fundamental solution of the heat equation on closed manifold - gradient convergence for $t\to 0$

Question

Let $M$ be a compact Riemannian manifold without boundary and $H$ its heat kernel. Given a smooth function $u\colon M\to \mathbb{R}$ let us consider

$v(t,x):=\int_{M} H(x,y,t)v_0(y) dy$

where $dy$ is short for integration on the Riemannian manifold $M$.

$v$ is a solution of $\partial_tv-\Delta v=0$.

Moreover $v(t,.)\to v_0$ in $C^0(M)$ as $t\to 0^+$, i.e. $v(t,.)\to v_0$ uniformly as $t\to 0^+$

Question: Does it hold that $\nabla v(t,.)\to \nabla v_0$ uniformly as $ t\to 0^+$? ($\nabla$ is the gradient with resprect to the Riemannian metric.)

Thanks for any answers.