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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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The Boundary Value Problem for Laplacian on Differential Forms and Conformal Einstein Infinity

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.

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Abstract

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.

Keywords

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

Introduction

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 equationand equation of degree equation , depending on spectral parameters. Their origin is motivated by the examples given in Subsection 3.2. We prove that equation satisfy a three step recurrence relation, cf. Proposition 2 and 2, while equation turns out to be a linear combination of equation for k=m, m−1,m−2,0, see Proposition 2. By a cascade of variable changes, we identify equation and equation as (a linear combination of) generalized hypergeometric functions, in particular equation 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 equation and equation 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.

Some Combinatorial Identities

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 Rk(y;α),

equation (1)

of degree equation and depending on a parameter equation Conventionally, we set R0(y;α):=1.

Remark: We notice that some versions of the polynomials Rk(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 equation in the variable y(y+1),

equation (2)

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

equation

equation (3)

equation

equation (4)

for equation and two parameters equation.

We shall now observe basic recurrence relations satisfied by equation and equation.

Proposition: The collection of polynomials equation satisfies the following recurrence relation

equation (5)

with equation

Proof: The identity

Rk+1(y;0)=[y−k(k+1)]Rk(y;0)

for equation, leads to

equation

equation

equation

equation

equation

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

equation

equation

equation

equation

equation

equation

equation

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

equation

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

Proposition: The collection of polynomials equation satisfies the recurrence relations

equation

with equation.

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

equation

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

equation

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

Furthermore, we introduce another set of polynomials:

equation (7)

of degree equation, and set the convention (1) equation.

Proposition: The polynomials equation andequation are related by

equation(8)

for all equation.

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 Rm(y;0). We note

equation

and the standard combinatorial identities

equation

allow to obtain

equation

Hence, our claim is equivalent to

equation

for all i=1,…,m. The identity

equation

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

equation

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

equation

which is exactly

equation , sinceequation.

This completes the proof.

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

equation (9)

with equation

Proof: The proof is straightforward. Starting from

equation

We substitute

equation

and shift the summation index. This yields

equation

which completes the proof.

Finally, we introduce the set of polynomials

equation (10)

of degree equation, and we remark that (1) equation.

Proposition: The set of polynomials equation satisfies

equation (11)

Notice that equation.

Proof: By Proposition 2 and Lemma 5, we have

equation

Comparision of the coefficients in Equation (11) by Rk(y;0) gives

equation

for all equation 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 equation as the dual Hahn polynomials, cf. Appendix 5. The key step is to apply the variable y(y+1) to the polynomials equation, cf. (2). Furthermore, it follows from Proposition 2 that equation 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 equation are given by generalized hypergeometric functions of type (3,2):

equation

equation

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

equation

When choosing equation for some equation 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 equation . 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 equation is a linear combination of generalized hypergeometric functions of type (3,2) and (2,1),

equation

equation (12)

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

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

Proposition: The set of polynomials equation, forequation has the following descriptions:

equation

equation (13)

where equation. Additionally it holds

equation

equation (14)

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

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

equation

equation

The standard Pochhammer identity equation allows to decompose our generalized hypergeometric function 4F3 into two summands, which lead to Equation (14). The proof is complete.

Boundary Value Problem for Conformal Infinity

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 equation has a formal adjoint given by the codifferential equationwhen acting on p+1-forms. Here we denoted by equation the Hodge operator on (M,h). The form Laplacian

equation

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

equation (15)

for differential forms equationand equationcharacterized 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

equation (16)

Here equation and equation 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 equation we consider the eigenequation

equation (17)

with ω a p-form on X. The boundary value problem for conformal infinity consists of finding an asymptotic solution ω∈Ωp(X) of Equation (17) with prescribed boundary value ?∈Ωp(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 equation be the euclidean space. Then the associated Poincaré-Einstein metric can be realized as the hyperbolic metric

equation

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

equation(18)

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

equation (19)

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

Remark: Due to the absence of curvature, the solution operators equation are given in terms of δh(dδh)j−1, (δhd)j and (dδh)j.

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

equation

equation

for all equation and equation

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

equation

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

equation

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

Conformally Einstein metrics

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 equation, respectively. In this case the Poincaré-Einstein metric is of the form

equation (20)

for equation. 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

equation

where

equation

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

equation

for equation .Hence, we get on p-forms

equation

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

equation

The result then follows from (??).

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

equation(21)

equation(22)

Using the polynomial type of equation , 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 r2, respectively. This is the key step to formulate

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

equation

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

equation

equation (23)

equation (24)

with coefficients

equation

depending on equation and the initial dataequation and equation. Furthermore, it holdsequation

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 equation Based onequation and eveness of involved coefficients in r, we get equation for all equation. 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

equation

and (24) for j=2 implies

equation

for

equation

Hence equation are well-defined forequation. As we will see later, for equation 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

equation

in terms of the operator

equation

The relation (24) for j=4 reduces to

equation (27)

where

equation

Hence equation are well-defined for equation 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

equation

in terms of

equation

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

equation (28)

expressed in terms of operators

equation

Hence equation are well-defined forequation. For equation 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:

equation

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

equation

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

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

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

equation

for all equation and boundary data equation.

Proof: In order to shorten the notation, we introduce

equation

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

equation

This can be decomposed, due to equation and dealing with dδh and δhd as independent commuting variables, into two independent claims. The first is

equation

while the second is given by

equation

equation

Note that

equation

equation

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

equation

Furthermore,

equation

for all k∈?. Applying Proposition 2 to equation, for k=m,m−1,m−2, allows to rewrite Equation (??) just in terms of equationfor 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

equation

which finally proves Equation (??).

Now we proceed to prove

equation

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

equation

and due to Lemma 5, we get

equation

we see that Equation (32) is equivalent to

equation

equation (33)

We replace the terms equationand equation , using Proposition 2, byequationfor 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

equation

This proves the theorem.

Applications: Branson-Gover operators on Einstein manifolds

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

equation

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

equation

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

equation

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

equation

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

equation

by two additional differential operators

equation

equation

called the gauge companion and the Q-curvature operator, respectively. Similarly to equation, 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 equation for a constant equation By ref. [5], it follows that one can recover Branson-Gover operators as residues of solution operators, see Equation (34). More precisely, we have

equation

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

equation (35)

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

equation(36)

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

Remark: The normalization factor equation 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 equation, or there exists an l∈{1,…,N−1} such that equation

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

equation

for equation

Proof: We use the statements

equation

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

equation

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

equation

The preparatory identities above ensure that this equals to

equation

and the proof is complete.

This recurrence relation for equation implies the result [13].

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

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

equation

In the setting of Theorem 4 it holds

equation(38)

Now we discuss the cases when equation. Let us introduce

equation(39)

when acting on p-forms. Due to d2=0=(δh)2 we have

equation(40)

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

equation

where

equation

is a fourth-order differential operator.

Proof: First note that equation decomposes, due tod2=0=(δh)2, as

equation(41)

For l∈{1,…,N−1} such that equation it holds

equation (42)

Similarily, using Proposition 2 we have

equation

equation(43)

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

equation

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

equation

which is exactly the definition of equation, 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 equation for some constantequation. The critical Branson-Gover operator equation is given by the product formula

equation

Proof: It follows from Equation (??) that

equation

Note that the last factor reduces to

equation

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

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

equation

and the gauge companion operator

equation

Obviously,

equation

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

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

equation

equation

equation

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