, and and p=0,…,n such that when n is even. The collection of normalized Branson-Gover operators , satisfies the recurrence relation

for

**Proof:** We use the statements

where the first is easily verified, while the second follows from Proposition 2. In addition, we need an elementary identity

We start with the evaluation of the right hand side of Equation (37). We have

The preparatory identities above ensure that this equals to

and the proof is complete.

This recurrence relation for implies the result [13].

**Theorem:** Let (M,h) be an Einstein manifold with normalization given by for some constant

Let when n is odd, and such that when n is even. The normalized Branson-Gover operators factorize as

In the setting of Theorem 4 it holds

(38)

Now we discuss the cases when . Let us introduce

(39)

when acting on p-forms. Due to d^{2}=0=(δh)^{2} we have

(40)

Proposition: Let n be even and . For such that β=2l, the 2Nth-order Branson-Gover operator factorizes by

where

is a fourth-order differential operator.

**Proof: **First note that decomposes, due tod^{2}=0=(δh)^{2}, as

(41)

For l∈{1,…,N−1} such that it holds

(42)

Similarily, using Proposition 2 we have

(43)

Now we rewrite the claim of Proposition 4 using equation (41) as

This is equivalent by using Equations (42) and (43) to

which is exactly the definition of , see Equation (35). This completes the proof.

From now on let n be even. We proceed with explicit formulas for the critical Branson-Gover operator, gauge companion operator and Q-curvature operator:

**Proposition:** Let (M,h) be an Einstein manifold with normalization given by for some constant. The critical Branson-Gover operator is given by the product formula

**Proof:** It follows from Equation (??) that

Note that the last factor reduces to

since β=n−2p. This completes the proof.

Consequently, we found the explicit formulas for the Q-curvature operator

and the gauge companion operator

Obviously,

which is the famous double factorization of the critical Branson-Gover operator.

Remark: Let be the euclidean space. The explicit formulas for and immediately imply after setting J=0 that [14,15]

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