# Hessian of the Zeta Function for the Laplacian on Forms

Let be a compact closed -dimensional manifold. Given a Riemannian metric on , we consider the zeta functions for the de Rham Laplacian and the Bochner Laplacian on -forms. The hessian of with respect to variations of the metric is given by a pseudodifferential operator . When the real part of is less than we compute the principal symbol of . This can be used to determine whether a general critical metric for has finite index, or whether it is an essential saddle point.

^{†}

^{†}support: The first author was supported by the National Science Foundation #DMS-9703329.

1. Introduction.

Over the last fifteen years, several results have been proved which identify maximal or minimal metrics for the determinant of a geometric elliptic operator on a compact manifold. In [OPS1], surfaces of constant curvature were shown to maximize the determinant of the Laplacian within the conformal class. Most higher dimensional cases studied involve extremizing a conformally covariant operator within a conformal class. In these cases local variation formulas are available, [BØ], [Po], and global results have been obtained, see [BCY], [Br], [CY]. (Some results also exist for manifolds which are non-closed or non-compact, see for example [BFKM], [CQ], [HZ], [OPS2], [OPS3].) In addition, extremal metrics for special values of the zeta function for conformally covariant operators have been studied, see [Mo]. The zeta function for the de Rham Laplacian on forms is interesting because of its connection to topology. For the Laplace-Beltrami operator , the standard -sphere is a local maximum for , but if is odd, the standard -sphere is a saddle for , see [Ok1], [Ri]. Beyond the scalar case, the determinant of the de Rham Laplacian was studied in relation to analytic torsion, [Ch], [Mü], [RS] and the variation of the zeta function was studied in [Ro], but there does not seem to have been progress on the subject of critical metrics since then. (In the non-scalar case there is not a great deal known about the spectrum of the Laplacian on forms. See [Ba], [BGV], [CT], [CC], [Fe], [Do], [Go], [Lo], [Ta] for results and further references.) In [Ok2] it was shown that for general geometric operators of Laplace type, the Hessian of the zeta function is given by a pseudodifferential operator , and when , a formula for the principal symbol was given. In the current paper we compute explicitly for the case of the de Rham and Bochner Laplacians on -forms. This can be applied to compute the behavior of the determinant and zeta function in the neighborhood of critical metrics. The computation of is somewhat lengthy for the de Rham Laplacian. The proof given here can serve as a blueprint for carrying out the calculation for any operator of Laplace type. Happily, the results of the calculations are simple to state, see Theorem 1.

In order to present the results we set up some notation. Let be a closed, compact -dimensional manifold, let be a character of the fundamental group of , let denote the complex line over twisted by , let denote the space of smooth -forms valued in , and let denote the exterior differential. For the Riemannian metric on , let be the volume form on and let be the volume of . For any tensor bundle over with values in , the metric naturally gives rise to an inner product on , an an inner product on the smooth sections of , and a connection on . We consider the de Rham Laplacian and the Bochner Laplacian on . These differ by a curvature operator. We use the notation to denote either of the two operators. Let

be the non-zero eigenvalues of . The spectral zeta function

converges for and extends to a meromorphic function with simple poles located at

The spectral determinant of is defined to be

Define the modified zeta function to be

which is entire in . We remark that the dimension, , of the kernel of the de Rham Laplacian is a topological invariant, and is consequently invariant under changes of the metric.

Let denote the bundle of symmetric tensor fields of type on , and let denote its space of smooth sections. The Hessian of the modified zeta function is the bilinear form on defined by

We recall the following result which holds for the general geometric operator of Laplace type defined on a tensor bundle.

###### [Ok2, Theorem 1]

For , there exists a unique symmetric pseudodifferential operator such that

The operator is analytic in . For , there exist polyhomogeneous pseudodifferential operators and of degrees and respectively such that . The operators and are meromorphic in with simple poles located in . (The poles of and cancel in the sum , but the symbol expansion of for involves logarithmic terms.) For general , the polyhomogeneous symbol expansion of is computable from the complete symbol of the operator . In particular, there is a simple algorithm to compute the term of homogeneity . Furthermore, we can differentiate in to obtain

and the principal symbol of is equal to the leading order term of (provided this does not vanish identically).

The purpose of this paper is to compute the symbol for the de Rham and Bochner Laplacians. This is the principal symbol of when , (providing it does not vanish identically) and so for in this range, governs to some extent how behaves in the neighborhood of a critical metric. In particular, generally governs the behavior of . We remark that the above theorem was stated in [Ok2] for the case when the Laplacian has been normalized to be scale invariant, but this assumption is easily seen to be unnecessary.

