# Lower Gaussian estimates for Dirichlet heat kernel on manifolds

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

https://projecteuclid.org/journals/journal-of-differential-geometry/volume-36/issue-2/Uniformly-elliptic-operators-on-Riemannian-manifolds/10.4310/jdg/1214448748.full

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 $$y,y'\in B(x,\delta r)$$, $$t' we have $$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.

• Aren't $y$ and $y'$ assumed to be in a smaller ball $B(x, \delta r)$ rather than in $\Omega = B(x, \delta)$? Mar 5, 2021 at 20:37
• @MateuszKwaśnicki I think the $\delta$ can be close to 1, so really no much restriction is put on $y,y'$? Or I misunderstood. Mar 5, 2021 at 20:38
• The constants depend on $\delta$, do they not? Mar 5, 2021 at 20:57