Minimal Surfaces In R3 Research Articles

The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends

Given an integer k ≥ 2 k\geq 2 , let S ( k ) {\mathcal S}(k) be the space of complete embedded singly periodic minimal surfaces in R 3 \mathbb {R}^3 , which in the quotient have genus zero and 2 k 2k Scherk-type ends. Surfaces in S ( k ) {\mathcal S}(k) can be proven to be proper, a condition under which the asymptotic geometry of the surfaces is well known. It is also known that S ( 2 ) {\mathcal S}(2) consists of the 1 1 -parameter family of singly periodic Scherk minimal surfaces. We prove that for each k ≥ 3 k\geq 3 , there exists a natural one-to-one correspondence between S ( k ) {\mathcal S}(k) and the space of convex unitary nonspecial polygons through the map which assigns to each M ∈ S ( k ) M\in {\mathcal S}(k) the polygon whose edges are the flux vectors at the ends of M M (a special polygon is a parallelogram with two sides of length 1 1 and two sides of length k − 1 k-1 ). As consequence, S ( k ) {\mathcal S}(k) reduces to the saddle towers constructed by Karcher.

Laguerre Geometry of Surfaces in R 3

Let f : M → R3 be an oriented surface with non–degenerate second fundamental form. We denote by H and K its mean curvature and Gauss curvature. Then the Laguerre volume of f, defined by L(f) = f (H2 – K)/KdM, is an invariant under the Laguerre transformations. The critical surfaces of the functional L are called Laguerre minimal surfaces. In this paper we study the Laguerre minimal surfaces in R3 by using the Laguerre Gauss map. It is known that a generic Laguerre minimal surface has a dual Laguerre minimal surface with the same Gauss map. In this paper we show that any surface which is not Laguerre minimal is uniquely determined by its Laguerre Gauss map. We show also that round spheres are the only compact Laguerre minimal surfaces in R3. And we give a classification theorem of surfaces in R3 with vanishing Laguerre form.

On metrics arising on surfaces of constant mean curvature

We formulate necessary and sufficient conditions on a Riemannian metric that ensure its embeddability in a three-dimensional space of constant curvature as a surface of constant mean curvature. This theorem is a generalization of a number of classical results, in particular, the Ricci theorem, which gives a description of metrics arising on minimal surfaces in ℝ3.

ON ALGEBRAIC MINIMAL SURFACES IN ℝ3 DERIVING FROM CHARGE 2 MONOPOLE SPECTRAL CURVES

We give formulae for minimal surfaces in ℝ3 deriving, via classical osculation duality, from elliptic curves in a line bundle over ℙ1. Specialising to the case of charge 2 monopole spectral curves we find that the distribution of Gaussian curvature on the auxiliary minimal surface reflects the monopole's structure. This is elucidated by the behaviour of the surface's Gauss map.

Dense solutions to the Cauchy problem for minimal surfaces

We show a general way to produce in explicit coordinates complete minimal surfaces in ℝ3 that lie densely in the whole space. This construction relies on solving the Bjorling problem for adequate initial data.

The Logarithmic Derivative for Minimal Surfaces in ℝ3

E. F. Beckenbach and collaborators have developed a value distribution theory for minimal surfaces which parallels the work of R. Nevanlinna and others for complex meromorphic functions. We continue in the development by establishing the Lemma of the Logarithmic Derivative for minimal surfaces in ℝ3.

The geometry of minimal surfaces of finite genus II; nonexistence of one limit end examples

We demonstrate that a properly embedded minimal surface in ℝ3 with finite genus cannot have one limit end.

On the asymptotic behavior of a complete bounded minimal surface in ℝ³

In this paper we construct an example of a complete minimal disk which is properly immersed in a ball of R 3 \mathbb {R}^3 .

The Geometry of Minimal Surfaces of Finite Genus I: Curvature Estimates and Quasiperiodicity

Let \mathcal M be the space of properly embedded minimal surfaces in ℝ3 with genus zero and two limit ends, normalized so that every surface M ∊ \mathcal M has horizontal limit tangent plane at infinity and the vertical component of its flux equals one. We prove that if a sequence {M(i)} i ∊ \mathcal M has the horizontal part of the flux bounded from above, then the Gaussian curvature of the sequence is uniformly bounded. This curvature estimate yields compactness results and the techniques in its proof lead to a number of consequences, concerning the geometry of any properly embedded minimal surface in ℝ3 with finite genus, and the possible limits through a blowing-up process on the scale of curvature of a sequence of properly embedded minimal surfaces with locally bounded genus in a homogeneously regular Riemannian 3-manifold.

