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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The functional determinant of a four-dimensional boundary value problem

Authors: Thomas P. Branson and Peter B. Gilkey
Journal: Trans. Amer. Math. Soc. 344 (1994), 479-531
MSC: Primary 58G26; Secondary 58G20
MathSciNet review: 1240945
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Abstract: 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 conformal change $ g \to {\Omega ^2}g$, $ \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 $ \{ {\Omega ^2}g\} $, and develop a formula for the quotient of the determinant at $ {\Omega ^2}g$ 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 $ {g_0}$ is explicitly computable, the result is a formula for the determinant at each metric $ {\Omega ^2}{g_0}$ (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 $ {B^4}$, using our general quotient formulas in the case of the conformal Laplacian, together with an explicit computation on the hemisphere $ {H^4}$.

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