Boundary operators associated to the sixth-order GJMS operator

Boundary operators associated to the sixth-order GJMS operator

Jeffrey S. Case 109 McAllister Building
Penn State University
University Park, PA 16801
jscase@psu.edu
 and  Weiyu Luo Department of Electrical Engineering and Computer Science
University of California, Irvine
Irvine, CA 92617
weiyul7@uci.edu
Abstract.

We describe a set of conformally covariant boundary operators associated to the sixth-order GJMS operator on a conformally invariant class of manifolds which includes compactifications of Poincaré–Einstein manifolds. This yields a conformally covariant energy functional for the sixth-order GJMS operator on such manifolds. Our boundary operators also provide a new realization of the fractional GJMS operators of order one, three, and five as generalized Dirichlet-to-Neumann operators. This allows us to prove some sharp Sobolev trace inequalities involving the interior -seminorm, including an analogue of the Lebedev–Milin inequality on six-dimensional manifolds.

Key words and phrases:
conformally covariant operator; boundary operator; fractional Laplacian; Sobolev trace inequality; Poincaré–Einstein manifold
2000 Mathematics Subject Classification:
Primary 58J32; Secondary 53A30, 58J40

1. Introduction

The GJMS operators [24] are conformally covariant differential operators with leading-order term an integer power of the Laplacian. These operators play a key role in many questions at the intersection of geometry and analysis. As one example, the GJMS operator of order two — more commonly known as the conformal Laplacian — controls the behavior of the scalar curvature within a conformal class and as such plays an important role in the resolution of the Yamabe Problem (see [32] and references therein). As another example, the Sobolev embedding can be seen as a consequence of the sharp Sobolev inequality

(1.1)

for all , where is an explicit constant and is the GJMS operator of order on flat Euclidean space. By conformal covariance, one can use stereographic projection to write (1.1) as an equivalent inequality on the round -sphere (cf. [5]).

In order to study a GJMS operator and its related scalar invariants on a manifold with boundary, one should first find conformally covariant boundary operators which are suitably adapted to the GJMS operator in question. For the case of the conformal Laplacian, Cherrier [14] and Escobar [15] showed that the trace (or restriction) operator and the conformal Robin operator serve as the appropriate boundary operators. In particular, Escobar proved the sharp Sobolev trace inequality

(1.2)

for all , where is an explicit constant and is the conformal Robin operator on flat Euclidean upper half space . The zeroth-order term of the conformal Robin operator is the mean curvature of the boundary, leading these operators to play an important role in the resolution of the boundary Yamabe problem (see, for example, [16, 17]).

The conformal Laplace operator and conformal Robin operator naturally give rise to a conformally covariant Dirichlet-to-Neumann operator on the boundary of a Riemannian manifold for which . The operator recovers in Euclidean space. On boundaries of asymptotically hyperbolic manifolds, there is another formally self-adjoint, conformally covariant pseudodifferential operator with leading term , namely the fractional GJMS operator of order ; see [26]. It turns out that , provided the latter is defined in terms of the Loewner–Nirenberg metric [28].

The purpose of this article is to identify the boundary operators associated to the sixth-order GJMS operator and use them both to prove sharp Sobolev trace inequalites involving the -seminorm and give a new realization of the fractional GJMS operators of order , , and . This work is motivated by recent developments in three directions. First, boundary operators for the fourth-order GJMS operator — more commonly known as the Paneitz operator — and their relations to sharp Sobolev trace inequalities and fractional GJMS operators are now well understood [1, 9, 11, 21, 27, 38]. Second, this understanding of the Paneitz operator and its corresponding boundary operators is yielding new insights into the Yamabe-type problem for the fractional third-order -curvature [10]. Third, an algorithmic approach to constructing boundary operators for the higher-order GJMS operators via tractor calculus has recently been established [8, 21]. Relative to this latter work, the benefits of our approach are (i) that it directly yields local formulas for the boundary operators which are valid in all settings where the sixth-order GJMS operator is defined and (ii) that the generalized Dirichlet-to-Neumann operators constructed by our method are automatically formally self-adjoint. Unfortunately, unlike the tractor approach [8, 21], it does not seem practical to extend our method to GJMS operators of arbitrarily high order.

