Abstract
In the present paper we study multiplicity results for a Yamabe-type problem proposed by Escobar in 1992. We consider the product of a compact Riemannian manifold without boundary and null scalar curvature with a compact Riemannian manifold with boundary, having null scalar curvature and constant mean curvature on the boundary. We use some standard results from the bifurcation theory to prove the existence of an infinite number of conformal classes with at least two non-homothetic Riemannian metrics of null scalar curvature and constant mean curvature on the boundary of the product manifold. In addition, we obtain a convergence result for bifurcating branches.
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Notes
Here \(\hat{H}_{g^{(2)}}=\frac{m_2-1}{(m-1)}H_{g^{(2)}}\), where \(m_2=\text {dim}\,(M_2)\) and \(m=\text {dim}\,(M_1)+\text {dim}\,(M_2)\).
With a slight abuse of terminology, we will say that a metric g on a manifold M with boundary has constant mean curvature meaning that the boundary \(\partial M\) has constant mean curvature with respect to g.
here \(\mathbf{1}\) denote the constant function 1 on M
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The authors wish to thank the Universidad del Cauca for supporting this work through research project VRI ID 5069.
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Cárdenas, E.D., Sierra, W. Bifurcation of metrics with null scalar curvature and constant mean curvature on the boundary of compact manifolds. manuscripta math. 169, 123–139 (2022). https://doi.org/10.1007/s00229-021-01332-4
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DOI: https://doi.org/10.1007/s00229-021-01332-4