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We study stable constant mean curvature (CMC) hypersurfaces $$\Sigma $$ Σ with free boundary in slabs in a product space $$M\times {\mathbb {R}},$$ M × R , where M is an orientable Riemannian manifold. We obtain a characterization of stable cylinders and prove that if $$\Sigma $$ Σ is not a cylinder then it is locally a vertical graph. Moreover, in case M is $${\mathbb {H}}^n,{\mathbb {R}}^n$$ H n , R n , or $${\mathbb {S}}_+^n,$$ S + n , if each component of $$\partial \Sigma $$ ∂ Σ is embedded, then $$\Sigma $$ Σ is rotationally invariant. When M has dimension 2 and Gaussian curvature bounded from below by a positive constant $$\kappa ,$$ κ , we prove there is no stable CMC with free boundary connecting the boundary components of a slab of width $$l\ge 4\pi /\sqrt{3\kappa }.$$ l ≥ 4 π / 3 κ . We also show that a stable capillary surface of genus 0 in a warped product $$[0,l]\times _f M$$ [ 0 , l ] × f M where $$M={\mathbb {R}}^2, {\mathbb {H}}^2$$ M = R 2 , H 2 , or $${\mathbb {S}}^2,$$ S 2 , is rotationally invariant. Finally, we prove that a stable CMC immersion of a closed surface in $$M\times {\mathbb {S}}^1(r),$$ M × S 1 ( r ) , where M is a surface with Gaussian curvature bounded from below by a positive constant $$\kappa $$ κ and $${\mathbb {S}}^1(r)$$ S 1 ( r ) the circle of radius r ,  lifts to $$M\times {\mathbb {R}}$$ M × R provided $$r\ge 4/\sqrt{3\kappa }$$ r ≥ 4 / 3 κ .

Stable Constant Mean Curvature Surfaces with Free Boundary in Slabs