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[3] viXra:2210.0104 [pdf] submitted on 2022-10-24 02:40:57
Authors: James A. Smith
Comments: 14 Pages.
To help fill the need for examples of introductory-level problems that have been solved via Geometric Algebra (GA), we derive the equation for a plane that is tangent to three given planes. The approach that we use determines the unit bivector of the tangent plane from the interior and exterior products of the vectors that connect the centers of the given spheres. A more-general version of this approach is presented in an appendix.
Category: Geometry
[2] viXra:2210.0061 [pdf] replaced on 2026-01-16 02:43:17
Authors: Urs Frauenfelder, Joa Weber
Comments: 69 Pages. v2 Reference [SX14] added. J. Symplectic Geom. 23 no.5, 1109-1177 (2025). https://dx.doi.org/10.4310/JSG.251228022011
Critical points of a function subject to a constraint can be either detected by restricting the function to the constraint or by looking for critical points of the Lagrange multiplier functional. Although the critical points of the two functionals, namely the restriction and the Lagrange multiplier functional are in natural one-to-one correspondence this does not need to be true for their gradient flow lines. We consider a singular deformation of the metric and show by an adiabatic limit argument that close to the singularity we have a one-to one correspondence between gradient flow lines connecting critical points of Morse index difference one. We present a general overview of the adiabatic limit technique in the article [FW22b]. The proof of the correspondence is carried out in two parts. The current part I deals with linear methods leading to a singular version of the implicit function theorem. We also discuss possible infinite dimensional generalizations in Rabinowitz-Floer homology. In part II [FW22a] we apply non-linear methods and prove, in particular, a compactness result and uniform exponential decay independent of the deformation parameter.
Category: Geometry
[1] viXra:2210.0057 [pdf] replaced on 2026-01-16 02:42:32
Authors: Urs Frauenfelder, Joa Weber
Comments: 56 Pages. 2 figures. J. Symplectic Geom. 23 no.5, 1179-1134 (2025). https://dx.doi.org/10.4310/JSG.251228022324
In this second part to [FW22a] we finish the proof of the one-to-one correspondence of gradient flow lines of index difference one between the restricted functional and the Lagrange multiplier functional for deformation parameters of the metric close to the singular one. In particular, we prove that, although the metric becomes singular, we have uniform bounds for the Lagrange multiplier of finite energy solutions and all its derivatives. This uniform bound is the crucial ingredient for a compactness theorem for gradient flow lines of arbitrary deformation parameter. If the functionals are Morse we further prove uniform exponential decay. We finally show combined with the linear theory in part I that if the metric is Morse-Smale the adiabatic limit map is bijective. We present a general overview of the adiabatic limit technique in the article [FW22b].
Category: Geometry
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