Given the heat equation $Lu=0$, a function $H(x,y,t)\colon M \times M \times (0,\infty)\to \Bbb R$ is called the fundamental solution to the heat equation in $M$ if it satisfies the following conditions:



$\bullet$ It is $C^2$ in $x\in M$ and $C^1$ in $t$;

$\bullet$ For any bounded continuous function $f\colon M\to \Bbb R$, $$\displaystyle u(x,t):=\int_M H(x,y,t)f(y)\ \mathrm dV_g(y)$$ satisfies $Lu=0$ and $$\displaystyle\lim_{t\to 0}\int_M H(x,y,t)f(y)\ \mathrm dV_g(y)=f(x)$$ holds.



We will discuss Heat kernel estimates also and show that data is important to resolve certain issues about solution of Heat equation.