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On the building dimension of closed cones and Almgren's stratification principle


Author: Andrea Marchese
Journal: Proc. Amer. Math. Soc. 143 (2015), 3041-3046
MSC (2010): Primary 49Q05, 49Q20
Published electronically: February 27, 2015
MathSciNet review: 3336628
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Abstract: In this paper we disprove a conjecture stated in Stratification of minimal surfaces, mean curvature flows, and harmonic maps by Brian White on the equality of two notions of dimension for closed cones. Moreover, we answer in the negative the following question, raised in the same paper. Given a compact family $ \mathcal {C}$ of closed cones and a set $ S$ such that every blow-up of $ S$ at every point $ x\in S$ is contained in some element of $ \mathcal {C}$, is it true that the dimension of $ S$ is smaller than or equal to the largest dimension of a vector space contained is some element of $ \mathcal {C}$?


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Andrea Marchese
Affiliation: Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstrasse 22, 04103 Leipzig, Germany
Email: marchese@mis.mpg.de

DOI: https://doi.org/10.1090/S0002-9939-2015-12497-5
Keywords: Building dimension, stratification
Received by editor(s): November 4, 2013
Received by editor(s) in revised form: March 13, 2014
Published electronically: February 27, 2015
Communicated by: Tatiana Toro
Article copyright: © Copyright 2015 American Mathematical Society