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As is well known $\sum e^{-\lambda_{k}t}\approx(4\pi t)^{dim(M)/2}\sum a_{j}t^{j}$, where $a_{j}$ are geometric properties of manifold M.

Moreover, the arbitrary order coefficients don't have closed form (though some progress in "Combinatorics of heat kernel coefficients").

However, say $\widetilde{g}=e^{f}g$ where g=Euc. or Sph. metric, then the curvature terms are simpler in dimension 2. Are the $a_{j}$ coefficients known explicitly for such surfaces?

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    $\begingroup$ This family should include hyperbolic surfaces, so I doubt it. Some results on the general question are in McKean-Singer 1967 (e.g., Euler characteristic is audible). They indicate the computations for higher-order coefficients are very difficult. I believe there is also a study of this made in Buser's book Geometry and Spectra of Compact Riemann Surfaces, though I do not have it handy to check. Another reference is Chavel's book Eigenvalues in Riemannian Geometry although I do not recall explicit computations for conformal metrics. $\endgroup$
    – Neal
    Commented Feb 27, 2017 at 2:52
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    $\begingroup$ I don't know how much is the following fact related to your question or interest; but the conformal variations of the zeta regularized determinant (which is defined precisely using the (analytic continuation of the) series $\sum e^{-\lambda t}$) are expressed by the Polyakov formula. $\endgroup$
    – Giovanni De Gaetano
    Commented Feb 27, 2017 at 14:59

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