Recently, I am reading Sturm's paper "Analysis on local Dirichlet form III: X is a locally compact separable Hausdorff space and m is a positive Radon measure with supp[m]=X.

$\varepsilon$ is a symmetric and strongly local Dirichlet form with domain F on a real Hilbert space $L^2(X,m)$. Any such form can be written as $$ \varepsilon(u,v)=\int_X d\Gamma(u,v) $$ where $\Gamma$ is a postive semidefinite, symmetric bilinear form on F with values in the signed Radon measures on X. Let $$ F_{loc}(X)=\{u \in L^2_{loc}(X,m):\Gamma(u,u)\mbox{ is a Radon measure}\} $$ The energy measure defines in an intrinsic way a pseudo metric $\rho$ on X by $$ \rho(x,y)=sup\{u(x)-u(y):u \in F_{loc}(X) \cap C(X),\Gamma(u,u) \leq \mbox{ m on X}\} $$ The Dirichlet form is called strongly regular if it's regular and if $\rho$ is a metric on X whose topology coincides with the original one.

Question 1: On which kind of spaces, $\rho$ is a metric on X whose topology coincides with the original one? Riemannian manifold (Alexandrov spaces) with Ricci curvature bounded below, $CD(K,N)$ and $CD(K, \infty)$ satisfy?

The paper says: If "completeness property", "doubling" and "Poincare inequality" hold globally on X, then there are both Gaussian lower and upper bounds for the heat kernel,i.e. $$ 1/C (\mu(B_{\sqrt(t)}(x))^{-1/2}(\mu(B_{\sqrt(t)}(y))^{-1/2}e^{-\frac{\rho(x,y)^2}{C_1t}} \leq p_t(x,y) $$ $$\leq C(\mu(B_{\sqrt(t)}(x))^{-1/2}(\mu(B_{\sqrt(t)}(y))^{-1/2}e^{-\frac{\rho(x,y)^2}{C_2t}} $$ But Yau's book "differential geometry" just give a lower bound for nonnegative curved Riemannian manifold, why?

Question 2: Does the heat kernel of Riemannian manifolds (possibly noncompact and with boundary) with $Ric \ge -(n-1)k(k \ge 0)$ have both Guassian upper and lower bounds?