孤立平坦点与无焦点性质的 $C^2$-稳健性
Isolated Flat Points and $C^2$-Robustness of the No Focal Points Property
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中文总结 AI 辅助
本文证明紧致黎曼曲面上非正曲率且零曲率集有限的度量属于无焦点度量集合的 $C^2$-内部，并构造了位于负曲率边界但仍在无焦点区域内的共形度量。
中文摘要 AI 辅助
我们证明了紧致黎曼曲面上无焦点性质的一个局部稳定性判据。更精确地说，若一个光滑度量 $g$ 具有非正高斯曲率且其零曲率集合是有限的，则 $g$ 属于无焦点度量集合的 $C^2$-内部。因此，$g$ 的每一个充分小的 $C^2$-扰动仍然没有焦点，尽管任意小的扰动可能产生正高斯曲率区域。根据 Ruggiero 对无共轭点度量集合的 $C^2$-内部的刻画，所得邻域中的所有度量都是 Anosov 的。我们还证明，从紧致曲面上的任意双曲度量出发，可以将任意有限集合指定为某个光滑共形非正曲率度量的高斯曲率零集。这些度量可以被负曲率度量光滑逼近，因此它们位于负曲率区域的边界上，同时仍是无焦点区域的内点。稳定性定理的证明结合了均匀局部凸性、Gulliver 关于包含在小球中的测地线段长度的界，以及对 Riccati 方程的归纳论证，该论证控制着依次通过可能出现正曲率的区域。
英文摘要
We prove a local stability criterion for the no focal points property on compact Riemannian surfaces. More precisely, if a smooth metric $g$ has non-positive Gaussian curvature and its zero-curvature set is finite, then $g$ belongs to the $C^2$-interior of the set of metrics without focal points. Thus, every sufficiently small $C^2$-perturbation of $g$ still has no focal points, although arbitrarily small perturbations may create regions of positive Gaussian curvature. By Ruggiero's characterization of the $C^2$-interior of the set of metrics without conjugate points, all metrics in the resulting neighborhood are Anosov. We also show that, starting from any hyperbolic metric on a compact surface, one can prescribe an arbitrary finite set as the zero set of the Gaussian curvature of a smooth conformal non-positively curved metric. These metrics can be approximated smoothly by negatively curved metrics, so they lie on the boundary of the negatively curved regime while remaining interior points of the no-focal-points regime. The proof of the stability theorem combines uniform local convexity, Gulliver's bound on the length of geodesic segments contained in small balls, and an inductive argument on the Riccati equation that controls successive passages through the regions where positive curvature may appear.
