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(tau A)) is locally minimized at g. for any given t > 0. The local condition is expressed in terms of an L(x) neighborhood of the set of conformal factors of g of the form au', with tau a Mobius 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. (C) 1996 Academic Press, Inc. [References: 22]
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