On the heat operators of normal singular algebraic surfaces (Q1118882)

Let X be a normal singular algebraic surface over C embedded in the projective space \(P^ N({\mathbb{C}})\). The singularity set of X is a finite set of isolated points. Restricting the Fubini-Study metric of \(P^ N(C)\) to \(\chi =X-S\), one obtains an incomplete Riemannian manifold (\(\chi\),g). The author considers the Laplace operator \(\Delta =\delta d\) acting on square integrable functions on (\(\chi\),g) and proves the following main theorem. (1) The Laplace operator is selfadjoint. (2) The heat operator \(e^{-t\Delta}\) is of trace class, and there exists a constant \(K>0\) such that Tr \(e^{-t\Delta}\leq Kt^{-2}\), \(0<t\leq t_ 0\). The proof goes as follows: Let dis(x) be the distance from \(x\in X\) to the singularity set. For sufficiently small \(\epsilon >0\), let \(X_{\epsilon}=\{x\in X:\quad dis(x)\geq \epsilon \},\) and let \(\Delta_{\epsilon}\) be the Laplace operator of Dirichlet type on \((X_{\epsilon},g|_{X_{\epsilon}})\). Then one has to show Tr \(e^{-t\Delta}\leq Kt^{-2}\), \(0<t\leq t_ 0\), \(0<\epsilon <\epsilon_ 0\). But this assertion holds for a whole quasi-isometry class or for no element of the class. For this the author decomposes the space into less- complicated parts, nonoverlapping except on the boundaries, and proves the assertion after passing from these parts to simpler quasi-isometric metrics. Describing the induced metric on the parts, the author uses results of Hsing and Pati.