Index, vision number and stability of complete minimal surfaces

Eigenvalues of elliptic operators and geometric applications

“…The main point of the above proof was to show that the function u = n (x) · v has at least k components of constant sign. Having proved that, we could have used instead of Corollary 6.14 a result of Choe [7] about a vision number, which says that if this particular function u has k components of constant sign then ind (X) ≥ k − 1. We have preferred a more general approach based on q-massive sets, because this approach does not need a function u to be defined on the entire manifold X.…”

Complete Embedded Minimal Surfaces of Finite Total Curvature

“…Thus we have The following results allows us to compute the index of stability of the Meeks-Jorge n-noids, the Chen-Gackstatter surface and its generalizations to higher symmetry (See Remark 4.2). Choe [12], proved a weaker version of this proposition.…”

“…We end this chapter with a discussion of the work of Choe [12] about the vision number of a minimal surface with respect to a vector field V on R 3 . For a vector field V on S ⊂ R 3 , Choe defines the horizon of V to be H(S, V ) = {s ∈ S | V ∈ T s S} , and the vision number, to be v(S, V ) = # components of S − H(S, V ) .…”

Shadows and Convexity of Surfaces

“…H u := σ −1 u (0) is called the horizon [5] in the direction u. It is easy to see that in general ∂S u = H u = ∂S −u , where ∂ denotes the boundary; however, using Sard's theorem, we can show Proposition 2.1.…”