For background, recall that the Jacobi equation (also known as the equation of geodesic deviation) determines the evolution of the Jacobi field, interpreted as a deviation vector between two "infinitesimally nearby" geodesics. Specifically, let $u^a$ be the tangent vector to a geodesic and let $\eta^a$ be a Jacobi field along it. Then the Jacobi equation says that $$ u^a\nabla_a (u^b \nabla_b \eta^c) + R_{abd}^{\phantom{abd}c} u^a u^d \eta^b = 0, $$ where $R_{abcd}$ is the Riemann tensor of the ambient space.

My question is the following: since a geodesic is just a special case of a minimal surface, is there some analogous equation for the deviation vector field between two "infinitesimally nearby" minimal (or more generally, extremal) surfaces? That is, let $\Sigma$ be a minimal surface (of any dimension and codimension) and let $\eta^a$ be a deviation vector field on $\Sigma$ to some infinitesimally nearby minimal surface (more rigorously: let $\Sigma_t$ be a one-parameter family of minimal surfaces with $\Sigma = \Sigma_0$; then define $\eta^a = (\partial_t)^a$). Is there an equation of the form $$ D^2 \eta^a + R_{bcd}^{\phantom{bcd}a} h^{bd} \eta^c = \mathrm{stuff}, $$ where $D^2$ is the Laplacian on $\Sigma$, $h^{ab}$ is the (inverse) induced metric on $\Sigma$, and the "stuff" terms are linear in $\eta^a$ and can contain things like the curvatures (intrinsic and extrinsic) of $\Sigma$?

I've only been able to find partial answers to this question. I think the equation I want may be related to the so-called Jacobi or stability operator of a minimal surface, related to the formula for the second variation of the area (as mentioned here), but as far as I can tell that Jacobi operator is defined to act on functions (not vectors, as I want), and most references I see refer to embedded surfaces in Riemannian 3-manifolds (i.e. they don't seem to refer to completely general dimensions). Moreover, the sources I've seen derive integral equations for the second derivative of the area under some perturbation, whereas I want a differential equation for the Jacobi field.

I have also found this reference (specifically section 5.1) which does seem to do things in general dimensions, but runs into the same problem that it only obtains an integrated expression.

(As you may be able to tell, I was trained in general relativity so I'm used to a different sort of notation than is usual in the math literature; it may be that I found what I wanted but couldn't translate it into familiar language!)


In case anyone should find this question and be curious about its answer, a collaborator and I have given a review of the derivation I was asking for in section 2 of our paper: https://arxiv.org/abs/1904.08423. In that review section, we tried to cite all the previous works we could find that addressed this issue, but see also https://arxiv.org/abs/1904.08012 for another review that came out at the same time as ours.


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