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In this paper we investigate the small time heat kernel asymptotics on thecut locus on a class of surfaces of revolution, which are the simplest2-dimensional Riemannian manifolds different from the sphere with non trivialcut-conjugate locus. We determine the degeneracy of the exponential map near acut-conjugate point and present the consequences of this result to the smalltime heat kernel asymptotics at this point. These results give a first examplewhere the minimal degeneration of the asymptotic expansion at the cut locus isattained.

Small time heat kernel asymptotics at the cut locus on surfaces of revolution