Rigidity of Free Boundary Surfaces in Compact 3-Manifolds with Strictly Convex Boundary

Abstract

Abstract. In this paper we obtain an analogue of Toponogov theorem in dimension 3 for compact manifolds M 3 with nonnegative Ricci curvature and strictly convex boundary ∂M . Here we obtain a sharp upper bound for the length L(∂Σ) of the boundary ∂Σ of a free boundary minimal surface Σ 2 in M 3 in terms of the genus of Σ and the number of connected components of ∂Σ, assuming Σ has index one. After, under a natural hypothesis on the geometry of M along ∂M , we prove that if L(∂Σ) saturates the respective upper bound, then M 3 is isometric to the Euclidean 3-ball and Σ 2 is isometric to the Euclidean disk. In particular, we get a sharp upper bound for the area of Σ, when M 3 is a strictly convex body in R 3 , which is saturated only on the Euclidean 3-balls (by the Euclidean disks). We also consider similar results for free boundary stable CMC surfaces.

Rigidity of capillary surfaces in compact 3-manifolds with strictly convex boundary

“…By [3, Theorem 7.2], there exists a proper conformal branched cover satisfying , where is the Euclidean unit disk. By [6, Lemma 2.1], we can assume . Then, using f i as a test function, we obtain …”

“…Let be a proper conformal branched cover as in the proof of Theorem 1. Taking in [6, Lemma 2.1], we can assume for . Because Σ is stable …”

“…Critical points for this problem are CMC or minimal hypersurfaces Σ ⊂ M meeting ∂M orthogonally along ∂Σ, and they are known as CMC or minimal hypersurfaces with free boundary. In the last few years, this subject have been studied by many authors, for example, [1,2,6,[8][9][10][11][12].…”

Area estimates and rigidity of non-compact 𝐻-surfaces in 3-manifolds

“…For example, assuming a lower bound on the scalar curvature and that the surface is minimizing or does have index 1 with respect to the functional Area − H · V olume, some rigidity results were obtained in [7,26,5,6,22,16]. There are also related results for free boundary minimal surfaces in 3-manifolds with boundary, see [3,19], and for MOTS in a spacetime, see [12,20]. See also [4,21] for results in higher dimensions.…”

Compactness of the Space of Free Boundary CMC Surfaces with Bounded Topology

“…Secondly, there is no suitable isoperimetric inequality that allows us to remove the bound on the area. One could exchange the length bound condition by stability of the surface (see [11]) but as seen on [2], in particular, stable free boundary CMC surfaces have bounded topology so this condition would be topologically restrictive. Thirdly, even under positive Ricci curvature of the ambient space a sequence of free boundary CMC surfaces may have a neck-pinching phenomenum where the norm of the second fundamental form blows-up at a point in the limit.…”