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In the Laplacian comparision theorem by Yau and Schoen, the proof assume a $n$ dimensional space form $N$ with sectional curvature $-k^{2}$. And we have a separate $M$ with $Ric<-(n-1)k^{2}$. But afte …

Since the question has been "hanging on" for a while, I think it makes sense to give an outline of Atiyah's argument in the paper. Note that the paper is short (page 1 introduction, page 5-6 reference …

As a graduate student, I found the old books of differential geometry used a different set of notation from modern textbooks. For example, Chern and Milnor defined the curvature 2-form by $d\omega-\om …

This turned out to be a highly non-trivial topic. For a detailed analysis, see Jorgenson and Kramer's paper Bounds on canonical Green functions. Their result can be summarized as $$ g_{\textrm{can},X …

I do not really follow the question. Since everything is local, the map $u$ can be effectively replaced by its derivative $D u$, which is a matrix of dimension $\dim(N)\times \dim(M)$. The map may be …

While reviewing the proof of Gauss-Bonnet in John Lee's book, I noticed the following paragraph: " ...In a certain sense, this might be considered a very satisfactory generalization of Gauss-Bonnet. …

The Ray-Singer paper "R-torsion and the Laplacian on Riemannian manifolds" claimed that one may prove the metric invariance of analytical torsion by forming a homotopy between metric $\rho_{0},\rho_{1 …

I take a very brief look at the paper and I did not see $\Psi DO$ on manifold with boundary being used heavily anywhere (no conormal distribution, multiple blow-ups, heavy handed symbol estimates, etc …

One possible reference is Seeley's paper on Complex powers of Elliptic Operators, where Seeley did it for the Laplacian (page 6). But the discussion carries over to all elliptic $\Psi DO$s without muc …

This type of questions has been investigated systematically by Melrose in the framework of 'c-calculus', where $c$ stands for the cusp. The basic idea, if I recall correctly is to blow up the boundary …

This is related to Li-Yau gradient estimates. I think the usual set up is for a complete manifold with $\textrm{Ric}(M)>-k$. You probably need some Harnark type inequalities for parabolic equations. I …

Let $M$ be a compact Riemann Surface. Let $P$ be a fixed point on $M$ and let $\delta_{P}$ be the Dirac point distribution at $M$. Consider the fundamental solution of the heat equation $$ (\partial_{ …

I do not know if this is really the end of the story. You may be interested in the following paper by Jay Jorgenson. Basically he extended the work by Ray-Singer by fixing all unknown invariants, bu …

I am not entirely familiar with $KK$-theory, so please correct me if there are mistakes. I think ultimately you are trying to show the topological $K$-theory class you get from taking the horizontal d …

I recall that Richard Wentworth's first paper on precise constants in bosonization formula (which is part of his PhD thesis?) extensively used computational methods from bosonic string theory. I am no …