The functional determinant of a four-dimensional boundary value problem

Thomas P. Branson, Peter B. Gilkey
1994 Transactions of the American Mathematical Society  
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 tensorspinor bundle; should depend in a universal way on a Riemannian metric g and be formally selfadjoint; should behave in an appropriate way under conformai change g -> Q2g , Í2 a smooth positive function; and the leading symbol of A should be positive definite. We view the functional
more » ... unctional determinant det Aß of such a problem as a functional on a conformai class {0.2g} , and develop a formula for the quotient of the determinant at Cl2g 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 go is explicitly computable, the result is a formula for the determinant at each metric Si2g0 (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 conformai Laplacian, together with an explicit computation on the hemisphere H\
doi:10.1090/s0002-9947-1994-1240945-8 fatcat:dva3cfdo4va3rnhe7lqc76vsh4