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Asymptotic Formulae in Spectral Geometry

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The following result provides the spectral theory of self-adjoint elliptic operators. Furthermore, the estimate of Assertion 2 shows the eigenvalues grow relatively rapidly. The same observation is true even in the non-self-adjoint setting. The following is a special case of a more general result.

This is the union of a large ball around the origin with a small wedge around the real axis which contains the set [0; 1 in its interior. If the hypotheses of Theorem 1. We may apply the L2 functional calculus to de ne the operator Z 1 e tP: This is now a bounded operator on L2. We shall omit the proofs in the interest of brevity.

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The heat trace asymptotics an F; P vanish for n odd. Let V be a vector bundle of rank r over M. The zeroth order part vanishes in this example. Thus in this instance, the weight simply counts the total number of derivatives of the metric which appear. This de nition of weight is also motivated by the Seeley calculus [] which gives explicit combinatorial formulae for the endomorphism valued invariants en x; P which were de ned above in Theorem 1. The following observation is then an immediate consequence of these formulae. We refer to the discussion in [] for further details. If P is an operator of Laplace type, then the situation simpli es considerably as we shall see in Section 1.

We shall discuss the weight further at that point in the context of dimensional analysis. We de ne scalar valued invariants an x; P by setting an x; P: There is an in nite product formula that can be used to show s never vanishes. The zeta function and the heat trace are related by the Mellin transform. We assume the bundles V1 and V2 have Hermitian structures. The index of A is given by setting index A: Then we may use Theorem 1. We de ne the supertrace heat asymptotics by an x; A: C 1 V2 be an elliptic complex over a closed m dimensional Riemannian manifold M.

We use an observation due to Bott in the proof; this observation is the foundation of heat equation proofs of the index theorem. The Theorem now follows by equating terms in the associated asymptotic series; only the constant term can be non-zero. This will lead to a heat equation proof of the classical ChernGauss-Bonnet theorem for manifolds with boundary in Theorem 1.

Assertion 1 can be established using the Seeley calculus []. We begin by introducing the classical Lopatinskij-Shapiro condition. We refer to Grubb [] for further details concerning the material of this section. Rather than working with the symbolic spectrum, it is convenient to work with a complementary subset. We suppose that U is a conical subset of C. Thus necessarily 0 2 U. Fix a connection r on V ; if P is an operator of Laplace type, then we shall take the connection of Lemma 1. To simplify the notation, we set W: One can use a partition of unity to see that Lemma 1.

C 1 W is surjective. This is a slight notational change from the convention employed in []. In the context of the singular structures de ned in Section 1. We suppress this additional technical fuss for the present in the interest of notational simplicity; we shall return to this point in Sections 1.

Both W and W are examples of graded vector bundles. However, a bit of care is needed with the initial condition. It is worth digressing brie y to provide an example of this phenomena. We will return to this example subsequently in Section 2. We will show presently in Example 1. Separation of variables now yields p X 22 2 tn p sin nx: Assume that P; B is elliptic with respect to the cone C.

Let F 2 C 1 End V be an auxiliary smooth endomorphism. Let the interior invariants en x; P be as described in Theorem 1. Let r be an auxiliary connection on V. If B denotes spectral boundary conditions see Section 1. The asymptotic series here can contain non-local terms and also log terms. However, these terms do not appear until the constant term and thus play no role in our analysis. In the interest of simplicity, we omit a fuller discussion of this context and refer instead to Grubb [, , ] and to Grubb and Seeley [, , ]. C 1 V2 be an elliptic complex of Dirac type.

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Impose spectral boundary conditions B. Let F be an auxiliary smooth endomorphism of V1. However the calculations of the parametrix discussed there generalize immediately to establish the existence of the appropriate asymptotics for the heat content function; we refer to Grubb [] for an excellent discussion of the heat equation in the context of elliptic boundary value problems. We also note that work of Kozlov [] provides an alternative approach to the matter at hand. Assume P; B is elliptic with respect to the cone C.

The interior invariants are not unique. Choose the branch of the square root function so that p 0: The solutions to Equation 1. The following operator will play a crucial role in our subsequent analysis. It will be used to express the Lopatinskij-Shapiro condition on the symbolic level when considering boundary conditions for operators of Laplace type.

