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Min-max constructions of minimal surfaces in closed Riemannian manifolds


Tasnady, D. Min-max constructions of minimal surfaces in closed Riemannian manifolds. 2011, University of Zurich, Faculty of Science.

Abstract

We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces
in closed smooth (n+1)-dimensional Riemannian manifolds, a
theorem proved first by Pitts for 2 ≤ n ≤ 5 and extended later by Schoen
and Simon to any n.
Our proof follows Pitts’ original idea to implement a min-max construction.
We introduce some new ideas that allow us to shorten parts of Pitts’
proof – a monograph of about 300 pages – dramatically.
Pitts and Rubinstein announced an index bound for the minimal surface
obtained by the min-max construction. To our knowledge a proof has
never been published. We refine the analysis of our interpretation of the
construction to draw some conclusions that could be helpful to prove the
index bound.

Abstract

We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces
in closed smooth (n+1)-dimensional Riemannian manifolds, a
theorem proved first by Pitts for 2 ≤ n ≤ 5 and extended later by Schoen
and Simon to any n.
Our proof follows Pitts’ original idea to implement a min-max construction.
We introduce some new ideas that allow us to shorten parts of Pitts’
proof – a monograph of about 300 pages – dramatically.
Pitts and Rubinstein announced an index bound for the minimal surface
obtained by the min-max construction. To our knowledge a proof has
never been published. We refine the analysis of our interpretation of the
construction to draw some conclusions that could be helpful to prove the
index bound.

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Additional indexing

Item Type:Dissertation
Referees:De Lellis C, Kappeler T
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2011
Deposited On:23 Jun 2011 11:32
Last Modified:17 Feb 2018 13:25
Number of Pages:122
OA Status:Green

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