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Starting with the most fundamental differential-geometric principles we derive here new explicit parameterizations of the Delaunay surfaces of revolution which depend on two real parameters with fixed ranges. Besides, we have proved that these parameters have very clear geometrical meanings. The first one is responsible for the size of the surface under consideration and the second one specifies its shape. Depending on the concrete ranges of these parameters any of the Delaunay surfaces which is neither a cylinder nor the plane is classified unambiguously either as a first or a second kind Delaunay surface. According to this classification spheres are Delaunay surfaces of first kind while the catenoids are Delaunay surfaces of second kind. Geometry of both classes is established meaning that the coefficients of their fundamental forms are found in explicit form. [ABSTRACT FROM AUTHOR]

Starting with the most fundamental differential-geometric principles we derive here new explicit parameterizations of the Delaunay surfaces of revolution which depend on two real parameters with fixed ranges. Besides, we have proved that these parameters have very clear geometrical meanings. The first one is responsible for the size of the surface under consideration and the second one specifies its shape. Depending on the concrete ranges of these parameters any of the Delaunay surfaces which is neither a cylinder nor the plane is classified unambiguously either as a first or a second kind Delaunay surface. According to this classification spheres are Delaunay surfaces of first kind while the catenoids are Delaunay surfaces of second kind. Geometry of both classes is established meaning that the coefficients of their fundamental forms are found in explicit form.

In the absence of gravity, a capillary surface within a circular cylinder will have constant mean curvature and will make a constant contact angle with the cylinder. Two types of surfaces with these properties are Delaunay surfaces and cylinders (i.e., the free surface of the liquid is another circular cylinder). Stability and energy minimality of these capillary surfaces are investigated. [ABSTRACT FROM AUTHOR]

A simple modification of the surface tension in the axisymmetric case leads to analogues of the Delaunay surfaces. Here we have derived an explicit parameterization of the most simple case of this new class of surfaces which can be considered as a generalization of the catenoids. The geometry of these surfaces depends on two real parameters and has been studied in some detail

We construct codimension 1 surfaces of any dimension that minimize a periodic nonlocal perimeter functional among surfaces that are periodic, cylindrically symmetric and decreasing. These surfaces may be seen as a nonlocal analogue of the classical Delaunay surfaces (onduloids). For small volume, most of their mass tends to be concentrated in a periodic array and the surfaces are close to a periodic array of balls (in fact, we give explicit quantitative bounds on these facts). [ABSTRACT FROM AUTHOR]

t ∈ N, where Mn is the Riemannian manifold Sn or Hn and n ≥ 2, bifurcating from the cylinder Bn×R (where Bn is a geodesic ball in Mn) for which the first eigenfunction of the Laplace-Beltrami operator with zero Dirichlet boundary condition also has constant Neumann data at the boundary. In other words, the overdetermined problem. [ABSTRACT FROM AUTHOR]

A family of embedded rotationally symmetric tori in the Euclidean 3-space consisting of two opposite signed constant mean curvature surfaces that converge as varifolds to a double round sphere is constructed. Using complete elliptic integrals, it is shown that their Willmore energy lies strictly below 8 π. Combining such a strict inequality with previous works by Keller–Mondino–Rivière and Mondino–Scharrer allows to conclude that for every isoperimetric ratio there exists a smoothly embedded torus minimising the Willmore functional under isoperimetric constraint, thus completing the solution of the isoperimetric-constrained Willmore problem for tori. Similarly, we deduce the existence of smoothly embedded tori minimising the Helfrich functional with small spontaneous curvature. Moreover, it is shown that the tori degenerate in the moduli space which gives an application also to the conformally-constrained Willmore problem. Finally, because of their symmetry, the Delaunay tori can be used to construct spheres of high isoperimetric ratio, leading to an alternative proof of the known result for the genus zero case. [ABSTRACT FROM AUTHOR]

We prove bifurcation results for (compact portions of) nodoids in ℝ3, whose boundary consists of two fixed coaxial circles of the same radius lying in parallel planes. Degeneracy occurs at an infinite discrete sequence of instants, that are divided into four classes. Different types of bifurcation and break of symmetry occur at each instant of three of the four classes; bifurcation does not occur at the degeneracy instants of the fourth class. [ABSTRACT FROM AUTHOR]

We obtain a classification result for rotational surfaces in Berger spheres, whose mean curvature is given as a prescribed function of their angle function. Under hypothesis on the prescribed function, we show that these surfaces behave like the Delaunay surfaces of constant mean curvature. The classification is done by means of a phase plane analysis of the ODE fulfilled by the profile curve of such surfaces. [ABSTRACT FROM AUTHOR]