Stationary liquid drops in Lorentz-Minkowski space

Abstract

This paper analyzes the configurations of shapes that shows a spacelike liquid drop in Minkowski space deposited over a spacelike plane $\Pi$. We assume the presence of a uniform gravity field directed toward $\Pi$ and that the volume of the drop is prescribed. Our interest are the liquid drops that are critical points of the energy of the corresponding mechanical system and we will say then that the liquid drop is stationary. In such case, the liquid-air interface is determined by the condition that the mean curvature is a linear function of distance from $\Pi$ and that the drop makes a constant hyperbolic angle of contact with the plate $\Pi$. As first result, we shall prove that the liquid drop must be rotational symmetric with respect to an axis orthogonal to $\Pi$. Then we prove the existence and uniqueness of symmetric solutions for a given angle of contact with $\Pi$. Finally, we shall study the shapes that a liquid drop can adopt in terms of its size. So, we shall derive estimates of its height, volume and area of the wetted surface on $\Pi$.