Let B be as above. A boundary condition is then an endomorphism B: We choose the branch of the square root function so p 0: We will use the following operator in our discussion of the Lopatinskij-Shapiro condition. Then P; B is elliptic with respect to the cone K if and only if B: Thus P; B is elliptic with respect to K if and only if B: At that point, the relevant sign conventions will be crucial.

Let B1 de ne a 0th order boundary condition; let B2: Adopt the notation of Equations 1. Then the following assertions are equivalent: P; B1 is elliptic with respect to the cone K. D; B2 is elliptic with respect to the cone C. We apply Lemmas 1. To prove that Assertion 1 implies Assertion 2 , we suppose that P; B1 is elliptic with respect to the cone K but that D; B2 is not elliptic with respect to the cone C and argue for a contradiction.

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This contradiction shows Assertion 1 implies Assertion 2. Conversely, suppose that D; B2 is elliptic with respect to the cone C. Let B be a boundary condition so that P; B is elliptic with respect to either the cone C or the cone K. Let V have a Hermitian innerproduct. Although the proofs are elementary, the formulae are central to the subject. We begin by establishing a basic identity.

The proper sign can be determined by making a computation on the half space in Rn ; we always use the inward unit normal. We can now derive the Green's formula for an operator of Dirac type: Let be a Clifford module structure on a vector bundle V over M. Then Dirac type on V. We use Equations 1. M ut It will be useful to have a similar Green's formula on the boundary for use in studying spectral boundary conditions in Section 1. Let be a Clifford module structure on a vector bundle V over M and let r be a compatible connection.

We apply Equation 1. We use Equation 1. Let feig be a local orthonormal frame. Let r be the Levi-Civita connection. Let B be a local boundary condition. Assume that P; B is elliptic with respect to the cone f0g and that PB is self-adjoint. Throughout this and the subsequent section, let D be an operator of Laplace type and let r be the associated covariant derivative. We adopt the notation of Equation 1. D; B is elliptic with respect to the cone C. The rst assertion now follows by Lemma 1.

To prove the second assertion, we use Lemma 1. Solving the heat equation with Neumann boundary conditions corresponds physically to insulating the boundary. Solving the heat equation with Robin boundary conditions corresponds to ensuring that the heat ow across the boundary is proportional to the temperature on the boundary. The same argument used to establish Assertion 2 of Lemma 1. If we take Dirichlet boundary conditions on CD and Robin boundary conditions on CR , the computations performed above show these boundary conditions are elliptic with respect to the cone C.

Furthermore as the structures don't interact, the dual boundary conditions are once again Dirichlet boundary conditions on CD and Robin boundary conditions on CR. Thus Assertion 1 follows from Lemma 1. To prove Assertion 2 , we use Lemma 1. They arise in index theory [] and also in the context of quantum gravity, see Luckock and Moss []. J dx The absolute boundary operator Ba is de ned by setting Ba: As in Section 1.

The form of the metric tensor given in Equation 1. Assertion 3 now follows from Assertion 1 and from Lemma 1. By Assertion 1 , S is self-adjoint. Assertion 4 follows from Lemma 1. Relative boundary conditions are de ned by dualizing with respect to? The following reformulation of absolute and relative boundary conditions is a useful one. Near the boundary, we may decompose any smooth form!

Let i be the inclusion of M into M. We de ne Ca! Let dT be the exterior derivative and let? We ignore the precise signs as they play no role in de ning the boundary conditions. Assertion 2 now follows dually from Assertion 1 and from Lemma 1. This establishes Assertion 3 ; Assertion 4 then follows by duality. We shall suppose that B de nes absolute boundary conditions; the corresponding assertion for relative boundary conditions will then follow by duality. Thus to complete the proof, we need only check the boundary conditions are preserved. We apply Lemma 1. This generalizes to the setting of smooth manifolds with boundary as: Br denote absolute resp.

We then have 1. Thus they induce the spectral resolution of the Laplacian with absolute boundary conditions on M given in Equation 1. The perhaps somewhat perverse sign convention for U that we shall adopt here is motivated by the notation established there. D; BU is elliptic with respect to the cone C. Thus the operator b of Equation 1. Consequently, Assertion 1 follows from Lemma 1.

The Green's formula given in Lemma 1. Conversely, if Display 1.

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Boundary conditions II 1. We increase the weight by 1 for every additional explicit covariant derivative which appears. It incorporates the work of many authors into the presentation, and includes a complete bibliography that serves as a roadmap to the literature on the subject. This is a slight notational change from the convention employed in []. Atiyah, Bott, and Patodi [7] used Weyl's rst theorem of invariants Theorem 1.

