In this paper, we study the Morse index of closed minimal submanifolds immersed into general Riemannian manifolds. Using the strategy developed by Ambrozio et al. (J Differ Geom 108(3):379–410, 2018) and under a suitable constrain on the submanifold, we obtain that the Morse index of the submanifold is bounded from below by a linear function of its first Betti’s number, as conjectured by Schoen and Marques-Neves. We also present many Riemannian manifolds and a sufficient condition to get the cited linear lower bound.

中文翻译：

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最小子流形的莫尔斯指数界限

在本文中，我们研究了浸入一般黎曼流形中的封闭最小子流形的莫尔斯指数。使用 Ambrozio 等人开发的策略。(J Differ Geom 108(3):379–410, 2018) 并且在对子流形的适当约束下，我们得到子流形的莫尔斯指数从下方受到其第一个 Betti 数的线性函数的限制，正如推测的那样舍恩和马克斯-内维斯。我们还提出了许多黎曼流形和获得所引用的线性下限的充分条件。