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

TL;DR

This paper proves the existence of two capillary embedded geodesics with specific contact angles on Riemannian 2-disks under certain convexity and curvature conditions, using min-max methods, and demonstrates the sharpness of these conditions.

Contribution

It introduces a min-max construction method for capillary geodesics on Riemannian disks with convex boundary, establishing existence results under sharp curvature conditions.

Findings

Existence of two capillary embedded geodesics with contact angle in (0, π/2).

Sharpness of total geodesic curvature condition for existence.

Existence of Morse index 1 and 2 capillary geodesics for generic metrics.

Abstract

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.