Genericity of Nondegenerate Free Boundary CMC Embeddings
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Genericity of Nondegenerate Free Boundary CMC Embeddings Carlos Wilson Rodr´ıguez C´ardenas Abstract. Let Σn and M n+1 be smooth manifolds with smooth boundary. In this paper, following the techniques developed by White (Indiana Univ Math J 40:161–200, 1991) and Biliotti–Javaloyes–Piccione (Indiana Univ Math J, 1797–1830, 2009), we prove that, given a compact manifold with boundary Σn and a manifold with boundary M n+1 , for a generic set of Riemannian metrics on M every free boundary CMC embedding φ : Σ → M is non-degenerate. Mathematics Subject Classification. 53A10, 58E12, 49Q05. Keywords. Constant mean curvature hypersurfaces, genericity of bumpy metrics, variational problem, Jacobi operator.
1. Introduction In calculus of variations, there is a class of problems called isoperimetric; the classical isoperimetric problem consists in finding the minimum area among all hypersurfaces of a Riemannian manifold enclosing a region with prescribed volume. We know that solutions to this problem are hypersurfaces with constant mean curvature (in short CMC). More precisely, if ϕ : Σ → M is an immersion of an orientable n-dimensional compact manifold Σ into the (n + 1)-dimensional Riemannian manifold M , the condition that ϕ has constant mean curvature H0 is equivalent to the fact that ϕ is a critical point of the area functional defined in the space of embeddings of Σ in M that bound a region of fixed volume (see, for instance, [5]). The solutions of the isoperimetric problem correspond to minima of the constrained variational problem, however, it is interesting to study all critical points of the problem. One of the interesting questions concerning general CMC hypersurfaces is establishing the non-degeneracy as constrained critical points, and this paper deals with aspects of this question. If ϕt is a smooth variation of ϕ, t ∈ (−, ), ϕ0 = ϕ, such that Vt = V0 , for all t ∈ (−, ), where Vt is the volume of the region bounded by ϕt (Σ), a standard approach to find the solution of such a isoperimetric problem 0123456789().: V,-vol
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is to look the critical points of functional f(t) = At + λVt , At the area of ϕt , λ = const., which is the classical method of Lagrange multipliers. When λ = nH0 we have the aforementioned equivalence. In the case where M is a manifolds with boundary ∂M and Σ is also a manifold with boundary, the isoperimetric problem can be described as follows. One wants to minimize the area among all compact hypersurfaces diffeomorphic to Σ in M with boundary contained in ∂M and whose interior lies in the interior of M , and which divide M in two regions such that the closure of one of them is compact and with prescribed volume. The solutions of this problem, called free boundary CMC hypersurfaces, are the so-called normal CMC hypersurfaces. Let H0 be denote the value (constant) of the mean curvature of one such hypersurface. If H0 = 0 then we say that ϕ(Σ) is a orthogonal free boundary minimal hypersurface. A. Ros and E. Vergasta obtain results on the stability of
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