We present new-curvature one-cycle sweepout estimates in Riemannian geometry, both on surfaces and in higher dimension. More precisely, we derive upper bounds on the length of one-parameter families of one-cycles sweeping out essential surfaces in closed Riemannian manifolds. In particular, we show that there exists a homotopically substantial one-cycle sweepout of the essential sphere in the complex projective space, endowed with an arbitrary Riemannian metric, whose one-cycle length is bounded in terms of the volume (or diameter) of the manifold. This is the first estimate on sweepout volume in higher dimension without curvature assumption. We also give a detailed account of the situation for compact Riemannian surfaces with or without boundary, in relation with questions raised by P. Buser and L. Guth.

中文翻译：

---

封闭黎曼流形中基本曲面的单循环扫描估计

我们在黎曼几何中提出了新曲率单周期扫描估计，包括曲面和更高维度。更准确地说，我们推导出了在封闭黎曼流形中扫除基本曲面的单循环的单参数族长度的上限。特别是，我们表明在复杂的射影空间中存在基本球体的同伦实质单循环扫掠，赋予任意黎曼度量，其单循环长度以球的体积（或直径）为界。多方面的。这是第一次在没有曲率假设的情况下对更高维度的清扫量进行估计。我们还详细说明了有边界或无边界的紧凑黎曼曲面的情况，与 P. Buser 和 L. Guth 提出的问题有关。