We prove a sharp area estimate for catenoids that allows us to rule out the phenomenon of multiplicity in min-max theory in several settings. We apply it to prove that i) the width of a three-manifold with positive Ricci curvature is realized by an orientable minimal surface ii) minimal genus Heegaard surfaces in such manifolds can be isotoped to be minimal and iii) the "doublings" of the Clifford torus by Kapouleas-Yang can be constructed variationally by an equivariant min-max procedure. In higher dimensions we also prove that the width of manifolds with positive Ricci curvature is achieved by an index 1 orientable minimal hypersurface.

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悬链线估计及其几何应用

我们证明了对链状体的清晰面积估计，这使我们能够在几种情况下排除 min-max 理论中的多重性现象。我们应用它来证明 i) 具有正 Ricci 曲率的三流形的宽度由可定向的最小曲面实现 ii) 这种流形中的最小属 Heegaard 曲面可以同位素化为最小和 iii) 的“加倍” Kapouleas-Yang 的 Clifford 圆环可以通过等变最小-最大程序变分构造。在更高维度上，我们还证明了具有正 Ricci 曲率的流形的宽度是通过索引 1 可定向的最小超曲面实现的。