Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds, II: Morse Theory
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 Mübius degeneration of the tori. In the first paper t...
| Egile Nagusiak: | , , |
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| Formatua: | Journal article |
| Hizkuntza: | English |
| Argitaratua: |
Johns Hopkins University Press
2017
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| _version_ | 1826269653486272512 |
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| author | Ikoma, N Malchiodi, A Mondino, A |
| author_facet | Ikoma, N Malchiodi, A Mondino, A |
| author_sort | Ikoma, N |
| collection | OXFORD |
| description | 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 Mübius 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 Mübius 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. |
| first_indexed | 2024-03-06T21:28:28Z |
| format | Journal article |
| id | oxford-uuid:43e25e42-bf09-4f10-b003-009342f0b4e4 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T21:28:28Z |
| publishDate | 2017 |
| publisher | Johns Hopkins University Press |
| record_format | dspace |
| spelling | oxford-uuid:43e25e42-bf09-4f10-b003-009342f0b4e42022-03-26T14:58:12ZEmbedded area-constrained Willmore tori of small area in Riemannian three-manifolds, II: Morse TheoryJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:43e25e42-bf09-4f10-b003-009342f0b4e4EnglishSymplectic Elements at OxfordJohns Hopkins University Press2017Ikoma, NMalchiodi, AMondino, AThis 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 Mübius 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 Mübius 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. |
| spellingShingle | Ikoma, N Malchiodi, A Mondino, A Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds, II: Morse Theory |
| title | Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds, II: Morse Theory |
| title_full | Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds, II: Morse Theory |
| title_fullStr | Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds, II: Morse Theory |
| title_full_unstemmed | Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds, II: Morse Theory |
| title_short | Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds, II: Morse Theory |
| title_sort | embedded area constrained willmore tori of small area in riemannian three manifolds ii morse theory |
| work_keys_str_mv | AT ikoman embeddedareaconstrainedwillmoretoriofsmallareainriemannianthreemanifoldsiimorsetheory AT malchiodia embeddedareaconstrainedwillmoretoriofsmallareainriemannianthreemanifoldsiimorsetheory AT mondinoa embeddedareaconstrainedwillmoretoriofsmallareainriemannianthreemanifoldsiimorsetheory |