FROM THE CATENOID-HELICOID DEFORMATION TO THE GEOMETRY OF LOOP GROUPS J.-H. ESCHENBURG Abstract. A formula for the angle between two curves on a surface. Show that a catenoid and helicoid are locally isometric. Definition & property of Conformal mapping. 45 o rotation. Created Date: The catenoid and Enneper’s surface are the unique complete minimal surfaces in R3 with finite total curvature −4π (Osserman). 5 Definition. The key obstruction to the existence oflocal isometries. For conformal mappings read §5.3 Nov 11 A diffeomorphism : S S' between regular surfaces is … Nevertheless, each of them can be continuously bent into the other: they are locally isometric. Rigidity Isometries arerare. These characterizations also depend on more recent work of Colding-Minicozzi 2008 and Collin 1997. But surfaces with the same curvature are so | locally. 3 Definition. isometry F b w the helicoid and the catenoid The previous example shows that isometries need not preserve It M has 4 0 N has H l to L Q Do isometries preserve K. Gauss.stheoremtgregihn.IT F M N is an isometry then Kmcp Ku Fcp for all peM. surfaces with di erent curvatures can’t be isometric. an isometry. the catenoid to part of the helicoid given by the image of "σ(sinhu,v). Greg Arone Introduction to Di erential Geometry Lecture #10 Read: §5.4 for area. It follows from Theorema Egregium that under this bending the Gaussian curvature at any two corresponding points of the catenoid and helicoid is always the same. Show that every local isometry of the helicoid H to the catenoid C must carry the axis of H to the central circle of C, … (b) Animate the series of plots in (a). Locally the helicoid is isometric to the catenoid. The Bonnet rotation is an isometry of the surface, that is, all distances within the surface are preserved; there is no stretching or wrinkling. I.e. As the Bonnet rotation angle increases, a continuous family of minimal surfaces is generated. Catenoid, 0 o rotation. of an isometry between a helicoid (a ruled surface) and a catenoid (a rotation surface) shows that a condition for the image surface to be ruled is not trivial. The catenoid and the helicoid are two very different-looking surfaces. Nov 6: Area of a portion of a surface. Figure 1: Straight helicoid References In fact such local isometry can be achieved as endpoint of a continuous one-parameter family of isometric deformations which are all minimal surfaces. See do Carmo, Problem 14 on page 213, and also Example 2 on pages 221-222. ... For example, the catenoid and helicoid are adjoints. (a) Plot the surface M t for at least six values of t from t = 0 (helicoid) to t < π / 2 (catenoid). Problem 3. The Catenoid and the Helicoid Are Isometric. Catenoid - conjugate surface to the helicoid Image by Matthias Weber The catenoid is the unique complete embedded minimal surface with nite topology and two ends (Schoen 1983) or of nite topology and genus zero (Lopez-Ros 1991). Example of an isometry, from the catenoid to the helicoid. I : R2!R3 is an isometry of R2 with its image. Fact: Curvature is alocal invariantunder isometry. The less known result states that if two ruled surfaces are locally isometric, then the local isometry preserves their rulings, unless the … The isometry from catenoid to helicoid is exercise 5.8. We then note that the parallels u constant on the catenoid get mapped helices on the heli-coid, and that the meridians v constant on the catenoid get mapped to the rulings t &→(tcosθ,tsinθ,θ). 7. For isometries read §5.2. Every rotation around the origin in C is an (intrinsic) isometry of the Enneper surface, but most of these isometries do not extend to ambient isometries. Lifting a \pacman region" to a cone is a local isometry (you were asked to verify this in homework assignment 3) There is a local isometry between a helicoid and a catenoid, \wrapping" the helicoid around the catenoid.
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