This establishes the second condition of De nition 1. We remark that this metric need not be smooth on M. We postpone until Lemma 3. We use Remark 1. Then B1 and BU de ne the same boundary conditions. Assertion 2 now follows. We complete the proof by applying Equations 1. As before, no matching condition is assumed on the normal derivatives of the metrics.

D; BS is elliptic with respect to the cone C. Since the ellipticity condition does not depend upon the lower order terms, Assertion 1 follows from Lemma 1. We use the the Green's formula given in Lemma 1. We follow the discussion in Carslaw and Jaeger []. This means that 1. This yields the boundary condition de ned by the boundary operator of Equation 1.

The boundary conditions of Equations 1. The chiral operator discussed in Section 1. We recall some notation de ned previously in Equations 1. We change notation slightly from that established earlier in the proof of Lemma 1. Then f m u1; Relative to the basis fu1; m u1 ; Assertion 1 now follows; Assertion 2 follows from Assertion 1 by Lemma 1.

Spectral boundary conditions were introduced by Atiyah, Patodi, and Singer [9, 10, 11] in their study of the signature and spin complexes as these complexes do not admit local boundary conditions. This is closely related to the obstruction described in Lemma 1. Let M be a compact Riemannian manifold with smooth boundary M.

In our discussion of bag boundary conditions in Section 1. Suppose to the contrary that P; B is elliptic with respect to the cone K. Fix a point y 2 M. Since P; B is elliptic with respect to the cone K, by Lemma 1. We then have, see for example [] Lemma 2. The matrices f m 1 ; We introduce some auxiliary notation. Let A be admissible with respect to P. Note that projection on the negative spectrum of A would not be elliptic with respect to the cone K.

We may now derive Assertion 3 from Equations 1. Even if A is not self-adjoint, an appropriate generalization of Theorem 1. This operator is said to be a non-minimal operator in the physics literature [83, ]. Let e and i be exterior and interior multiplication. Thus the leading symbol of D is self-adjoint. This shows that the symbolic spectrum of D is 0; 1 and thus D is elliptic with respect to the cone C.

With a bit more work, the same arguments as those used in the discussion of Section 1. We omit details in the interest of brevity. We consider a slightly more general notion of ellipticity. The operator of Equation 1. We determine the associated Green's formula to prove the second assertion.

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Let r be the connection de ned by D0 on V using Lemma 1. This is not de ned by the functional calculus. Let dx0 be the Riemannian metric of the initial background metric g0. We must express D tensorially. Since our study is motivated by the ordinary heat equation in the context of variable geometries, we must express the scalar Laplacians for a family of time-dependent metrics in this setting. Let Dt be the associated scalar Laplacians. We follow the discussion in [] to prove this Lemma. We correct a minor mistake in the computation which was given there. We may now compute: The Lemma now follows as the point P was arbitrary.

We shall not do this as the whole point is to work in a basis free fashion and thereby derive results which can be used to determine local invariants of a Riemannian manifold. Following Weyl, we shall say that a polynomial map f: Let Ak;V be the commutative unital algebra of all such orthogonal invariants. We then have by Weyl []: Every orthogonal invariant which depends on k vectors v1 ; We also have by Weyl []: These two results can be given a slightly more algebraic avor as follows.

The evaluation e Aij v1 ; Ak;V which is surjective by Theorem 1. If f is multi-linear, then f v1 ; Thus there are no multilinear maps which are orthogonally invariant if k is odd so we assume k is even henceforth. We apply Theorem 1. Set fI v1 ; We may then use Theorem 1. Since p is multi-linear, p x1 v1 ; Since p v1 ; We then have fI x1 v1 ; Consequently Dk fI x1 v1 ; Thus we may restrict the sum in Equations 1.

Choose an inner product preserving inclusion i: We de ne a restriction map R: Ik;W which is characterized dually by the property R p w1 ; Ik;W is injective if k 1 A If p is given by contractions of indices which range from 1 to m, then 1. R p is given by restricting the range of summation to range from 1 to m Consequently, by Theorem 1.

To prove Assertion 2 , we use Theorem 1. We use the second main theorem of invariants 1. The desired result now follows from Equation 1. In light of Theorem 1. Such invariants can be constructed as follows. This is a formal object that can be evaluated on a Riemannian metric g once a local orthonormal frame e is chosen.

