THE FUNCTIONAL DETERMINANT OF A 4-DIMENSIONAL BOUNDARY-VALUE PROBLEM

Working on four-dimensional manifolds with boundary, we consider, elliptic boundary value problems (A, B), A being the interior and B the boundary operator. These problems (A, B) should be valued in a tensor-spinor bundle; should depend in a universal way on a Riemannian metric g and be formally selfadjoint; should behave in an appropriate way under conformal change g --> OMEGA2g, OMEGA a smooth positive function; and the leading symbol of A should be positive definite. We view the functional determinant det A(B) of such a problem as a functional on a conformal class {OMEGA2g}, and develop a formula for the quotient of the determinant at OMEGA2g by that at g. (Analogous formulas are known to be intimately related to physical string theories in dimension two, and to sharp inequalities of borderline Sobolev embedding and Moser-Trudinger types for the boundariless case in even dimensions.) When the determinant in a background metric g0 is explicitly computable, the result is a formula for the determinant at each metric OMEGA2g0 (not just a quotient of determinants). For example, we compute the functional determinants of the Dirichlet and Robin (conformally covariant Neumann) problems for the Laplacian in the ball B4, using our general quotient formulas in the case of the conformal Laplacian, together with an explicit computation on the hemisphere H-4.