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

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