= Normal-coordinate divergence-form heat parametrix
{title2=$B_k(x)=-\int_0^1s^{k-1}(LB_{k-1})(sx)\,ds$}
For $L=-\partial_j(g^{ij}\partial_i)$ in <geodesic normal coordinates>, the <Gauss lemma> gives $g^{ij}x_i=x_j$. Substituting $F=(4\pi t)^{-n/2}e^{-|x|^2/(4t)}\sum_{k\geq0}t^kB_k$ into $(\partial_t+L)F=0$ therefore yields $(x\cdot\nabla+k)B_k=-LB_{k-1}$, with $B_0=1$. Integration along radial rays gives the displayed formula. Every coefficient is smooth at the centre. Any smooth solution of $(x\cdot\nabla+k)h=0$ is zero for $k>0$, since $s^kh(sx)$ is constant and tends to zero at $s=0$. This proves uniqueness. In this special divergence-form case $L1=0$, so every $B_k$ for $k>0$ is zero: the Gaussian itself solves the equation exactly on the coordinate ball. The <positive Laplace-Beltrami operator> instead has a metric volume-density term and a nonconstant leading amplitude.
Back to article page
