Local Extrema of Traces of Heat Kernels onS2

OnS2we consider metrics conformal to the standard round metricgand of area 4π. We show that among such metrics the trace of the heat kernelTr(etΔ) is locally minimized atg, for any givent>0. The local condition is expressed in terms of anL∞neighborhood of the set of conformal factors ofgof the form |τ′|, withτa 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.