We study sharp asymptotics of the first eigenvalue on Riemannian surfaces obtained from a fixed Riemannian surface by attaching a collapsing flat handle or cross cap to it. Through a careful choice of parameters this construction can be used to strictly increase the first eigenvalue normalized by area if the initial surface has some symmetries. If these symmetries are not present we show that the first eigenvalue normalized by area strictly decreases for the same range of parameters. These results are the main motivation for the construction in \cite{MS3}, where we show a monotonicity result for the normalized first eigenvalue without any symmetry assumptions.

中文翻译：

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一些退化曲面上第一特征值的锐渐近

我们研究了从固定黎曼曲面获得的黎曼曲面上第一个特征值的锐渐近性，方法是将折叠平手柄或十字帽连接到它。如果初始表面具有一些对称性，则通过仔细选择参数，此构造可用于严格增加按面积归一化的第一特征值。如果这些对称性不存在，我们将表明对于相同的参数范围，按面积归一化的第一个特征值会严格减小。这些结果是在 \cite{MS3} 中构建的主要动机，我们在没有任何对称性假设的情况下展示了归一化第一特征值的单调性结果。