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July  2009, 23(3): 655-683. doi: 10.3934/dcds.2009.23.655

On the prescribed scalar curvature on $3$-half spheres: Multiplicity results and Morse inequalities at infinity

M. Ben Ayed 1, and  Mohameden Ould Ahmedou 2,

1. 

Département de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, Sfax, Tunisia
2. 

Mathematisches institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tubingen, Germany

Received  August 2007 Revised  July 2008 Published  November 2008

We consider the existence and multiplicity of riemannian metrics of prescribed mean curvature and zero boundary mean curvature on the three dimensional half sphere $(S^3_+,g_c)$ endowed with its standard metric $g_c$. Due to Kazdan-Warner type obstructions, conditions on the function to be realized as a scalar curvature have to be given. Moreover the existence of critical point at infinity for the associated Euler Lagrange functional makes the existence results harder to be proved. However it turns out that such noncompact orbits of the gradient can be treated as a usual critical point once a Morse Lemma at infinity is performed. In particular their topological contribution to the level sets of the functional can be computed. In this paper we prove that, under generic conditions on $K$, this topology at infinity is a lower bound for the number of metrics in the conformal class of $g_c$ having prescribed scalar curvature and zero boundary mean curvature.
Keywords: Prescribed scalar curvature, Topology at Infinity, Gradient flow, Morse inequalities, Critical point at infinity.
Mathematics Subject Classification: Primary: 58E05, 35J65; Secondary: 53C21.
Citation: M. Ben Ayed, Mohameden Ould Ahmedou. On the prescribed scalar curvature on $3$-half spheres: Multiplicity results and Morse inequalities at infinity. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 655-683. doi: 10.3934/dcds.2009.23.655
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