A capillary surface is a two dimensional surface in \({\mathbb{R}}^ 3\), whose mean curvature H(x), \(x\in {\mathbb{R}}^ 3\), \(x=(x_ 1,x_ 2,x_ 3)\), is a given function and which meets the prescribed boundary walls of the container C in a prescribed angle \(\gamma\) ; a volume constraint is also included. For more details and the presentations of the more recent results on capillary surfaces see by the same author: Equilibrium capillary surfaces (1986; Zbl 0583.35002). In this paper the author considers the physical surfaces in the earth's gravitational field, such that \(H(x)=kz+\lambda,\quad k>0,\quad z=x_ 3\) and in the absence of gravity \((k=0)\). He studies the property of uniqueness of such surfaces for prescribed volume and contact angle \(\gamma\). In particular he can prove that for any gravity field g (including \(g=0)\) and any angle \(\gamma\) a container C can be found which can be partially filled with liquid in a continuum of distinct ways in order to have a parameter family \({\mathcal F}\) of capillary surfaces, not mutually congruents, all with the same enclosed volume and all with the same energy (according to the principle of virtual works). The case \(g=0\), \(\gamma =\pi /2\) was considered by \textit{R. Gulliver} and \textit{S. Hildebrandt} [Manuscr. Math. 54, 323-347 (1986; Zbl 0589.53008)]. The author can also prove that for a particular configuration in \({\mathcal F}\), the second variation of energy can be made negative. As a consequence, he can construct an example of a rotationally symmetric container C that differs only locally and as little as desired from a circular cylinder, so that an energy minimizing configuration filling half the container exists but cannot be symmetric. In the case \(g=0\) and C a sphere, the author proves that the capillary surface of the type of the circular disk is uniquely determined by the volume enclosed and the contact angle \(\gamma\), using a result of \textit{J. C. C. Nitsche} [Arch. Ration. Mech. Anal. 89, 1-19 (1985; Zbl 0572.52005)]