论文标题
边界问题的低维度中的聚类现象
Clustering phenomena in low dimensions for a boundary Yamabe problem
论文作者
论文摘要
我们考虑了通过在$ n- $ dimensional commensional compact riemannian歧管上对标量的标量和边界平均曲率开出标量和边界平均值的经典几何问题。我们处理负标态曲率和正边界平均曲率的情况。众所周知，如果$ n = 3 $，所有爆炸点都是孤立且简单的。在这项工作中，我们证明，对于线性扰动，在低维度中不再是正确的$ 4 \ leq n \ leq 7 $。特别是，我们构建了一个具有聚类爆破边界点（即非分离）的解决方案，该解决方案是非企业的，是边界的无痕迹第二基本形式的局部最小化器。
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.