To describe our results, recall that the sixth-order GJMS operator of , , is given by

(1.3)

where is the Schouten tensor, is its trace, is the Bach tensor,

(1.4)

for the square of as an endomorphism, and is the sixth-order -curvature

We emphasize that (1.3) defines as an operator, so that the right-hand side consists of sums of compositions of operators. See Section 2 for a more detailed explanation of our notation. The conformal covariance of was proven independently by Branson [6] and Wünsch [41], though this can also be deduced via Juhl’s recursive formula [30] for the GJMS operators. In fact, is well-defined so long as is odd or is either locally conformally flat or Einstein. However, not every four-dimensional manifold admits a sixth-order conformally covariant operator with leading-order term ; see [23].

It is clear from (1.3) that is a sixth-order operator which is formally self-adjoint in the interior of . We thus expect there to be a set of six operators , , of total and normal order which give rise to formally self-adjoint boundary problems for . The following theorem in fact gives a stronger statement about the boundary operators.

Theorem 1.1.

Let , , be a compactification of a Poincaré–Einstein manifold . There exist explicit operators ,

where denotes the outward-pointing normal vector field along the boundary, denotes the Laplacian defined in terms of the induced metric on the boundary, and “” in denotes terms of order at most in , such that

  1. the operator is conformally covariant of bidegree ; i.e.

    for all , where is defined with respect to ; and

  2. the bilinear form ,

    is symmetric.

In fact, we prove a stronger version of Theorem 1.1 which requires only that is defined and that the boundary satisfy certain conformally invariant assumptions involving only the extrinsic geometry of the boundary ; see Section 3 for this version and explicit formulae for the operators .

By definition, is formally self-adjoint if for all , where is a -tuple of boundary operators. It follows from Theorem 1.1 that each of the eight possible -tuples formed by choosing an operator from each of , , and is such that is formally self-adjoint. The operators constructed by Gover and Peterson [21], which are defined whenever is defined and under no assumptions on the geometry of the boundary, also have the property that such triples are formally self-adjoint, though it is not yet known whether the corresponding bilinear form is symmetric. Similar operators constructed earlier by Branson and Gover [8] are such that the corresponding bilinear form is symmetric, but their construction does not work in the critical dimension .

One reason to desire the symmetry of , rather than just the formal self-adjointness of , is that it implies the formal self-adjointness of the generalized Dirichlet-to-Neumann operators associated to and its boundary operators from Theorem 1.1. As we show in Proposition 3.6 below, under the (conformally invariant) assumption that

for any triple , there is a unique solution of

(1.5)

In particular, the generalized Dirichlet-to-Neumann operators , , and are well-defined, and the symmetry of implies that these operators are formally self-adjoint. Indeed, , , is also conformally covariant with leading order term a multiple of ; see Proposition 3.6.

The operators constructed above have the same properties as the fractional GJMS operators constructed by Graham and Zworski [26], leading one to wonder how these operators are related. When is a compactification of a Poincaré–Einstein manifold, it turns out that and are proportional. Indeed, even more is true:

Theorem 1.2.

Let be a compactification of a Poincaré–Einstein manifold such that for . Suppose additionally that is such that . Then

where , , are the boundary operators of Theorem 1.1 and are the fractional GJMS operators of order .

In other words, for compactifications of Poincaré–Einstein manifolds, it holds that for all . The proof of Theorem 1.2 uses heavily the fact that the GJMS operators factor at Einstein metrics [18, 20]. We do not know if the Poincaré–Einstein assumption can be relaxed. Note also that requiring in Theorem 1.2 yields a curved analogue of the higher-order Caffarelli–Silvestre-type extension theorem of R. Yang [13].

Another reason to desire the symmetry of in Theorem 1.1 is that it gives rise to variational characterizations of solutions of with various boundary conditions. For example, a function is a solution of (1.5) if and only if it is a critical point of the functional

when constrained to the set

(1.6)

Under an additional spectral assumption on the Laplacian of the Poincaré–Einstein metric , one can in fact minimize the functional in .

Theorem 1.3.

Let be a compactification of a Poincaré–Einstein manifold such that . Given any , it holds that

(1.7)

for all . Moreover, equality holds in (1.7) if and only if is the unique solution of (1.5).

Note that the spectral assumption of Thoerem 1.3 holds automatically when the conformal boundary has nonnegative Yamabe constant [31].

In Section 5, we prove a more general version of Theorem 1.3 which only requires conformally invariant assumptions on the spectrum of and the extrinsic geometry of the boundary .

Theorem 1.3 gives a sharp norm inequality for the well-known embedding

(1.8)

as well as an explicit right inverse. By combining (1.8) with the Sobolev embedding , , one obtains the embedding

(1.9)

One can deduce a sharp norm inequality for the embedding (1.9) from Theorem 1.3 and a sharp norm inequality for the embedding . Three particular cases of interest are upper half space, a closed Euclidean ball, and a round hemisphere.

