We study non-reversible Finsler metrics with constant flag curvature $1$ on $S^2$ and show that the geodesic flow of every such metric is conjugate to that of one of Katok’s examples, which form a 1-parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We also show, in any dimension, that the geodesic flow of a Finsler metric with constant positive flag curvature is completely integrable.

Finally, we give an example of a Finsler metric on $S^2$ with positive flag curvature such that no two closed geodesics intersect and show that this is not possible when the metric is reversible or has constant flag curvature.

In this paper, we prove that the open Gromov–Witten invariants defined in [20] on K3 surfaces satisfy the Kontsevich–Soibelman wall-crossing formula. One hand, this gives a geometric interpretation of the slab functions in Gross–Siebert program. On the other hands, the open Gromov–Witten invariants coincide with the weighted counting of tropical discs. This is an analog of the corresponding theorem on toric varieties [26][27] but on compact Calabi–Yau surfaces.

We use a min-max procedure on the Allen–Cahn energy functional to construct geodesics on closed, $2$‑dimensional Riemannian manifolds, as motivated by the work of Guaraco [Gua18]. Borrowing classical blowup and curvature estimates from geometric analysis, as well as novel Allen–Cahn curvature estimates due to Wang–Wei [WW19], we manage to study the fine structure of potential singular points at the diffuse level, and show that the problem reduces to that of understanding “entire” singularity models constructed by del Pino–Kowalczyk–Pacard [dPKP13] with Morse index $1$. The argument is completed by a Morse index estimate on these singularity models.

The algebras of valuations on $S^6$ and $S^7$ invariant under the actions of $G_2$ and $\operatorname{Spin}(7)$ are shown to be isomorphic to the algebra of translation-invariant valuations on the tangent space at a point invariant under the action of the isotropy group. This is in analogy with the cases of real and complex space forms, suggesting the possibility that the same phenomenon holds in all Riemannian isotropic spaces. Based on the description of the algebras the full array of kinematic formulas for invariant valuations and curvature measures in $S^6$ and $S^7$ is computed. A key technical point is an extension of the classical theorems of Klain and Schneider on simple valuations.