Let $x:\Sigma^n\to\mathbb{R}^{n+1}$ be a CMC (constant mean curvature) immersion of an orientable compact smooth $n$-dimensional manifold $\Sigma$. Denote the Gauss map by $\nu$.
In Barbosa-do Carmo, a Jacobi field is defined to be a normal vector field $\varphi\nu$ where $\varphi:\Sigma\to\mathbb{R}$ is a smooth function satisfying the following:
- $\varphi\equiv 0$ on the boundary $\partial\Sigma$.
- $\displaystyle\int_{\Sigma}\varphi d\mu=0$, where $d\mu$ is the area element in the induced metric.
- $\varphi$ satisfies \begin{align} \Delta\varphi+|A|^2\varphi=0 \end{align} (where $\Delta$ is the Laplacian in the induced metric, and $A$ is the second fundamental form) which was called the Jacobi equation in the paper.
This looks quite different from the usual definition of Jacobi field (and Jacobi equation) that we've learnt, which is defined using geodesic. Thus I would like to ask if there is any relation between these two seemingly distinct notions of Jacobi field? Or it just happens to be a mere coincidence that both are named the same?
My gut feeling is that the former is probably the case, since the Jacobi equation in item 3 appears in the second variation formula of some functional (e.g. area functional) at a critical point. Thus it seems to be a higher-dimensional analogue of the usual Jacobi equation. But this is just a gut feeling. I'm not sure if this is true, or if there's any detail I've missed.
Any comment or answer are welcomed and greatly appreciated.
There are indeed related as you described. Whenever one has a functional $A$ defined on a space (of mappings) $\mathscr M$, then the second variations (at a critical point $P$) defines the Jacobi operator $\mathscr J$:
$$\mathscr J : T_p\mathscr M \to T_p\mathscr M, \ \ \ \langle \mathscr J X, X\rangle = \frac{d^2}{ds^2}\bigg|_{s=0} A(X_s).$$
and an element in the Kernel of $\mathscr J$ is called a Jacobi field. This terminology is at least used for geodesics, minimal surface, CMC surfaces and harmonic mappings.