Spectral invariants of affine hypersurfaces (Q2774638)

Assume that \((M,g)\) is a compact \(m\)-dimensional Riemannian manifold with boundary \(\partial M\) (possible empty). For an affine connection \(\nabla\), \(D=D(g,\nabla)\) is the trace of the normalized Hessian. Although the Laplace type operator \(D\) need not be self-adjoint in general, it satisfies an important transformation rule with respect to conformal change of the metric \(g\). It is considered the heat equation \((\partial_t+D)u(x;t)=0\), \(u(x;0)=\phi(x)\), and \(u(y;t)=0\) for \(y\in\partial M\), and the asymptotic expansion \(\operatorname{tr}_{L^2}(e^{-tD})\sim\sum_{n\geq 0}a_n(D)t^{(n-m)/2}\). The coefficients \(a_n(D)\) are locally computable invariants. The main topic is to study the characterizations of some geometric properties in the terms of the properties of the operator \(D\) and the coefficients \(a_n(D)\). The work is based on the result of \textit{T. P. Branson} and \textit{P. B. Gilkey} [Commun. Partial Differ. Equ. 15, 245-272 (1990; Zbl 0721.58052)], where the coefficients \(a_0(D),\ldots,a_4(D)\) are explicitly computed. In Theorem 2.5 these coefficients are determined in the setting of relative normalization \(\{x,X,y\}\) of a regular embedding \(x\) for a positive definite metric \(g\). A very interesting characterization is obtained for the Euclidean normalization and a compact centroaffine hypersurface. It is also obtained a characterization of equiaffine normalization if \(x\) is a compact, locally strongly convex Blaschke hypesurface with boundary. In Section 5, for a Blaschke hyperovaloid \(x\) hyperellipsoids are characterized by the condition that \({}^1\bar{D}\) is self-adjoint.