Genus bounds for minimal surfaces arising from min-max constructions

By Camillo De Lellis, F. Pellandini. 2009

58 citations recorded by the source corpus.

Abstract In this paper we prove genus bounds for closed embedded minimal surfaces in a closed 3-dimensional manifold constructed via min-max arguments. A stronger estimate was announced by Pitts and Rubinstein but to our knowledge its proof has never been published. Our proof follows ideas of Simon and uses an extension of a famous result of Meeks, Simon and Yau on the convergence of minimizing sequences of isotopic surfaces. This result is proved in the second part of the paper.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/612f8d5dd2a9d293cc574b56311a63224fd50ea4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Statistics and Probability

Citations in the corpus

Morse index of multiplicity one min-max minimal hypersurfaces (2015, unknown): “Genus bounds were proven recently by Ketover [15] (see also De Lellis-Pellandini [12]), as conjectured by Pitts and Rubinstein [26].”
A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary (2012, methodology): “Following [5], for sufficiently small, there exists natural numbers N. / andmi .”
Min-max for phase transitions and the existence of embedded minimal hypersurfaces (2015, methodology): “We refer the reader to [7, 17, 20, 23, 43, 44] for applications to compact manifolds; [6, 26] for applications to non-compact manifolds; [18] for applications to a free-boundary situation; and to Marques-Neves [22, 21] for the solution of the long-standing Willmore conjecture and the existence of in”
Rigidity of min-max minimal spheres in three-manifolds (2011, background): “Conditions (b) and (d) are stated slightly differently from the corresponding conditions in [12] in order to allow Σt to be a graph for some t ∈ T .”
Min-max theory for minimal hypersurfaces with boundary (2016, background): “Part of this program was carried out later in the papers [8], [9], [19] and [7], whereas several other questions raised in [28,29] constitute an active area of research right now (see for instance [25]).”