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DCDS

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.## Year of publication

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