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Calibrations and laminations

Published online by Cambridge University Press:  15 June 2016

VICTOR BANGERT  and
XIAOJUN CUI
VICTOR BANGERT
Affiliation:
Mathematisches Institut der Universität Freiburg, Eckerstr. 1, D-79104 Freiburg i. Br., Germany. e-mail: victor.bangert@math.uni-freiburg.de
XIAOJUN CUI
Affiliation:
Department of Mathematics, Nanjing University, 22 Hankou Road, Nanjing, 210093, Jiangsu Province, People's Republic of China. e-mail: xjohncui@yahoo.com
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Abstract

A calibration of degree k ∈ ℕ on a Riemannian manifold M is a closed differential k-form θ such that the integral of θ over every k-dimensional, oriented submanifold N is smaller or equal to the Riemannian volume of N. A calibration θ is said to calibrate N if θ restricts to the oriented volume form of N. We investigate conditions on a calibration θ that ensure the existence of submanifolds calibrated by θ. The cases k = 1 and k > 1 turn out to be essentially different. Our main result says that, on a compact manifold M, a calibration θ calibrates a lamination if θ is simple, of class C1, and if θ has minimal comass norm in its cohomology class.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 1 , January 2017 , pp. 151 - 171
Copyright
Copyright © Cambridge Philosophical Society 2016 

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