WHEN THE LEADING TERMS IN THE HEAT-EQUATION ASYMPTOTICS ARE COERCIVE

Let M be a compact Riemannian manifold without boundary of dimension m greater than or equal to 2. Let a(n)(D) be the asymptotics of the heat equation for n greater than or equal to 3. If D is the p form valued Laplacian for 0 less than or equal to p less than or equal to m, the a(n) lead to coercive estimates for the highest order jets of the covariant derivatives of the curvature tenser for any n. If D is the conformal Laplacian, the a(n) lead to coercive estimates if and only if 2n is not an element of {m - 2, m - 1, m}. If D is the spinor Laplacian, the a(n) lead to coercive estimates if and only if 2n > m.