Let $(M,g)$ be a Riemannian manifold, and let $F_s:\Sigma\to M$ be a smooth one-parameter family of immersions whose images are two-sided minimal hypersurfaces for $s$ near $0$, with $F_0(\Sigma)=\Sigma$. Choose the unit normals smoothly near $s=0$, and write $\nu=\nu_0$. Let $A$ be the second fundamental form of $F_0$ with respect to $\nu$, and define the Jacobi operator $L:C^\infty(\Sigma)\to C^\infty(\Sigma)$ by

\begin{align*} L\psi=\Delta_\Sigma\psi+\left(|A|^2+\operatorname{Ric}_M(\nu,\nu)\right)\psi. \end{align*}

If the variational vector field at $s=0$ has smooth normal part $\phi\nu$, then $\phi\in C^\infty(\Sigma)$ satisfies

\begin{align*} L\phi=0. \end{align*}