For a point , we write for the endomorphisms of the cotangent space , and we write “” for the trace function on . It is convenient to introduce the variable , and set

###### Theorem 1

Suppose is a metric on , is a point of , and . Set

For a non-zero covector , let

denote the -orthogonal projection onto the space spanned by , and let

Let be the de Rham or Bochner Laplacian on -forms, and in the case of the Bochner Laplacian assume that the dimension of its kernel has constant dimension when is deformed. Then at the metric ,

where

Before discussing applications, we point out some features of the result. Firstly one notices that for , if is given by

then

Hence for any , writing we have

This reflects on a micro-local level the fact that if is a one parameter family of diffeomorphisms of with equal to the identity and we set

then vanishes for every . It is easy to see that if does not have the form in (1.12) then

Secondly, we remark that instead of considering the full de Rham Laplacian, we can consider the operator acting on -forms. The zeta function for this operator is

From Theorem 1, we find that the symbol for the Hessian of is given by

Thirdly we remark that the analytic torsion defined in the acyclic case by

is a topological invariant. This implies that the corresponding alternating sum of symbols for on forms must vanish, and this is easily seen to be the case. In fact a stronger result is easily seen to be true from Theorem 1, namely for both the de Rham and the Bochner Laplacian and for all and all ,

Lastly we remark that the character plays no role in our results which come from local calculations. The only thing one sees in the results related to the line bundle is the factor “” in the right hand side of (1.7) which comes from the fact that the complex line has real dimension .

In a subsequent paper, we will show how Theorem 1 implies following results.

###### Theorem 2

Suppose , is real and , and is either the de Rham or the Bochner Laplacian. For , we consider the real numbers .

(a). If and and are both positive or both negative, then every critical metric for

has finite index. (That is, there exists a submanifold of of finite codimension, containing , such has a local minimum at on .)

(b). Conversely, if and have opposite signs, then every critical metric for is an essential saddle point. (That is, there are two infinite dimensional submanifolds , of , both containing , such that has a local minimum at on and a local maximum at on .)

###### Corollary

Let . If , every critical metric for has finite index. If , every critical metric for has finite index. The same is true if the de Rham Laplacian is replaced by the operator or the operator acting on forms. However, for the Bochner Laplacian on forms of degree , the determinant may have an essential saddle point. The lowest dimension in which this happens is .

We remark that because we have not normalized the Laplacian to be scale invariant, a critical metric in the results above is one which is critical for the given functional under variations of the metric which preserve the total volume.

The rest of the paper is devoted to proving Theorem 1. We work throughout in the real case, where the Laplacian is acting on real valued -forms. Because the calculation is entirely local, the character plays no role. The effect of going from the real case to the case to the case of the complex line is just to multiply by the factor , and this is the factor which appears in (1.7). In Section 2, we discuss the general formula for which was given in [Ok2], and we rewrite it in terms of the coefficient symbols of the first variation of the Laplacian. In Section 3, we compute the leading order terms in the first variation of , where is the Bochner or de Rham Laplacian. This is carried out in coordinates utilizing (3.7) as suggested by a referee for an early version of the paper which dealt only with the de Rham operator. Our change in notation has the advantage that the Bochner and de Rham Laplacians can be dealt with in the same way, and it will also hopefully make the proofs easier to follow. In Section 4, we identify the matrix entries of the first variation of the Laplacian. In Sections 5 and 6, we substitute these matrix entries into our formula for , thus proving Theorem 1. For case of the de Rham Laplacian, it is essential to work in coordinates in which is diagonal, since otherwise the calculation becomes extremely lengthy. It is justified to choose to work in such coordinates because is the principal symbol of a pseudodifferential operator, and hence does not depend on the choice of coordinates. Section 7 contains some observations on the calculation of for the de Rham Laplacian in general coordinates.

2. The general formula for .

Our starting point to prove Theorem 1, is the following result which gives in the general case.

###### [Ok2, Theorem 2]

The symbol for the general geometric operator of Laplace type on the tensor bundle of real dimension can be computed as follows. Write and let . Take coordinates on which are orthonormal in the metric at the point , and take a local trivialization of on a neighborhood of to obtain coordinates for . Suppose that in these coordinates, the operator is given at the point by

where and are multiindices and is an real valued matrix. Set

Then at , the value of is given by

where writing for the identity operator on , the terms are given as follows:

It is convenient to introduce the following notation.

Note that the matrix for is

so

We will now write as a sum

To define the symbols , choose coordinates which are orthonormal for the metric at the point . We notice that (2.3) only involves those terms in (2.1) which have total degree , where the total degree of a term

is . We write

where the notation “” means that the expressions are equal up to terms having lower total degree, and where for fixed, and are matrix valued functions. The term is obtained by considering the terms in (2.4)-(2.9) of the form . We define the coefficient symbols of by replacing each occurrence of in (2.14) by .

These are matrix valued. We also define the coefficient components of by

Then is obtained by collecting all multiples of from (2.4)–(2.9), and its quadratic form is given by

The symbol is obtained from (2.6) after subtracting the term which goes into . Its quadratic form is given by

The term is computed from (2.7) and (2.8) and represents the interaction of with the other terms after subtracting the terms which go into :

The term is computed from (2.9) and represents the interaction of with itself after has been taken into account:

The fact that (2.11) holds may be verified by explicitly computing by substituting (2.14) into (2.3).

3. Calculation of .

To apply these formulas to the Bochner and de Rham Laplacians , we must compute the terms of of total degree in the form (2.14). Let be local coordinates on , and let denote the bundle of alternating tensors on . Then the subsets of size index a basis for . The th basis element is , where are chosen so that and . For a -form , we write in components

where , with defined so that and . It is convenient to write for .

###### Lemma 3.1

In coordinates which are normal at a point ,

and

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