Uniqueness Results for Free Boundary Minimal Hypersurfaces in Conformally Euclidean Balls and Annular Domains

Abstract

In this work, we investigate the existence of compact free-boundary minimal hypersurfaces immersed in several domains. Using an original integral identity for compact free-boundary minimal hypersurfaces that are immersed in a domain whose boundary is a regular level set, we study the case where this domain is a quadric or, more generally, a rotational domain. This existence study is done without topological restrictions. We also obtain a new gap theorem for free boundary hypersurfaces immersed in an Euclidean ball and in a rotational ellipsoid.

Gap phenomena for constant mean curvature surfaces

“…We point out that higher dimensional versions of this result where recently obtained by the first and the third named authors with Gonçalves in [4], and topological versions were established by the second named author, Mendes and Vitório in [10].…”

“…Therefore, $$\gamma (t)$$ is a critical point of $$v:\Sigma \rightarrow \mathbb {R}$$, for all $$t\in [0,1]$$. Since the function $$\Psi$$ is radially increasing (see [4]), we have that the set $$\mathcal {C}$$ is contained in the interior of $$\Sigma$$. By [11, Theorem 2.5], the critical points in the nodal set $$v^{-1}(0)$$ of a nontrivial solution to Equation (3.1) are isolated.…”

“…Therefore, 𝛾(𝑡) is a critical point of 𝑣 ∶ Σ → ℝ, for all 𝑡 ∈ [0, 1]. Since the function Ψ is radially increasing (see [4]), we have that the set  is contained in the interior of Σ. By [11, Theorem 2.5], the critical points in the nodal set 𝑣 −1 (0) of a nontrivial solution to Equation (3.1) are isolated.…”

“…The submanifolds satisfying ‖𝐴‖ 2 = 𝑛∕(2 − 1∕𝑘) were latter classified by the works of Lawson [16], when 𝑘 = 1, and Chern, do Carmo, and Kobayshi [13], for any 𝑘. They are the Veronese minimal surface in 𝕊 4 and the Clifford tori.…”

Heintze–Karcher and Jellett‐type theorems in conformally flat spaces

“…It is important to point out that Jellett already used the so-called Minkowisky formula to obtain his result. Around one century after, Hopf [12] presented a generalization of this result showing that an immersed constant mean curvature surface 𝑀 ⊂ ℝ 3 with the topology of a sphere is also a round sphere. In the eighties of the last century, Hsiang et al [13] presented a new class of spherical immersed hypersurfaces of the constant mean curvature in ℝ 2𝑛 that are not round spheres.…”