Clustering phenomena in low dimensions for a boundary Yamabe problem

Sergio Cruz-Blázquez; Angela Pistoia; Giusi Vaira

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arXiv · 2211.08219

Clustering phenomena in low dimensions for a boundary Yamabe problem

Sergio Cruz-Blázquez·Angela Pistoia·Giusi Vaira

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Abstract

We consider the classical geometric problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a $n-$dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature and positive boundary mean curvature. It is known that if $n=3$ all the blow-up points are isolated and simple. In this work we prove that, for a linear perturbation, this is not true anymore in low dimensions $4\leq n\leq 7$. In particular, we construct a solution with a clustering blow-up boundary point (i.e. non-isolated), which is non-umbilic and is a local minimizer of the norm of the trace-free second fundamental form of the boundary.

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Sergio Cruz-Blázquez, Angela Pistoia, Giusi Vaira. 2022-11-15. Clustering phenomena in low dimensions for a boundary Yamabe problem. https://arxiv.org/abs/2211.08219

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