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Transformations and singularities of polarized curves
1Vienna, Austria.
Annals of Global Analysis and Geometry
|April 23, 2019
Summary
Darboux and Calapso transforms of polarized curves near singularities exhibit distinct limiting behaviors. First-order poles lead to convergence, while second-order poles result in convergence or limit points/circles for Calapso transforms.
Area of Science:
- Differential Geometry
- Geometric Analysis
- Topology
Background:
- Darboux and Calapso transforms are key tools in analyzing curves and surfaces.
- Understanding the behavior of these transforms near singularities is crucial for geometric studies.
- Polarized curves in conformal spheres present complex geometric structures.
Purpose of the Study:
- To investigate the limiting behavior of Darboux and Calapso transforms of polarized curves.
- To analyze the impact of first and second-order poles on these transforms.
- To extend findings to isothermic surfaces and singular umbilics.
Main Methods:
- Analysis of polarized curves in the conformal n-dimensional sphere.
- Study of Darboux and Calapso transforms with first and second-order poles.
- Examination of convergence properties and limit behaviors.
Main Results:
- For a first-order pole, all Darboux and Calapso transforms converge to the original curve.
- For a second-order pole, generic Darboux transforms converge, while Calapso transforms yield limit points or circles.
- Results are applicable to isothermic surfaces approaching singular umbilics.
Conclusions:
- The order of the pole significantly influences the limiting behavior of Darboux and Calapso transforms.
- These findings provide insights into the geometry of singular points on curves and surfaces.
- The study contributes to the understanding of geometric transformations in higher dimensions.
Keywords:
Calapso transformationDarboux transformationIsothermic surfacesPolarized curveSingular umbilicsMore Related Videos
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