UZH-Logo

Maintenance Infos

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.

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.

Downloads

190 downloads since deposited on 23 Jun 2011
68 downloads since 12 months
Detailed statistics

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:05 Apr 2016 14:55
Number of Pages:122
Permanent URL: https://doi.org/10.5167/uzh-48318

Download

[img]
Preview
Content: Published Version
Filetype: PDF
Size: 1MB

TrendTerms

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.
You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.

Author Collaborations