Isolated Flat Points and C2-Robustness of the No Focal Points Property
Abstract
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 C2-interior of the set of metrics without focal points. Thus, every sufficiently small C2-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 C2-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.
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BibTeX
@misc{cantoral2026isolated,
title = {{Isolated Flat Points and $C^2$-Robustness of the No Focal Points Property}},
author = {Alexander Cantoral and Sergio Romaña},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15185},
eprint = {2609.15185},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15185},
note = {CC0},
}
RIS
TY - UNPB
TI - Isolated Flat Points and C^2-Robustness of the No Focal Points Property
AU - Alexander Cantoral
AU - Sergio Romaña
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15185
UR - https://arxiv.org/abs/2609.15185
AB - 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.
ER -