## 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 ${H^4}$.

## References

- David R. Adams,
*A sharp inequality of J. Moser for higher order derivatives*, Ann. of Math. (2)**128**(1988), no. 2, 385–398. MR**960950**, DOI 10.2307/1971445 - William Beckner,
*Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality*, Ann. of Math. (2)**138**(1993), no. 1, 213–242. MR**1230930**, DOI 10.2307/2946638 - David Bleecker,
*Determination of a Riemannian metric from the first variation of its spectrum*, Amer. J. Math.**107**(1985), no. 4, 815–831. MR**796904**, DOI 10.2307/2374358 - Thomas P. Branson,
*Conformally covariant equations on differential forms*, Comm. Partial Differential Equations**7**(1982), no. 4, 393–431. MR**652815**, DOI 10.1080/03605308208820228 - Thomas P. Branson,
*Differential operators canonically associated to a conformal structure*, Math. Scand.**57**(1985), no. 2, 293–345. MR**832360**, DOI 10.7146/math.scand.a-12120 - Thomas P. Branson,
*Group representations arising from Lorentz conformal geometry*, J. Funct. Anal.**74**(1987), no. 2, 199–291. MR**904819**, DOI 10.1016/0022-1236(87)90025-5 - Thomas P. Branson,
*Harmonic analysis in vector bundles associated to the rotation and spin groups*, J. Funct. Anal.**106**(1992), no. 2, 314–328. MR**1165857**, DOI 10.1016/0022-1236(92)90050-S - Thomas P. Branson, Sun-Yung A. Chang, and Paul C. Yang,
*Estimates and extremals for zeta function determinants on four-manifolds*, Comm. Math. Phys.**149**(1992), no. 2, 241–262. MR**1186028** - Thomas P. Branson and Peter B. Gilkey,
*The asymptotics of the Laplacian on a manifold with boundary*, Comm. Partial Differential Equations**15**(1990), no. 2, 245–272. MR**1032631**, DOI 10.1080/03605309908820686 - Thomas P. Branson and Bent Ørsted,
*Conformal indices of Riemannian manifolds*, Compositio Math.**60**(1986), no. 3, 261–293. MR**869104** - Thomas P. Branson and Bent Ørsted,
*Conformal geometry and global invariants*, Differential Geom. Appl.**1**(1991), no. 3, 279–308. MR**1244447**, DOI 10.1016/0926-2245(91)90004-S - Thomas P. Branson and Bent Ørsted,
*Explicit functional determinants in four dimensions*, Proc. Amer. Math. Soc.**113**(1991), no. 3, 669–682. MR**1050018**, DOI 10.1090/S0002-9939-1991-1050018-8 - Constantine Callias and Clifford H. Taubes,
*Functional determinants in Euclidean Yang-Mills theory*, Comm. Math. Phys.**77**(1980), no. 3, 229–250. MR**594302** - Michael Eastwood and Michael Singer,
*A conformally invariant Maxwell gauge*, Phys. Lett. A**107**(1985), no. 2, 73–74. MR**774899**, DOI 10.1016/0375-9601(85)90198-7 - José F. Escobar,
*Sharp constant in a Sobolev trace inequality*, Indiana Univ. Math. J.**37**(1988), no. 3, 687–698. MR**962929**, DOI 10.1512/iumj.1988.37.37033 - José F. Escobar,
*The Yamabe problem on manifolds with boundary*, J. Differential Geom.**35**(1992), no. 1, 21–84. MR**1152225** - Howard D. Fegan and Peter Gilkey,
*Invariants of the heat equation*, Pacific J. Math.**117**(1985), no. 2, 233–254. MR**779919** - Peter B. Gilkey,
*Recursion relations and the asymptotic behavior of the eigenvalues of the Laplacian*, Compositio Math.**38**(1979), no. 2, 201–240. MR**528840** - Peter B. Gilkey,
*Invariance theory, the heat equation, and the Atiyah-Singer index theorem*, Mathematics Lecture Series, vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984. MR**783634** - Peter B. Gilkey and Lance Smith,
*The eta invariant for a class of elliptic boundary value problems*, Comm. Pure Appl. Math.**36**(1983), no. 1, 85–131. MR**680084**, DOI 10.1002/cpa.3160360105 - Yvette Kosmann,
*Dérivées de Lie des spineurs*, Ann. Mat. Pura Appl. (4)**91**(1972), 317–395 (French, with English summary). MR**312413**, DOI 10.1007/BF02428822 - Yvette Kosmann,
*Degrés conformes des laplaciens et des opérateurs de Dirac*, C. R. Acad. Sci. Paris Sér. A-B**280**(1975), no. 5, Aii, A283–A285 (French, with English summary). MR**391187** - E. Onofri,
*On the positivity of the effective action in a theory of random surfaces*, Comm. Math. Phys.**86**(1982), no. 3, 321–326. MR**677001** - Bent Ørsted,
*The conformal invariance of Huygens’ principle*, J. Differential Geometry**16**(1981), no. 1, 1–9. MR**633620** - B. Osgood, R. Phillips, and P. Sarnak,
*Extremals of determinants of Laplacians*, J. Funct. Anal.**80**(1988), no. 1, 148–211. MR**960228**, DOI 10.1016/0022-1236(88)90070-5 - B. Osgood, R. Phillips, and P. Sarnak,
*Compact isospectral sets of surfaces*, J. Funct. Anal.**80**(1988), no. 1, 212–234. MR**960229**, DOI 10.1016/0022-1236(88)90071-7
S. Paneitz, - Steven Rosenberg,
*The determinant of a conformally covariant operator*, J. London Math. Soc. (2)**36**(1987), no. 3, 553–568. MR**918645**, DOI 10.1112/jlms/s2-36.3.553 - Robert S. Strichartz,
*Linear algebra of curvature tensors and their covariant derivatives*, Canad. J. Math.**40**(1988), no. 5, 1105–1143. MR**973512**, DOI 10.4153/CJM-1988-046-7 - Ilan Vardi,
*Determinants of Laplacians and multiple gamma functions*, SIAM J. Math. Anal.**19**(1988), no. 2, 493–507. MR**930041**, DOI 10.1137/0519035 - William I. Weisberger,
*Normalization of the path integral measure and the coupling constants for bosonic strings*, Nuclear Phys. B**284**(1987), no. 1, 171–200. MR**879081**, DOI 10.1016/0550-3213(87)90032-0 - E. T. Whittaker and G. N. Watson,
*A course of modern analysis*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR**1424469**, DOI 10.1017/CBO9780511608759

*A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds*, preprint, 1983.

## 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