preprint

Inserted: 2 apr 2026

Year: 2023

Abstract:

In this paper, we prove the existence of two capillary embedded geodesics with a contact angle $θ\in (0,π/2)$ on Riemannian $2$-disks with strictly convex boundary, where the absence of a simple closed geodesic loop based on a point of boundary is given. In particular, our condition contains the cases of Riemannian $2$-disks with strictly convex boundary, nonnegative Gaussian curvature and total geodesic curvature lower bound $π$ of the boundary. Moreover, by providing examples, we prove that our total geodesic curvature condition is sharp to admit a capillary embedded geodesic with a contact angle $θ\in (0,π/2)$ under the nonnegative interior Gaussian curvature condition. We also prove the existence of Morse Index $1$ and $2$ capillary embedded geodesics for generic metric under the assumptions above.