The algebra of such formulae is not a free polynomial algebra as we must take into account the symmetries of the covariant derivatives curvature tensor; see, for example, the relations in Equations 1. We say that P is invariant if the value of P g; e is independent of the particular orthonormal frame e which is chosen and depends only on the underlying metric g.

Let Pm be the space of all such invariants. The space Pm inherits a natural grading. We de ne the weight of Rijkl to be 2 and increase the weight by 1 for every explicit covariant derivative present; weight Ri1 i2 i3 i4 ;j1 ; We can also apply dimensional analysis to see that the weight is well de ned. We rescale the Riemannian metric to de ne gc: Since the Levi-Civita connection is unchanged, we may compute Rijkl gc ; ec: More generally, a similar computation shows that Ri1 i2 i3 i4 ;j Atiyah, Bott, and Patodi [7] used Weyl's rst theorem of invariants Theorem 1.

We follow their approach to show: Then P 2 P2;m is linear in the components of the curvature tensor. We regard P as a map from W to R which is orthogonally invariant. Extending P to be zero on W? Thus by Equation 1. Assertion 2 now follows since, after taking into account the symmetries of Display 1. The relation of Equation 1. As W is orthogonally invariant, extending P to be zero on W? On the boundary, the normal vector eld plays a distinguished role and thus the structure group is O m 1 and not O m.

We use Theorem 1. Weyl's theorem then applies. Reducing by the appropriate relations is a bit more complicated as relations of the form given in Lemma 1. The following Lemma then follows: In Chapter 2 and in Chapter 3, we will study more complicated spaces of invariants, but the basic approach will be similar. We refer, for example, to Lemmas 2. We can use Theorem 1. We introduce formal endomorphism valued invariants rk E and rk.

Set weight Ri1 i2 i3 i4 ;j We note that Equation 1. The following observation provides a useful restatement of Lemma 1. As the proof is a straightforward application of the techniques we have discussed above, we shall omit the proof. We shall return to this question again in Section 3. Then the heat invariant en x; D is given by a universal non-commutative polynomial of total weight n in the variables Lemma 1. For example, it will follow from Theorem 3.

We refer to Section 3. It is convenient, however, to postpone this analysis for a bit and to have Lemma 1. Pn;m 1 Restricting the indices to range from 1 through m 1 instead of from 1 through m induces a natural restriction map r: This map is surjective by Theorem 1. We can also describe the restriction map geometrically. If N; gN is an m 1 dimensional Riemannian manifold, then we let M: If x 2 N is the point of evaluation, we take the corresponding point x; 1 2 M for evaluation; which point on the circle chosen is, of course, irrelevant as S 1 has a rotational symmetry.

We de ne the totally anti-symmetric tensor " by setting "IJ: If I and J are permutations of a set of k distinct indices, then " is the sign of the permutation which sends I to J. This observation will play a crucial role in the next section. We will conclude our study of these invariants in Section 3. We follow the discussion in [, , ].

The Witten Laplacian, introduced in Section 1. The metric g and the dimension m are introduced into the notation for later convenience. Let f be a smooth scalar function on M. We have adn;m Since the invariants in question are local, global questions of orientability play no role and we may without loss of generality assume that the manifold in question is orientable. Assertions 1 and 2 now follow. We may now use Theorem 1. There are some additional properties of these invariants that will be useful. Introduce the parity operator ": The restriction map r of Section 1.

If m is odd, then adn;m Lemma 1. This shows that curvature R appears in each Weyl monomial of adn;m Proof: In particular, adn;m vanish if m is odd. Let I and J be m tuples of indices indexing an orthonormal frame for T M. The weight n of A is then given by Proof: Let k be total number of indices present in A. This proves Assertion 2. Thus the inequalities of Display 1.

The argument given above shows we can write Ri1 j1 i2 j2 in terms of Ri1 i2 j2 j1. We omit certain invariants from the list if m is small. We do not use the Witten co-derivative at this stage. This is a crucial technical point. Let N be a Riemannian manifold of dimension m 1. Qn;m 1 is de ned dually by the identity: Qn;m 1 is surjective.

It is straightforward to extend Theorem 1. We refer to [7] for further details. The invariants that belong to Qpn;m are de ned by letting the indices in question range from 1 to m. We de ne rQ by restricting the range of summation to be from 1 to m 1. Assertion 1 now follows.

