Let $(\mathcal{M},g)$ be a Riemannian manifold. In Normal coordinates around a point $p$, we have the equality: $$ R(p) = -\frac{3}{2} \langle g , \Delta_\text{LB}g \rangle(p) \tag{1} $$ Where $R$ is the Ricci scalar, $\langle , \rangle$ is the tensor contraction and, $\Delta_\text{LB} = d\circ \delta+\delta \circ d$ is the Laplace-Beltrami operator. Now take a boundary $(\partial \mathcal{M}, \gamma)$, with $\gamma$ the first fundamental form on $\mathcal{M}$. We have for the trace $K$ of the second fundamental form: $$ K(p) = \frac{1}{2} \langle \Gamma , \mathcal{L}_\textbf{n} \Gamma \rangle(p) \tag{2} $$ With $\mathcal{L}_\textbf{n} = d\circ \iota_\textbf{n}+\iota_\textbf{n}\circ d$ the Lie derivative along the unit normal $n$ to the boundary, and $\Gamma$ is just $\gamma$ but with the same number of components as $g$ (so $\Gamma = g-n\otimes n$).
Now, let $\mathbb{B}_p^\epsilon$ be a closed subset of $\mathcal{M}$ with infinitesimal measure $\mu_g(\mathbb{B}_p^\epsilon)=\epsilon^d$ ($d$ is the dimension of $\mathcal{M}$), and whose boundary $\partial \mathbb{B}_p^\epsilon$ contains $p$. I am searching for a theorem that would link $(1)$ and $(2)$ in the following form: $$ \int_{\mathbb{B}_p^\epsilon}{}^{(n)}\star \langle g , \Delta_\text{LB}g \rangle \propto \int_{\partial \mathbb{B}_p^\epsilon}{}^{(n-1)}\star\langle \Gamma , \mathcal{L}_\textbf{n} \Gamma \rangle \tag{3} $$ For ${}^{(i)}\star$ the Hodge dual in $i$ dimensions. In fact, I have done calculations that lead to the conclusion of this proportionality, and a couple of simulations in Python that tend to show $(3)$ (but not enough to be sure it is no coincidence). So is there any known theorem that states the proportionality $(3)$?