Min-max construction of two capillary embedded geodesics on Riemannian $2$-disks

In this paper, we prove the existence of two capillary embedded geodesics with a contact angle $\theta \in (0,\pi/2)$ 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 $\pi$ 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 $\theta \in (0,\pi/2)$ 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.