Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On minimal Lagrangian surfaces in the product of Riemannian two
  manifolds   Let $(\Sigma_1,g_1)$ and $(\Sigma_2,g_2)$ be connected, complete and
orientable Riemannian two manifolds. Consider the two canonical K\"ahler
structures $(G^{\epsilon},J,\Omega^{\epsilon})$ on the product 4-manifold
$\Sigma_1\times\Sigma_2$ given by $ G^{\epsilon}=g_1\oplus \epsilon g_2$,
$\epsilon=\pm 1$ and $J$ is the canonical product complex structure. Thus for
$\epsilon=1$ the K\"ahler metric $G^+$ is Riemannian while for $\epsilon=-1$,
$G^-$ is of neutral signature. We show that the metric $G^{\epsilon}$ is
locally conformally flat iff the Gauss curvatures $\kappa(g_1)$ and
$\kappa(g_2)$ are both constants satisfying $\kappa(g_1)=-\epsilon\kappa(g_2)$.
We also give conditions on the Gauss curvatures for which every
$G^{\epsilon}$-minimal Lagrangian surface is the product
$\gamma_1\times\gamma_2\subset\Sigma_1\times\Sigma_2$, where $\gamma_1$ and
$\gamma_2$ are geodesics of $(\Sigma_1,g_1)$ and $(\Sigma_2,g_2)$,
respectively. Finally, we explore the Hamiltonian stability of projected rank
one Hamiltonian $G^{\epsilon}$-minimal surfaces.

29
The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
26
The categories of this paper are: bayes-an
14
The categories of this paper are: math-ph