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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The functional determinant of a four-dimensional boundary value problem
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by Thomas P. Branson and Peter B. Gilkey PDF
Trans. Amer. Math. Soc. 344 (1994), 479-531 Request permission

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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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 344 (1994), 479-531
  • MSC: Primary 58G26; Secondary 58G20
  • DOI: https://doi.org/10.1090/S0002-9947-1994-1240945-8
  • MathSciNet review: 1240945