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We prove a local rigidity result for infinitesimally rigid capillary surfacesin some Riemannian $3$-manifolds with mean convex boundary. We also derivebounds on the genus, number of boundary components and area of any compacttwo-sided capillary minimal surface with low index under certain assumptions onthe curvature of the ambient manifold and of its boundary.

Low index capillary minimal surfaces in Riemannian $3$-manifolds