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We study the stability of capillary surfaces without gravity for anisotropic free surface energies. For a large class of rotationally symmetric energy functionals, it is shown that the only stable equilibria supported on parallel planes are either cylinders or a part of the Wulff shape.

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We study the stability of capillary surfaces for anisotropic energies having boundaries supported in horizontal planes. A wetting energy term for the surface to plane interface is included.

For a CMC immersion from a two-dimensional compact smooth manifold with boundary into the Euclidean three-space, we give sufficient conditions under which it has a CMC deformation fixing the boundary. Moreover, we give a criterion of the stability for CMC immersions. Both of these are achieved by using the properties of eigenvalues and eigenfunctions of an eigenvalue problem associated to the second variation of the area functional. In a certain special case, by combining these results, we obtain a 'visible' way of judging the stability. 1.Introduction. An immersion X : M → R 3 of a two-dimensional orientable compact connected C ∞ manifold M with boundary ∂M into the Euclidean three-space R 3 has constant mean curvature if and only if X is a critical point of the area functional for all volumepreserving variations of X that fix the boundary (cf. Barbosa-do Carmo [1, Proposition 2.7]). When the mean curvature of X is constant (we will say that X is a CMC immersion), X is said to be stable if the second variation of the area functional is nonnegative for all such variations of X as above.The objective of this paper includes two themes. One of them is on the possibility of (not necessarily isometric) CMC deformation of CMC immersions that fix the boundary (Theorems 1.1 and 1.2). The other is on the determination of the stability or the unstability of CMC immersions (Theorem 1.3, Corollary 1.1). These two themes are related to each other in the following sense. First, both of these are achieved by using the properties of eigenvalues and eigenfunctions of an eigenvalue problem associated to the second variation of the area functional (Theorems 1.1, 1.2, and 1.3). Second, in the most characteristic case as a solution of a certain variational problem with constraint, a criterion of the stability of a CMC immersion is represented by the property of its CMC deformation fixing the boundary (Corollary 1.1).Given a CMC immersion X : M → R 3 , consider a volume-preserving variation X t of X that fixes the boundary. Denote by A(t) the area of X t . Then

We show that for elliptic parametric functionals whose Wulff shape is smooth and has strictly positive curvature, any surface with constant anisotropic mean curvature which is a topological sphere is a rescaling of the Wulff shape.

Abstract. Let Σ be a compact immersed stable capillary hypersurface in a wedge bounded by two hyperplanes in R n+1 . Suppose that Σ meets those two hyperplanes in constant contact angles and is disjoint from the edge of the wedge. It is proved that if ∂Σ is embedded for n = 2, or if ∂Σ is convex for n ≥ 3, then Σ is part of the sphere. And the same is true for Σ in the half-space of R n+1 with connected boundary ∂Σ.

Abstract. In this paper, we shall prove that space-like surfaces with bounded mean curvature functions in real analytic Lorentzian 3-manifolds can change their causality to time-like surfaces only if the mean curvature functions tend to zero. Moreover, we shall show the existence of such surfaces with nonvanishing mean curvature and investigate their properties.