Free boundary constant mean curvature surfaces in a strictly convex three-manifold

Gap phenomena for constant mean curvature surfaces

“…Therefore, $$\mathcal {C}$$ is a geodesic circle lying in a totally geodesic plane $$\Pi$$ passing through the center of $$\mathbb {B}^3_r$$. Using [20, Lemma 4.2], we conclude that $$\Sigma$$ is a Delaunay surface meeting $$\Pi$$ orthogonally along the $$\mathcal {C}$$. Note that as the rotation axis of $$\Sigma$$ passes through at the center of the circle $$\mathcal {C}$$ and $$\mathcal {C}$$ is a great circle of a sphere $$\mathbb {S}^2_{r_0}$$ centered at the origin, we conclude that $$\Sigma$$ is centered at the origin.…”

Rigidity and vanishing theorems for submanifolds with free boundary in the unit ball

“…Theorem 1.1 was generalized by Barbosa-Cavalcante-Pereira [3] to constant mean curvature surfaces with free boundary in 𝑩 3 under the pinching condition |𝑥 ⟂ | 2 |𝜙| 2 (𝑥) ⩽ 1 2 (2 + ⟨𝑯, 𝑥⟩) 2 for all 𝑥 ∈ 𝑀, where 𝑯 is the mean curvature vector and 𝜙 is the trace-free second fundamental form. Min-Seo [32] proved a similar gap theorem for free boundary minimal surfaces in a geodesic ball of 3-dimensional space forms. The higher codimension analogue of Theorem 1.1 was proved by Barbosa-Viana [8], which states that an oriented compact minimal surface 𝑀 with free boundary in 𝑩 2+𝑝 satisfying |𝑥 ⟂ | 2 |𝒉| 2 (𝑥) ⩽ 2 for all 𝑥 ∈ 𝑀 is either an equatorial disk or the critical catenoid inside a 3-dimensional linear subspace of 𝑩 2+𝑝 .…”

“…Theorem 1.1 was generalized by Barbosa–Cavalcante–Pereira [3] to constant mean curvature surfaces with free boundary in $$\bm{B}^3$$ under the pinching condition $$\vert x^{\perp } \vert ^2 \vert {\phi } \vert ^2 (x) \leqslant \frac{1}{2} (2 + \langle \bm{H}, x \rangle)^2$$ for all $$x \in M$$, where $$\bm{H}$$ is the mean curvature vector and $${\phi }$$ is the trace‐free second fundamental form. Min–Seo [32] proved a similar gap theorem for free boundary minimal surfaces in a geodesic ball of 3‐dimensional space forms. The higher codimension analogue of Theorem 1.1 was proved by Barbosa–Viana [8], which states that an oriented compact minimal surface $$M$$ with free boundary in $$\bm{B}^{2 + p}$$ satisfying $$\vert x^{\perp } \vert ^2 \vert \bm{h} \vert ^2 (x) \leqslant 2$$ for all $$x \in M$$ is either an equatorial disk or the critical catenoid inside a 3‐dimensional linear subspace of $$\bm{B}^{2 + p}$$.…”