Corollary 1.4.

Let denote the (closed) upper half space

equipped with the Euclidean metric. For all , it holds that

where the -norms on the left-hand side are taken with respect to the Lebesgue measure on ,

for all , and

(1.10)

Moreover, equality holds if and only if and there are points , constants , and positive constants such that

(1.11)

for all .

Corollary 1.5.

Let denote the closed unit ball

equipped with the Euclidean metric. For all , it holds that

where

for the distance to . Moreover, equality holds if and only if and there are constants and points such that

(1.12)

for all .

Corollary 1.6.

Let denote the closed upper hemisphere

equipped with the round metric induced by the Euclidean metric on . For all , it holds that

where

for the outward-pointing unit normal along . Moreover, equality holds if and only if

and there are constants and points

such that

(1.13)

for all .

Corollary 1.4, Corollary 1.5 and Corollary 1.6 are equivalent by stereographic projection, though it is useful to have them written out explicitly in all cases. Since sharp Sobolev (trace) inequalities are useful when studying (boundary) Yamabe problems (e.g. [4, 16, 19]), we expect these corollaries to find applications in studies of the higher-order fractional Yamabe problem (cf. [19]).

One replacement of the embedding in the critical case is the Orlicz embedding . There is a sharp Onofri-type inequality [5] which establishes this embedding. The critical cases of Corollary 1.4, Corollary 1.5 and Corollary 1.6 are as follows:

Corollary 1.7.

Let denote the closed upper half space. For all , it holds that

where ,

for all , the -norms on the left-hand side are taken with respect to the Riemannian volume element of , and is the average of with respect to . Moreover, equality holds if and only if and there are points , constants , and positive constants such that

for all .

Corollary 1.8.

Let denote the closed Euclidean unit ball. For all , it holds that

where is the probability measure on induced by ,

for the distance to , the -norms on the right-hand side are taken with respect to the Riemannian volume element of , and is the average of with respect to . Moreover, equality holds if and only if there are constants and points such that

for all .

Corollary 1.9.

Let denote the closed hemisphere. For all , it holds that

where is the probability measure on induced by ,

for the outward-pointing normal along . Moreover, equality holds if and only if

and there are constants and points

such that

for all .

These inequalities generalize the Lebedev–Milin inequality for closed surfaces with boundary [36] and closed four-manifolds with boundary [1, 9]. For this reason, we expect Corollary 1.7, Corollary 1.8 and Corollary 1.9 to be useful when studying variational problems involving the functional determinant (cf. [12, 36]) and the fifth-order fractional -curvature (cf. [1, 9]) on six-manifolds with boundary.

This article is organized as follows:

In Section 2 we recall the definitions of Poincaré–Einstein manifolds and the fractional GJMS operators, pointing out in particular some useful conformally invariant properties of the boundaries of compactifications of Poincaré–Einstein manifolds. We also recall Branson’s method [7] for finding conformally covariant operators and give some computational lemmas which are useful in proving the conformal covariance of our boundary operators.

In Section 3 we give the full formulas for our boundary operators and prove Theorem 1.1. We also discuss the pseudodifferential operators they determine on the boundary.

In Section 4 we prove Theorem 1.2.

In Section 5 we study the Sobolev trace embeddings (1.8) and (1.9) in dimensions at least six. This includes deriving explicit sharp norm inequalities in the Euclidean upper half space, the Euclidean ball, and the round hemisphere.

Acknowledgments

JSC was supported by a grant from the Simons Foundation (Grant No. 524601).

2. Background

2.1. Important tensors on Riemannian manifolds with boundary

We begin by recalling some important tensors defined on a Riemannian manifold . The Schouten tensor of is

where is the Ricci tensor and is the scalar curvature of . We denote , so that is a constant multiple of the scalar curvature. The significance of the Schouten tensor comes from the decomposition

of the Riemann curvature tensor into the totally trace-free Weyl tensor and the Kulkarni–Nomizu product of the Schouten tensor and the metric. We sometimes use abstract index notation to represent tensors; for example,

We use the metric to raise and lower indices. For example, , which is regarded as a section of either or , depending on context. We denote by the composition of with itself; i.e.

and similarly for other compositions. The Cotton tensor is

and the Bach tensor is

where denotes the Levi-Civita connection of . Recall that each of , and are trace-free. Finally, the divergence of a