Equilibrium of Surfaces in a Vertical Force Field

Abstract

In this paper, we study $$\varphi $$ φ -minimal surfaces in $$\mathbb {R}^3$$ R 3 when the function $$\varphi $$ φ is invariant under a two-parametric group of translations. Particularly those which are complete graphs over domains in $$\mathbb {R}^2$$ R 2 . We describe a full classification of complete flat-embedded $$\varphi $$ φ -minimal surfaces if $$\varphi $$ φ is strictly monotone and characterize rotational $$\varphi $$ φ -minimal surfaces by its behavior at infinity when $$\varphi $$ φ has a quadratic growth.

Geometry of $[φ,\vec{e}_{3}]$-minimal surfaces in $\mathbb{R}^{3}$

“…In this case, we can assert (see [34,Proposition 4.1]) that the problem (2.7)-(2.8) has a unique solution u ∈ C 2 ([0, R]) for some R > 0, which depends continuously on the initial data. Now, once the existence of solution is guaranteed, we want to describe [ϕ, e 3 ]minimal surfaces that are invariant under the one-parameter group of rotations that fix the e 3 direction.…”

“…The proof of Theorem 5.1 is based on the use of the Alexandrov reflection principle (see [1]) to prove that the surface is symmetric with respect to any vertical plane through the origin. Although this principle is applied in a standard way, it is crucial in the proof (see [34,Lemmas 6.3 and 6.4]) to show that it is possible to start the reflexion respect to any vertical plane far enough from the origin.…”

“…a heavy surface in a gravitational field that, according to the architect F. Otto [40, p. 290] are of importance for the construction of perfect domes). We refer to [4,11,12,14,28,29,30,31,34,39] for some progress in this family.…”

“…[34, Theorem 4.5]). Under the conditions (2.14), the curve γ is the graph of a strictly convex symmetric function u(x) defined on a maximal interval ] − ω + , ω + [ which has a minimum at 0 and lim x→±ω+ u(x) = +∞.…”

“…[34, Theorem 4.11]). For every x 0 > 0, there exists a complete embedded rotational [ϕ, e 3 ]-minimal surface, see Figure2.4 (right) with the annulus topology whose distance to axis of revolution is x 0 and whose generating curve γ is of winglike type seeFigure 2.4 (left).…”