Thomas P. Branson

The asymptotics of the Laplacian on a manifold with boundary

Let P be a second-order differential operator with leading symbol given by the tensor on a compact Riemannian manifold with boundary. We compute the asymptotics of the heat equation for Dirichlet, Neumann, and mixed boundary conditions.

Sharp inequalities, the functional determinant, and the complementary series

Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequalities, are combined in a study of functional determinants of natural differential operators. The setting is that of compact Riemannian manifolds. We concentrate especially on the conformally flat case, and obtain formulas in dimensions 2, 4, and 6 for the functional determinants of operators which are well behaved under conformai change of metric. The two-dimensional formulas are due to Polyakov, and the four-dimensional formulas to Branson and Orsted; the method is sufficiently streamlined here that we are able to present the sixdimensional case for the first time. In particular, we solve the extremal problems for the functional determinants of the conformai Laplacian and of the square of the Dirac operator on S2 , and in the standard conformai classes on S4 and S6 . The S2 results are due to Onofri, and the S4 results to Branson, Chang, and Yang; the S6 results are presented for the first time here. Recent results of Graham, Jenne, Mason, and Sparling on conformally covariant differential operators, and of Beckner on sharp Sobolev and Moser-Trudinger type inequalities, are used in an essential way, as are a computation of the spectra of intertwining operators for the complementary series of SOo(m + 1, 1), and the precise dependence of all computations on the dimension. In the process of solving the extremal problem on S6 , we are forced to derive a new and delicate conformally covariant sharp inequality, essentially a covariant form of the Sobolev embedding L2(S6) «-» L3(S6) for section spaces of trace free symmetric two-tensors. 0. INTRODUCTION Some very recent work in analysis and geometry has revealed strong new connections among fields which, while never completely separate, have at least been studied in very different ways. Part of the stimulus for this has been physical string theory, which led in the last decade to a fresh look at Riemann surfaces, always a meeting ground for different disciplines in analysis. In this paper, we would like to clarify some of these connections as they have manifested themselves in the study of string theoretic principles in higher ( > 2 ) dimensions. Broadly speaking, the fields in question are: (I) the spectral theory of differential operators; (II) conformai geometry; and Received by the editors June 30, 1992 and, in revised form, July 16, 1994. 1991 Mathematics Subject Classification. Primary 58G25; Secondary 22E46, 26D10. Partially supported by the Danish Research Council. ©1995 American Mathematical Society License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Explicit functional determinants in four dimensions

4 2 2 ABSTRACT. Working on the four-sphere S , a flat four-torus, S x S2, or a compact hyperbolic space, with a metric which is an arbitrary positive function times the standard one, we give explicit formulas for the functional determinants of the conformal Laplacian (Yamabe operator) and the square of the Dirac operator, and discuss qualitative features of the resulting variational problems. Our analysis actually applies in the conformal class of any Riemannian, locally symmetric, Einstein metric on a compact 4-manifold; and to any geometric differential operator which has positive definite leading symbol, and is a positive integral power of a conformally covariant operator.

Estimates and extremals for zeta function determinants on four-manifolds

AbstractLetA be a positive integral power of a natural, conformally covariant differential operator on tensor-spinors in a Riemannian manifold. Suppose thatA is formally self-adjoint and has positive definite leading symbol. For example,A could be the conformal Laplacian (Yamabe operator)L, or the square of the Dirac operator. Within the conformal class

PROLONGATIONS OF GEOMETRIC OVERDETERMINED SYSTEMS

\left\{ {g = e^{2w} g_0 |w \in C^\infty (M)} \right\}

Conformally Invariant Operators, Differential Forms, Cohomology and a Generalisation of Q-Curvature

of an Einstein, locally symmetric “background” metricgo on a compact four-manifoldM, we use an exponential Sobolev inequality of Adams to show that bounds on the functional determinant ofA and the volume ofg imply bounds on theW2,2 norm of the conformal factorw, provided that a certain conformally invariant geometric constantk=k(M, goA) is strictly less than 32π2. We show for the operatorsL and that indeedk < 32π2 except when (M, go) is the standard sphere or a hyperbolic space form. On the sphere, a centering argument allows us to obtain a bound of the same type, despite the fact thatk is exactly equal to 32π2 in this case. Finally, we use an inequality of Beckner to show that in the conformal class of the standard four-sphere, the determinant ofL or of is extremized exactly at the standard metric and its images under the conformal transformation groupO(5,1).

Conformal geometry and global invariants

Abstract We calculate the coefficient a 5 of the heat kernel asymptotics for an operator of Laplace type with mixed boundary conditions on a general compact manifold.

Vacuum expectation value asymptotics for second order differential operators on manifolds with boundary

We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on the dimension of the solution space.