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This establishes Assertion 2. The analysis of the formal cohomology groups of the spaces of invariants of Riemannian manifolds, which was given in [], then extends immediately to this more general setting to yield: Assertion 1 shows that any scalar invariant which always integrates to zero is canonically a divergence. Assertion 2 implies that any 1 form valued invariant which is co-closed is canonically co-exact. We use this result to show Lemma 1.

Thus by Lemma 1. Since by Lemma 1. Let k be the total number of indices appearing in A. As in the proof of Lemma 1. We distinguish two cases: Suppose that Display 1. Then all the 2m indices present in A are contracted using the " tensor. We can commute covariant derivatives at the cost of introducing additional curvature terms. Furthermore, by the rst and second Bianchi identity, at most 2 indices can be alternated in Rijkl;.

Since the inequality in Display 1. Case 2 Suppose that Display 1. We shall expand the determinant "UV in minors around various rows and columns. We sum these invariants to expand Equation 1. Next, we expand about row 2. We expand around the rst row. This establishes the lemma. If m is even, then adm;m J;1. Assertions 1 , 2 , and 3 follow directly from Lemmas 1. To prove Assertion 4 , let m be even.

Rm 1;mm;m 1 and where? Let an;k be the heat trace invariants de ned in Equation 1. On the boundary, consider polynomials in the components of the tensors fR; rR; r2 R; We let weight rk R: We de ne RA;t 1. Rat 1 at bt bt 1 ; A;t LB;s: Lat bt ; 2k A;m 1 Fmk 1;m: Consequently if t 1. Let A be a monomial of Q of weight n for Proof: Let kT be the total number of tangential indices in A, and let km be the total number of times the normal index m appears in A. Since every index is tangential and all are contracted using the tensor ", after 1;k and Fm;m 2;k.

Unfortunately, rPm1 1;m need not vanish and we must adjust Pm1 1;m in exactly the same fashion as was done previously. This means that every inequality in Display 1. We adopt the notation of Display 1. There exist constants so that: Let M be a compact Riemannian manifold of dimension m with smooth boundary and let Ba de ne absolute boundary conditions. Subsequently, additional proofs were given by Atiyah, Bott, and Patodi [7] and by Gilkey []. The extension to manifolds with boundary was established later in []. We will discuss similar results subsequently in Section 3.

Impose absolute boundary conditions. Let M be a compact Riemannian manifold which has smooth boundary M. Subsequently, the local index density for the classical elliptic complexes was identi ed by Atiyah, Bott, and Patodi [7] who used this to give a heat equation proof of the Atiyah-Singer index theorem in complete generality. The case of manifolds with boundary was studied in []. We also refer to [61, 63, , ] for other treatments of the local index theorem.

Patodi's approach involved a direct calculation analyzing cancellation formulae for the fundamental solution of the heat equation. Atiyah, Bott, and Patodi used invariance theory to identify the local index density for the twisted signature and twisted spin complexes. By expressing the de Rham complex, at least locally, in terms of the twisted spin complex, they gave a second proof of Patodi's result for the de Rham complex. We rst analyze the interior invariants. We use this relation to establish Assertion 1 of Theorem 1. Assertion 2 of Theorem 1. We solve this equation for ckm;m;0 to complete the proof of Theorem 1.

We adopt the following notational conventions throughout. Let M be a compact Riemannian manifold of dimension m with smooth boundary M. Let DB be the realization of D with respect to the boundary conditions de ned by a suitable boundary operator B. The initial condition is to be understood in the distributional sense if M is non-empty. M By Theorem 1. M M The interior invariants are completely determined by Equation 1. Thus we shall focus on computing the boundary integrands for1 various boundary conditions. This requires changing more symmetric integrand 2 M the formula for 4 to include the appropriate correction term in the Green's formula.

This change is motivated by preserving the symmetry, established in Lemma 2. There are many functorial properties which are independent of the particular boundary condition studied. The symmetry property of Lemma 2. There are also relations between the heat content asymptotics for various boundary conditions. For example Lemma 2.

There are also many results which are speci c to the particular boundary condition studied.

In subsequent sections, we treat various boundary conditions separately. The results of Section 2. In addition to terms that arise from decoupling the problem as a direct sum of Dirichlet and Robin boundary conditions, there are new contributions which re ect the failure of the problem to decouple. In principle, 4 could be computed, but the essential new features are already manifest at the 3 level. Much of the computation relies upon the functorial properties which relate the heat content asymptotics with these boundary conditions to the corresponding heat content asymptotics with Dirichlet and Robin boundary conditions.

