Let $(M,g)$ be a Riemannian $n$-manifold with $Ric_g\ge -Kg$, $\Omega\subset M$ be an open subset. We can define Dirichlet heat kernel on $\Omega$, $p_{\Omega}(y,t,y',t')$ as the minimal fundamental solution to heat equation with Laplace-Beltrami operator on $\Omega\times(t',+\infty)$. The following paper

gives a 2-side Gaussian bound, especially a lower Gaussian bound in Theorem 6.1 in terms of Riemannian distance. I will copy down the lower bound part, for simplicity I only state this theorem for Laplace-Beltrami operator, the author did it in much more general context. In that paper's notation $\beta=0, \alpha=1,\mu=1$ for Laplace-Beltrami operator.

$\bf Theorem$ Let $0<\delta<1$ be fixed and suppose $B(x,r)\subset \Omega$. Then for all $y,y'\in B(x,\delta r)$, $t'<t<t'+r^2$ we have $$e^{-C_1(1+K\tau)}V(y,\sqrt{t})^{-1/2}V(y',\sqrt{t})^{-1/2}\exp(-C_2\rho^2(y,y')/t)\le p_{\Omega}(y,t,y',t')$$ where $\tau=t-t'$, $\rho$ is the Riemannian distance on $M$,and $V(x,r)$ is the volume of $B(x,r)$ the constants depend on dimension $n$, $\delta$, possibly $K$, but (seem to) do not depend on $\Omega$.

The author claims that this is a consequence of the Harnack inequality, i.e. Theorem 5.3 in the same paper. I did not understand the proof and I am a bit surprised by the fact that this lower Gaussian estimate is in terms of the Riemannian distance but not the intrinsic distance on $\Omega$. In general the intrinsic distance can be quite different from the Riemmanian distance on $M$.

For example take $M=S^1$ with large radius, and $\Omega=S^1\setminus[0,\varepsilon]$ be a major arc that is almost the full circle(when $\varepsilon$ is small). Then we can take $B=\Omega$ since arcs are geodesic balls in $S^1$, the heat kernel on $\Omega$ behaves like a Gaussian with intrinsic distance on the arc. If we take $y,y'\in \Omega$ to be close to the 2 endpoints, then $d_{S^1}(y,y')$ is close to $\epsilon$ but $d_\Omega(y,y')$ can be arbitrarily large, it is hard to imagine that the heat kernel on $\Omega$ can be bounded below by the Gaussian with distance on $S^1$.

So could anybody tell me why and how this kind of Gaussian bound is possible? Any help is appreciated.