Affine Minimal Surfaces Research Articles

On Analogues of B\" acklund Theorem in Affine Differential Geometry of Surfaces

We recall the well-known Chern-Terng theorem concerning affine minimal surfaces. Next we formulate some complementary (with transversal fields necessarily not parallel) affine B\"acklund theorem. We describe some geometrical conditions which imply the local symmetry of both induced connections. We give also some necessary and sufficient conditions under which the affine fundamental forms are proportional.

Deformation and singularities of maximal surfaces with planar curvature lines

Minimal surfaces with planar curvature lines in the Euclidean space have been studied since the late 19th century. On the other hand, the classification of maximal surfaces with planar curvature lines in the Lorentz-Minkowski space has only recently been given. In this paper, we use an alternative method not only to refine the classification of maximal surfaces with planar curvature lines, but also to show that there exists a deformation consisting exactly of all such surfaces. Furthermore, we investigate the types of singularities that occur on maximal surfaces with planar curvature lines. Finally, by considering the conjugate of maximal surfaces with planar curvature lines, we obtain analogous results for maximal surfaces that are also affine minimal surfaces.

A characterization of hyperbolic affine flat, affine minimal surfaces in [formula omitted

We investigate the geometric properties of hyperbolic affine flat, affine minimal surfaces in the equiaffine space A3. We use Cartan's method of moving frames to compute a complete set of local invariants for such surfaces. Using these invariants, we give a complete local classification of such surfaces and construct new examples.

Smooth surfaces from bilinear patches: Discrete affine minimal surfaces

Motivated by applications in freeform architecture, we study surfaces which are composed of smoothly joined bilinear patches. These surfaces turn out to be discrete versions of negatively curved affine minimal surfaces and share many properties with their classical smooth counterparts. We present computational design approaches and study special cases which should be interesting for the architectural application.

Cauchy problems for discrete affine minimal surfaces

In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes, we show how to solve discrete Cauchy problems analogous to the Cauchy problems for smooth analytic improper affine spheres and smooth analytic affine minimal surfaces.

Discrete affine minimal surfaces with indefinite metric

Inspired by the Weierstrass representation of smooth affine minimal surfaces with indefinite metric, we propose a constructive process producing a large class of discrete surfaces that we call discrete affine minimal surfaces. We show that they are critical points of an affine area functional defined on the space of quadrangular discrete surfaces. The construction makes use of asymptotic coordinates and allows defining the discrete analogs of some differential geometric objects, such as the normal and co-normal vector fields, the cubic form and the compatibility equations.

Affine minimal hypersurfaces of rotation

We give a complete list of affine minimal surfaces inA3 with Euclidean rotational symmetry, completing the treatise given in [1] and prove that these surfaces have maximal affine surface area within the class of all affine surfaces of rotation satisfying suitable boundary conditions. Besides we show that for rotationally symmetric locally strongly convex affine minimal hypersurfaces inA n ,n≥4, the second variation of the affine surface area is negative definite under certain conditions on the meridian.

An algebraic representation of the affine B�cklund transformation

In 1980 Chern and Terng defined a Backlund transformation for affine minimal surfaces. In this paper we show that this Backlund transformation can be simply represented by an involution and translation of the affine conormal.

Translationsfl�chen in der �quiaffinen Differentialgeometrie

We discuss with equiaffine methods the surfaces of translation with plane generating curves in the three-dimensional affine space. Using (pseudo-) isothermic parameters we determine in this class all the affine minimal surfaces (which include the affine spheres, the quadrics and the ruled surfaces), all the surfaces with vanishing affine Gauss curvature (which include the surfaces with constant non vanishing affine mean curvature), and all the surfaces with only one family of affine lines of curvature.

Affine variation formulas and affine minimal surfaces.

Affine variation formulas and affine minimal surfaces.

An analogue of Bäcklund's theorem in affine geometry

It is well-known that there is a correspondence between solutions of the Sine-Gordon equation (SGE) -~—,~Hr= sin q> dx dt Y and the surfaces of constant curvature -1 in R (see below). The classical Backlund transformation of such surfaces furnishes a way to generate new solutions of the SGE from a given solution. This has received much attention in recent studies of the soliton solutions of the SGE, and the technique has been used successfully in the study of other non-linear evolution equations. In the first section of this paper we present a simple derivation of the classical Backlund theorem and its applications by using the method of moving frames. Our main result concerns affine minimal surfaces. They arise as the solution of the variation problem for affine area. The corresponding Euler-Lagrange equation is a fourth order partial differential equation. In §2,we develop the basic properties of affine minimal surfaces.In §3 we study the transformation of affine surfaces by realizing them as the focal surfaces of a line congruence. The natural conditions that the congruence be a W-congruence and that the affine normals at corresponding points be parallel lead to the conclusion that both surfaces are affine minimal. This is the content of Theorem 4, the main result of our paper. As in the classical case, the Theorem leads to the construction of new affine minimal surfaces from a given one by the solution of a completely integrable system of first order partial differential equations. 1. The classical Backlund theorem and its consequences. Let M be a surface in R. We choose a local field of orthonormal frames v\, v O3 be the dual coframe of Vi, v2, #3. We can write dx 2^0ava a s Here and throughout this paper we shall agree on the index ranges (1.2) 1 £ij,k£2, 1 £a,s,r H 3. The structure equations of R are *Work done under partial support of NSF grant MCS74-23180. **Work done under partial support of NSF Grant MCS76-01692. Received by the editors on April 13, 1978. Copyright © 1980 Rocky Mountain Mathematics Consortium 105 106 S. S. CHERN AND C. L. TERNG dda = 2] Os A Osa, 6as + Osa = 0 ddas = 2 0«r A 0ri8. Restricting these forms to the frames defined above, we have (1.4) 03 = 0 and hence (1.5) 0 = </03 = L 0,-A 0,3. By Cartan's lemma we may write (1.6) o« = S M y A/y = V The first equation of (1.3) gives (1.7) d0< = S 0 y A0„y (where 012 is the Levi-Civita connection form on M which is uniquely determined by these two equations). The Gauss equation is (1.8) <*0,7 = E 0» A 0*y + e „ where ©12 = flu» A 023 = (det(/j,7))0i A 02, (1.9) K^LL det(Af-y) = the Gaussian curvature of M. The Codazzi equations are (1.10) rf0f-3 = £ 0l7 A 0y3. y The first fundamental form of the surface is (1.11) I = (0i) + (02) , and the second fundamental form is (1.12) II = £ 0,0,3 = E h fi fi j . i ij Near a non-umbilical point M can be parametrized by its lines of curvature, i.e., there are coordinates utin which I and II are both diagonaHzed. Explicitly we write I = (*i)(<fai) + (a2f(du2f II = bfaWutf + b2(a2)\du2)