Functional determinants and geometry. (English) Zbl 0602.58044

We study the zeta-function determinant in the context of elliptic boundary value problems. Our main technique is to relate the determinant of an operator, or a ratio of determinants, to the boundary values of the solutions of the operator. This has the advantage of restricting attention to the solutions of the operator, which do not depend on the boundary conditions and can often be written down explicitly, rather than the eigenvalues, which are usually difficult to work with. In addition, the problem is reduced to a calculation over the boundary of the manifold which is a closed manifold of dimension one less than the original manifold. This has special significance in the case that the manifold is a finite interval. In this case the boundary is a pair of points and the determinant of an ordinary differential operator is expressed in terms of the determinant of a finite matrix.
The results are then applied to some geometric operators. In section 4 we study the Jacobi operator acting along a geodesic segment and the covariant derivative operator acting along a loop. In section 2 we calculate the determinant of the Laplacian acting on sections of a flat bundle over a flat torus. This can be related to an Eisenstein series and thus we have presented a new geometric method of summing such series. This sum is known as Kronecker’s second limit formula. We then consider operators on a product manifold \(M\times S^ 1\).


58J32 Boundary value problems on manifolds
35J25 Boundary value problems for second-order elliptic equations
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