The symmetric minimal surface equation

Abstract

and, geometrically, A (u) represents the area functional for S(u); that is, A (u) is the (n+m−1)-dimensional Hausdorff measure H n+m−1(S(u)). This is clear because the integrand √ 1+|Du|2 um−1 for A (u) is the Jacobian of the map (x,ω) ∈Ω×Sm−1 7→ (x, u(x)ω)∈Ω×R, and this map is a local coordinate representation for the symmetric graph S(u). Since 1.1 1.1 expresses the fact that u is stationary with respect to A , we see that S(u) is stationary with respect to smooth symmetric deformations, and hence stationary with respect to all deformations by a well-known principle (see e.g. [Law72]). (The latter principle here is just the natural generalization of the fact that if a smooth hypersurface Σ is rotationally symmetric about an axis and if Σ is stationary with respect to smooth rotationally symmetric compactly supported perturbations, then Σ is minimal—i.e. stationary with respect to all smooth compactly supported perturbations whether symmetric or not.) Thus the smooth submanifold S(u) is stationary as a multiplicity 1 varifold in Ω× (R \ {0}) and hence is a smooth minimal submanifold of Ω× (R \{0}) as claimed.