There are, however, signi cant new phenomena which relate to the singular structures involved and which relate to the results in Lemmas 2. We give in Theorems 2. The ellipticity condition is not automatic in this setting, but it does follow for small tangential operators by Lemma 1.

It is somewhat surprising that the heat content asymptotics in this setting are not sensitive to the loss of ellipticity; the formulae of Theorem 2. By contrast, the formulae for the heat trace asymptotics given in [25, ] become singular when the ellipticity condition breaks down. We consider either Dirichlet boundary conditions or a smooth 1 parameter family of Robin boundary conditions. Here p is a smooth time-dependent potential function describing heat sources and sinks in the interior and is a smooth time-dependent function giving the heat ow on the boundary.

Both theorems relate the resulting heat content asymptotics to previously computed invariants. The most interesting phenomena relate to the heat transfer de ned by. In the proof of Theorem 2. These are not operators of Laplace type. We follow the discussion of [] to determine the invariants 0, 1, and 2 in this setting for relative and absolute boundary conditions. In our study of the boundary invariants for various boundary conditions, we shall have to introduce various universal constants. We clear the previous notation in each new Section; thus the constants ci of Section 2.

Until much later in this section, we work in great generality. We suppose M is a compact Riemannian manifold with smooth boundary M. The following observation will be fundamental to our study. Furthermore, Assertions 2 and 3 follow from Equation 1. Let u" x; t: Let u x; t: We shall use the same argument as that given in the proof of Assertions 2 and 3 of Theorem 1.

M M We apply Lemma 2. Normalize the interior integrands for n as in Lemma 2. For example, if B de nes Dirichlet boundary conditions, then we can let B1 be the Robin boundary operator for any S ; if B de nes Robin boundary conditions, then we can let B1 be the Dirichlet boundary operator. We assume further that DB is self-adjoint.

If DB is self-adjoint, then: Assertion 1 is now immediate since u satis es the de ning relations given in Display 1. The series in question converge uniformly. We use Lemma 2. Thus we recover Lemma 2. M Consequently, we may give another derivation of Lemma 2. The heat content asymptotics are additive with respect to direct sums. It is immediate that D; B is admissible. Assertion 2 follows from Assertion 1 by equating terms in the asymptotic expansions. We begin by studying Cartesian product structures: Let M1 be closed so no boundary condition is needed for D1.

De ne D and B by setting Dt: It is clear D; B is again admissible. The rst assertion now follows; the second follows from the rst by equating terms in the asymptotic expansions. We introduce some notational conventions. Let B de ne Dirichlet or Robin boundary conditions. Clearly D; B is admissible on M. M The Lemma now follows from these two identities. Assume D; B is admissible. The same argument as that given to establish Lemma 2. M M It then follows from the Greiner-Seeley calculus [, ] that: The operator Dc is of Laplace type with respect to the Riemannian metric gc: Furthermore, Dc; B is admissible.

The Riemannian measures dx on M and dy on M which are de ned by g are then given by q q dx: The operators D1 and D2 are formally self-adjoint. The operator B2 de nes Dirichlet boundary conditions. Thus the discussion in Sections 1. Thus, by Lemma 1. The second equality of Assertion 1 follows by equating terms in the asymptotic expansions.

Assertion 2 follows from Assertion 1 by duality, since the Hodge operator interchanges relative and absolute boundary conditions. We double a model problem M0; V0 ; D0 to obtain our rst functorial property by considering the singular structures. Decompose u into even and odd pieces by setting ueven x; t: This proves the rst assertion; the second now follows. The rst assertion of the Lemma is now immediate; the second follows from the rst.

M0 This establishes Assertion 1 ; Assertion 2 follows from Assertion 1. We adopt the notation of Section 1. The Lemma now follows. Since the operators and boundary conditions decouple, the following Lemma is immediate: The rst property deals with a rather trivial variation of the geometry.

Consider the family Dt: The rst assertion now follows. Consequently, we may compute: Let " be an auxiliary real parameter. We prove the rst assertion by computing: Thus is independent of the parameter t. We rst deal with Dirichlet boundary conditions as there are no additional structures present. We recall our previous notational conventions.

