PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.)
Vol. 64(78), pp. 133--145 (1998)
SPECTRAL INVARIANTS OF AFFINE HYPERSURFACES
Neda Bokan, Peter Gilkey, Udo SimonMatemati\v cki fakultet, Beograd, Yugoslavia and Mathematics Department, University of Oregon, Eugene, Oregon 97403, USA and Technische Universitat Berlin, Strasse des 17. Juni 135 D-10623 Berlin, Deutschland
Abstract: Let $M$ be a smooth compact manifold of dimension $m$ with smooth, possibly empty, boundary $\partial M$. If $g$ is a Riemannian metric on $M$ and if $\nabla$ is an affine connection, let $D=D(g,\nabla)$ be the trace of the normalized Hessian; if $\partial M$ is empty, then we impose Dirichlet boundary conditions. The structures $(g,\nabla)$ arise naturally in the context of affine differential geometry and we give geometric conditions which ensure that $D$ is formally self-adjoint in this setting. We study the asymptotics of the heat equation trace; we have that $a_m(D)$ is an affine invariant. We use the asymptotics of the heat equation to study the affine geometry of affine hypersurfaces.
Keywords: Operators of Laplace type, the Hessian, hypersurface immersed in affine space, the heat equation
Classification (MSC2000): 53A15, 58G25
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