Eric Toubiana

Symmetry of properly embedded special Weingarten surfaces in

In this paper we prove some existence and uniqueness results about special Weingarten surfaces in hyperbolic space.

Sur les surfaces de Weingarten spéciales de type minimal

We derive a classification of special Weingarten rotation surfaces of minimal type in Euclidean space. We prove existence and uniqueness, and we give a necessary and sufficient condition to have a complete surface. Futhermore, we prove that under some further simple condition there is a 1- parameter family of complete special surfaces with the same geometrical behaviour as the minimal catenoids family. We remark that there is in our context of special Weingarten minimal type surfaces related “half space theorem”, of Hoffman and Meeks, and “Bernstein theorem”.

Some remarks on deformations of minimal surfaces

We consider complete minimal surfaces (c.m.s.s) in R3 and their deformations. M1 is an s-deformation of Mo if M1 is a graph over Mo in an E tubular neighborhood of Mo and M1 is E C1-close to Mo. A minimal surface M is isolated if all c.m.s.s which are sufficiently small deformations of M are congruent to M. In this paper we construct an example of a nonisolated c.m.s. It is modelled on a 4-punctured sphere and is of finite total curvature. On the other hand, we prove that a c.m.s. discovered by Meeks and Jorge, modelled on the sphere punctured at the fourth roots of unity, is isolated. Introduction. We consider complete minimal surfaces (c.m.s.s) in R3 and their deformations. M1 in an s-deformation of Mo if M1 is a graph over Mo is an E tubular neighborhood of M1 and M1 is E C1-close to Mo. A c.m.s. Mo is isolated if all minimal surfaces M1, which are sufficiently small deformations of Mo, are congruent to Mo. Many of the classical minimal surfaces in R3 are known to be isolated [2], however, no example was known of a nonisolated minimal surface. In this paper we construct such an example; it is modelled on a 4-punctured sphere and is of finite total curvature. On the other hand, we prove that a c.m.s. discovered by Meeks and Jorge, modelled on the sphere punctured at the four roots of unity, {1, -1, i, -i}, is isolated. The analogous surface modelled on the sphere punctured at the cube roots of unity was shown to be isolated in [2]. The question is raised there of whether deformations of the four puncture case can be realized by deforming the conformal structure; i.e., changing the cross ratio of the four points. Thus for the Meeks-Jorge example in question, the answer is no. As we shall see, the conformal structure of our example that admits deformations does not change either. Perhaps the conformal structure never changes by small deformations? We wish to thank W. Meeks and B. Morin for helpful conversations and greatly simplifying suggestions. I. A deformable surface. Let M be a c.m.s. of finite total curvature, so that M is conformally equivalent to a compact Riemann surface M punctured at a finite number of points. An end E of M is said to be bounded if E is a bounded distance from a plane. If E corresponds to the puncture p and P is the plane orthogonal to g(p), g the Gauss map, then E bounded means E is a bounded distance from P. If E is embedded, then it can be expressed as the graph of a log R + O ({Rl), where R is the distance from the origin in P, and this holds for R large. Then E is bounded if and only if a = O. Let (9,) be a Weierstrass representation of M and 1,X2,X3 the associated analytic differentials on M. We know the real periods of each Xk are zero on M Received by the editors December 2, 1983 and, in revised form, May 1, 1984. 1980 Mathematics Subject Classification (1985 Revision). Primary 53A10. ({)1986 American Mathematical Society 0002-9947/86

Simply Connected Minimal Surfaces in R3 Transverse to Horizontal Planes

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Discrete and Nondiscrete Isometric Deformations of Surfaces in ℝ3

.25 per page

General curvature estimates for stable H-surfaces in 3-manifolds applications

We consider simply connected minimal surfaces in Euclidean space and we give a characterisation of the helicoid.

Screw motion surfaces in

AbstractWe prove existence of closed infinitely differentiable surfaces M of

and

\mathbb{R}^3