Let indices i; j; k range from 1 through m and index this frame eld. We let indices a; b; c range from 1 to m 1 and index the induced orthonormal frame for the tangent bundle of the boundary. Because the second fundamental form L is de ned only on M , Lab;c is not de ned and we use instead the tensor Lab: Let Rijkl be the components of the curvature tensor of the Levi-Civita connection, let ij be the components of the curvature operator of the connection de ned by D, let E be the endomorphism de ned by D, and let Lab be the components of the second fundamental form.

On the other hand, the second fundamental form L is de ned only on M. An appropriate generalization of Theorem 1. The analysis performed in Section 1.

The Levi-Civita connections of the metric g and gc: The connections determined by D and Dc: We may then use Theorem 2. M M M For the remaining boundary conditions, there are additional structures present. Taking into account this additional invariant, we have: Let B de ne Robin boundary conditions for an operator D Laplace type. Again, the weight is increased by 1 for every explicit covariant derivative that is present.

We illustrate this principle for Dirichlet boundary conditions. There exist universal constants ci which are independent of the dimension of the underlying manifold M and of the vector bundle V so that if B de nes Dirichlet boundary conditions for an operator D of Laplace type, then: Give M the product metric ds2M: We adopt the notation of Lemma 2. We may therefore apply Lemma 2. We follow the discussion of Section 1. The assertion that the constants ci can be chosen independent of the dimension m now follows from this discussion. For Robin, mixed, transmission, and transfer boundary conditions, additional invariants must be introduced which re ect the auxiliary structures which are involved.

With oblique boundary conditions, the leading symbol of the tangential operator has weight 0 and more care needs to be exercised. There is a Z2 symmetry involved using the involution P! The following observation is immediate: The next functorial property relates the heat content asymptotics for spectral boundary conditions to the corresponding heat content asymptotics for mixed boundary conditions in certain settings.

We recall the notation of Section 1. In the next result, we will relate spectral and mixed boundary conditions. Before giving the statement, it is necessary to introduce some notation. We assume a0 anti-commutes with m. We show that B0 de nes mixed boundary conditions for D0: Adopt the notational conventions and assumptions described above. We equate terms in the asymptotic expansions to derive Assertion 2 from Assertion 1. We refer to the subsequent discussion in Ansatz 2. Express n with respect to a suitable Weyl basis. Let PN be an operator of Dirac type on an m 1 dimensional manifold N.

De ne analogous structures on M by setting P: The desired independence of the universal constants now follows. We begin with two special case computations which are of interest in their own right and which deal with this situation. Recall the de nition of the Gamma function Z1 s: Let Dm be the unit disk in Rm. Next, we present some results for the hemisphere from [47]: Let H m be the upper hemisphere of the unit sphere S m. Let D be an operator of Laplace type on a compact Riemannian manifold with smooth boundary M.

Let B denote Dirichlet boundary conditions. If M is the upper hemisphere of the unit sphere in at space, then 0 M , 1 M , and 2 M were computed by van den Berg [37]; we refer to [47] for later work which establishes Theorem 2. There are related results due to Phillips and Janson []. Subsequently, Savo [, , , ] gave a closed formula for all the heat content asymptotics k M in this setting. We also refer to related work by McDonald and Meyers []. The remainder of this section is devoted to the proof of Assertions 1 , 2 , 3 , and 4 of Theorem 2.

Assertion 1 of Theorem 2. We shall use the arguments used to established Lemma 2. We shall complete the proof of Theorem 2. We begin our investigation by determining 1 in this setting: Already read this title? Please accept our apologies for any inconvenience this may cause. Exclusive web offer for individuals. Add to Wish List. Toggle navigation Additional Book Information. Description Table of Contents.

Summary A great deal of progress has been made recently in the field of asymptotic formulas that arise in the theory of Dirac and Laplace type operators. Asymptotic Formulae in Spectral Geometry collects these results and computations into one book. Written by a leading pioneer in the field, it focuses on the functorial and special cases methods of computing asymptotic heat trace and heat content coefficients in the heat equation.

It incorporates the work of many authors into the presentation, and includes a complete bibliography that serves as a roadmap to the literature on the subject. Geometers, mathematical physicists, and analysts alike will undoubtedly find this book to be the definitive book on the subject.

Table of Contents A great deal of progress has been made recently in the field of asymptotic formulas that arise in the theory of Dirac and Laplace type operators. Geometers, mathematical physicists, and analysts alike will undoubtedly find this to be the definitive book on the subject. Request an e-inspection copy.