Second variation of one-sided complete minimal surfaces

Abstract

Abstract. The stability and the index of complete one-sided minimal surfaces of certain 3-dimensional Riemannian manifolds with positive scalar curvature are studied.

Comparing the Morse index and the first Betti number of minimal hypersurfaces

“…However, it is still possible to obtain similar bounds for the Morse index of their embedded hypersurfaces. The case of closed embedded minimal surfaces in S 1 × S 2 was already studied in detail by Urbano in [38] (see Theorem 4.8 in [38] for the precise statement of his result). We remark that the proof in [38] used the coordinates of harmonic one-forms as test functions for the index form, i.e., the proof used formula (8) specialized to the case of the canonical product embedding of…”

“…In particular, he proved that the index of such surfaces is bounded from below by an affine function (whose coefficients do not depend on the particular surface) of their first Betti numbers (see Theorem 16 in [29]). In [38], Urbano used essentially the same method to prove the analogous result for closed minimal surfaces inside the product of the unit circle with a two-dimensional sphere of radius greater or equal than one.…”

Gaussian Harmonic Forms and two-dimensional self-shrinking surfaces

“…Instead of using harmonic one forms to represent the cohomology class of ω 0 as in Ros [6] and Urbano [7], we will use the form minimizing the L 2 λ 2 (Σ) norm where ω = ω 0 + df (Remember, λ 2 ≡ e −|x| 2 /4 ). To find the Euler-Lagrange equation, assume the minimum is achieved by a closed form ω.…”

“…Following the work of Ros [6] and Urbano [7] on the Jacobi operator on minimal surfaces, we may give lower bounds for the index of L if we have a condition on the principal curvatures. That is, we have Theorem 4.1.…”

Compact embedded minimal surfaces in $\mathbb{S}^2 \times \mathbb{S}^1$

“…They are also singly periodic and produce embedded minimal tori in S 2 × S 1 (r) for all r > 0 (see also [20, §7] for a beautiful description). Note that, thanks to [24,Theorem 4.8], a compact minimal surface Σ immersed in S 2 × S 1 (r), r ≥ 1, foliated by circles has index greater than or equal to one (the index is one only for r = 1 and Σ = Γ × S 1 (1), Γ ⊂ S 2 being a great circle). A similar estimation of the index for r < 1 remains an open question (see [24,Remark 4.9]).…”