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- Title
Multiplicity results for the scalar curvature problem on half spheres.
- Authors
Ayed, Mohamed Ben; Mehdi, Khalil El
- Abstract
Given a smooth positive function $ K $ defined on the standard half sphere endowed with its standard metric $ g $, we consider the problem of finding a metric $ \tilde{g} $ conformally equivalent to $ g $ and whose scalar curvature is equal to $ K $ and the boundary mean curvature is equal to zero. Using careful study of Hopf-Poincaré counting index formulae, we prove the existence of many of such metrics $ \tilde{g} $ provided that $ K $ is close to the scalar curvature of $ g $. Such Counting index formulae are related to the topological contribution of solutions at the level sets of the associated approximate variational functional. The location of the levels of these solutions are given very precisely which leads to a new type of multiplicity results on half spheres. Such multiplicity results are proved without any assumptions of symmetry or periodicity on the function $ K $.
- Subjects
CURVATURE; SPHERES; MULTIPLICITY (Mathematics); SMOOTHNESS of functions; SYMMETRY
- Publication
Discrete & Continuous Dynamical Systems: Series A, 2024, Vol 44, Issue 7, p1
- ISSN
1078-0947
- Publication type
Article
- DOI
10.3934/dcds.2024013