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Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds II : Morse Theory
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Ikoma, Norihisa, Malchiodi, Andrea and Mondino, Andrea (2017) Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds II : Morse Theory. American Journal of Mathematics, 139 (5). pp. 1315-1378.
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Official URL: https://doi.org/10.1353/ajm.2017.0033
Abstract
The synthesis of cationic rhodium and iridium complexes of a bis(imidazol-2-thione) functionalised calix[4]arene ligand and their surprising capacity for potassium binding is described. In both cases uptake of the alkali metal into the calix[4]arene cavity occurs despite adverse electrostatic interactions associated with close proximity to the transition metal fragment (Rh+ ···K+ = 3.715(1) Å, Ir+ ···K+ = 3.690(1) Å). The formation and constituent bonding of these unusual heterobimetallic adducts has been interrogated through extensive solution and solid-state characterisation, examination of the host-guest chemistry of the ligand and its upper-rim unfunctionalised calix[4]arene analogue, and computationally using DFT-based energy decomposition analysis (EDA).This is the second part of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three-manifolds. In both papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty being the Moebius degeneration of the tori. In the first paper the construction was performed via minimization, here by Morse Theory. To this aim we establish new geometric expansions of the derivative of the Willmore functional on small Clifford tori (in geodesic normal coordinates) which degenerate to small geodesic spheres with a small handle under the action of the Moebius group. By using these sharp asymptotics we give sufficient conditions, in terms of the ambient curvature tensors and Morse inequalities, for having existence/multiplicity of embedded tori which are stationary for the Willmore functional under the constraint of prescribed (sufficiently small) area.
| Item Type: | Journal Article | ||||||
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| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science > Mathematics | ||||||
| Library of Congress Subject Headings (LCSH): | Riemannian manifolds, Differential equations, Partial, Morse theory | ||||||
| Journal or Publication Title: | American Journal of Mathematics | ||||||
| Publisher: | The Johns Hopkins University Press | ||||||
| ISSN: | 0002-9327 | ||||||
| Official Date: | 1 October 2017 | ||||||
| Dates: |
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| Volume: | 139 | ||||||
| Number: | 5 | ||||||
| Page Range: | pp. 1315-1378 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Funder: | Nihon Gakujutsu Shinkōkai [Japan Society for the Promotion of Science] (JSPS), Fondo per gli Investimenti della Ricerca di Base (FIRB), Eidgenössische Technische Hochschule Zürich (ETH) | ||||||
| Grant number: | Research Fellowship 24-2259 (JSPS) |
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