Local extrema of traces of heat kernels on \(S^2\) (Q1816713)

The author's summary: ``On \(S^2\) we consider metrics conformal to the standard round metric \(g\) and of area \(4\pi\). We show that among such metrics the trace of the heat kernel Tr\((e^{t\Delta})\) is locally minimized at \(g\), for any given \(t>0\). The local condition is expressed in terms of an \(L^\infty\) neighborhood of the set of conformal factors of \(g\) of the form \(|\tau' |\), with \(\tau\) a Möbius transformation. To prove this result we use a power series expansion of the trace in terms of its conformal variations. We derive a combinatorial formula for the second variation and prove that it is positive definite except along the first eigenspace, where it vanishes. We estimate the higher variations and use a center of mass argument to complete the proof''.