## Functional determinants and geometry.(English)Zbl 0602.58044

We study the zeta-function determinant in the context of elliptic boundary value problems. Our main technique is to relate the determinant of an operator, or a ratio of determinants, to the boundary values of the solutions of the operator. This has the advantage of restricting attention to the solutions of the operator, which do not depend on the boundary conditions and can often be written down explicitly, rather than the eigenvalues, which are usually difficult to work with. In addition, the problem is reduced to a calculation over the boundary of the manifold which is a closed manifold of dimension one less than the original manifold. This has special significance in the case that the manifold is a finite interval. In this case the boundary is a pair of points and the determinant of an ordinary differential operator is expressed in terms of the determinant of a finite matrix.
The results are then applied to some geometric operators. In section 4 we study the Jacobi operator acting along a geodesic segment and the covariant derivative operator acting along a loop. In section 2 we calculate the determinant of the Laplacian acting on sections of a flat bundle over a flat torus. This can be related to an Eisenstein series and thus we have presented a new geometric method of summing such series. This sum is known as Kronecker’s second limit formula. We then consider operators on a product manifold $$M\times S^ 1$$.

### MSC:

 58J32 Boundary value problems on manifolds 35J25 Boundary value problems for second-order elliptic equations
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### References:

 [1] [Ag] Agmon, S.: On the eigenfunctions and eigenvalues of general elliptic boundary value problems. Commun. Pure Appl. Math.15, 119-147 (1962) · Zbl 0109.32701 [2] [At] Atiyah, M.F.: Circular symmetry and stationary phase approximation. Proceedings of the Conference in Honor of L. Schwartz. Astérisque131, 43-60 (1985) · Zbl 0578.58039 [3] [A-P-S] Atiyah, M.F., Patodi, V., Singer, I.M.: Spectral asymmetry and Riemannian geometry I. Math. Proc. Camb. Philos. Soc.77, 43-69 (1975) · Zbl 0297.58008 [4] [Bo] Bott, R.: On the iteration of closed geodesics and the Sturm intersection theory. Commun. Pure Appl. Math.9, 171-206 (1956) · Zbl 0074.17202 [5] [Ca] Calderon, A.P.: Boundary value problems for elliptic operators. Outline of the Joint Soviet-American Symposia on Partial Differential Equations, Novosibrush, August 1963 [6] [C-E] Cheeger, J., Ebin, D.G.: Comparison theorems in Riemannian geometry. New York: North Holland 1975 · Zbl 0309.53035 [7] [D-D] Dreyfuss, T., Dym, H.: Product formulas for the eigenvalues of a class of boundary value problems. Duke Math. J.45, 15-37 (1978) · Zbl 0387.34021 [8] [El] Elie, L.: Équivalent de la densité d’une diffusion en temps petit. Cas des points proches. Géodésiques et Diffusion en Temps Petit. Astérique84-85, 55-71 (1981) · Zbl 0507.60072 [9] [Fo] Forman, R.: Functional determinants and applications to geometry. Doctoral Thesis, Harvard University 1985 [10] [G-K] Gohberg, I.C., Krein, M.G.: Introduction to the theory of linear nonselfadjoint operators, Vol. 18 (Translation of Mathematical Monographs). Providence, Rhode Island, American Mathematical Society 1969 · Zbl 0181.13504 [11] [Hö] Hörmander, L.: Pseudo-differential operators and non-elliptic boundary problems. Ann. Math.83, 129-209 (1966) · Zbl 0132.07402 [12] [L-S] Levit, S., Smilansky, U.: A theorem on infinite products of eigenvalues of Sturm-Liouville type operators. Proc. Am. Math. Soc.65, 299-302 (1977) · Zbl 0374.34016 [13] [Pa] Palais, R.: Seminar on the Atiyah-Singer index theorem. Princeton, Princeton University Press 1965 · Zbl 0137.17002 [14] [R-S1] Ray, D.B., Singer, I.M.:R-Torsion and the Laplacian on Riemannian manifolds. Adv. Math.7, 145-210 (1971) · Zbl 0239.58014 [15] [R-S2] Ray, D.B., Singer, I.M.: Analytic torsion for complex manifolds. Ann. Math.98, 154-177 (1973) · Zbl 0267.32014 [16] [Se1] Seeley, R.: Singular integrals and boundary value problems. Am. J. Math.88, 154-177 (1966) · Zbl 0178.17601 [17] [Se2] Seeley, R.: Complex powers of an elliptic operator. Singular Integrals (Proceedings of the Symposia in Pure Mathematics, Chicago, Illinois 1966), Providence, Am. Math. Soc. pp. 288-307 [18] [Se3] Seeley, R.: Analytic extension of the trace associated with an elliptic boundary problems. Am. J. Math.91, 963-983 (1969) · Zbl 0191.11901
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