Some Picard theorems for minimal surfaces

This paper deals with the study of those closed subsets F ⊂ R 3 F \subset \mathbb {R}^3 for which the following statement holds: If S S is a properly immersed minimal surface in R 3 \mathbb {R}^3 of finite topology that is eventually disjoint from F , F, then S S has finite total curvature. The same question is also considered when the conclusion is finite type or parabolicity.

On properly embedded minimal surfaces with three ends

We classify all complete embedded minimal surfaces in ℝ3; with three ends of genus g and at least 2g+2 symmetries. The surfaces in this class are the Costa-Hoffman-Meeks surfaces that have 4g+4 symmetries in the case of a flat middle end. The proof consists of using the symmetry assumptions to deduce the possible Weierstrass data and then studying the period problems in all cases. To handle the 1-dimensional period problems, we develop a new general method to prove convexity results for period quotients. The 2-dimensional period problems are reduced to the 1-dimensional case by an extremal length argument.

Classification of doubly-periodic minimal surfaces of genus zero

We prove that if the quotient surface of a properly embedded doubly–periodic minimal surface in ℝ3 has genus zero, then the surface is one of the classical Scherk examples.

Existence result for minimal hypersurfaces¶with a prescribed finite number of planar ends

Paralleling what has been done for minimal surfaces in ℝ3, we develop a gluing procedure to produce, for any k≥ 2 and any n≥ 3 complete immersed minimal hypersurfaces of ℝn+1 which have k planar ends. These surfaces are of the topological type of a sphere with k punctures and they all have finite total curvature.

Embedded, Doubly Periodic Minimal Surfaces

We consider the question of existence of embedded doubly periodic minimal surfaces in R3 with Scherk-type ends, surfaces that topologically are Scherk'sdoubly periodic surface with handles added in various ways. We extend the existence results of H. Karcher and F. Wei to more cases, and we find other cases where existence does not hold.

The space of complete minimal surfaces with finite total curvature as lagrangian submanifold

-The space M of nondegenerate, properly embedded minimal surfaces in R3 with finite total curvature and fixed topology is an analytic lagrangian submanifold of Cn, where n is the number of ends of the surface. In this paper we give two expressions, one integral and the other pointwise, for the second fundamental form of this submanifold. We also consider the compact boundary case, and we show that the space of stable nonflat minimal annuli that bound a fixed convex curve in a horizontal plane, having a horizontal end of finite total curvature, is a locally convex curve in the plane C. Mathematics Subject Classification: 53A10, 53C42.

Half-space theorems for mean curvature one surfaces in hyperbolic space

We give conditions which oblige properly embedded constant mean curvature one surfaces in hyperbolic 3-space to intersect. Our results are inspired by the theorem that two disjoint properly immersed minimal surfaces in R 3 \mathbf {R}^3 must be planes.

Some remarks on complete simply connected minimal surfaces meeting the planes x3 = Constant Transversally

The helicoid and plane are the only known complete embedded minimal surfaces inR 3 that are simply connected. We prove the helicoid and plane are the only surfaces of this type that have bounded curvature and meet each plane x3 = constant in (at most) one smooth connected curve.

Embedded minimal surfaces: Forces, topology and symmetries

We prove topological uniqueness theorems for embedded minimal surfaces in ℝ3 under the assumption that certain forces associated to these surfaces are vertical. We give applications to minimal surfaces with symmetries and with free boundary.

Maximum principles for minimal surfaces in ℝ3 having noncompact boundary and a uniqueness theorem for the helicoidhaving noncompact boundary and a uniqueness theorem for the helicoid

Maximum principles for minimal surfaces in ℝ3 having noncompact boundary and a uniqueness theorem for the helicoidhaving noncompact boundary and a uniqueness theorem for the helicoid

Intrinsic third derivatives for Plateau's problem and the Morse inequalities for disc minimal surfaces in ?3

A simply branched minimal surface in ℝ3 cannot be a non-degenerate critical point of Dirichlet's energy since the Hessian always has a kernel. However such minimal surface can be non-degenerate in another sense introduced earlier by R. Bohme and the author. Such surfaces arise as the zeros of a vector field on the space of all disc surfaces spanning a fixed contour. In this paper we show that the winding number of this vector field about such a surface is ±2 p , wherep is the number of branch points. As a consequence we derive the Morse inequalities for disc minimal surfaces in ℝ3, thereby completing the program initiated by Morse, Tompkins, and Courant. Finally, this result implies that certain contours in ℝ4 arbitrarily close to the given contour must span at least 2 p disc minimal surfaces.