Abstract
We present some geometric applications, of global character, of the bubbling analysis developed by Buzano and Sharp for closed minimal surfaces, obtaining smooth multiplicity one convergence results under upper bounds on the Morse index and suitable lower bounds on either the genus or the area. For instance, we show that given any Riemannian metric of positive scalar curvature on the three-dimensional sphere the class of embedded minimal surfaces of index one and genus gamma is sequentially compact for any gamma ge 1. Furthemore, we give a quantitative description of how the genus drops as a sequence of minimal surfaces converges smoothly, with mutiplicity mge 1, away from finitely many points where curvature concentration may happen. This result exploits a sharp estimate on the multiplicity of convergence in terms of the number of ends of the bubbles that appear in the process.
Highlights
Let (N 3, g) be a compact Riemannian manifold of dimension three, without boundary
We shall be concerned here with certain global phenomena related to the convergence of a sequence of closed minimal surfaces, smoothly embedded in N, of bounded area and index
In [31], the fourth-named author proved a compactness theorem for the set M(, I ), and in later joint work Ambrozio-Carlotto-Sharp [2] proved a similar compactness theorem for Mp(, μ): given a sequence of minimal surfaces {Mk} in M(, I ) (or Mp(, μ)), there is some smooth limit in the same class to which the sequence sub-converges smoothly and graphically away from a discrete set Y on the limit, where one witnesses the formation of necks
Summary
Let (N 3, g) be a compact Riemannian manifold of dimension three, without boundary. We shall be concerned here with certain global phenomena related to the convergence of a sequence of closed minimal surfaces, smoothly embedded in N , of bounded area and index. 2. We will first employ the identity above together with another ancillary result, Lemma 12, to prove a strong compactness theorem for minimal surfaces of bounded index inside a 3manifold of positive scalar curvature. Colding and De Lellis presented in [10] a method to construct 3-manifolds of positive scalar curvature containing sequences of embedded, orientable minimal surfaces of any fixed genus γ , that converge to a minimal lamination with two singular points on a strictly stable minimal sphere. The topological lower bound one has to assume needs to be stronger due to the lack of classification results for complete (embedded) minimal surfaces ⊂ R3 of index equal to any natural number greater or equal than four We will circumvent this obstacle by exploiting the index estimates obtained by Chodosh and Maximo in [7,8]. The two results above can both be regarded as instances of a local-to-global correspondence, meaning that the understanding of complete minimal surfaces in the Euclidean space R3 is exploited to extract information for the blow-up analysis at the singular points of
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