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The Decomposition of Global Conformal Invariants (AM-182)$
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Spyros Alexakis

Print publication date: 2012

Print ISBN-13: 9780691153476

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691153476.001.0001

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The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I

The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I

Chapter:
(p.297) Chapter Six The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I
Source:
The Decomposition of Global Conformal Invariants (AM-182)
Author(s):

Spyros Alexakis

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691153476.003.0006

This chapter takes up the proof of Lemma 4.24, which has two cases, A and B. The strategy goes as follows: It first repeats the ideas from Chapter 5 and derives a new local equation from the assumption of Proposition 4.13. However, it finds that this new equation is very far from proving the claim of Lemma 4.24. It then has to return to the hypothesis of Proposition 4.13 and extract an entirely new equation from its conformal variation. It proceeds with a detailed study of this new equation (again using the inductive assumption of Proposition 4.13); the result is a second new local equation which again is very far from proving the claim of our lemma. Next, it formally manipulates this second new local equation and adds it to the first one, and observes certain miraculous cancellations, which yield new local equations that can be collectively called the grand conclusion. Lemma 4.24 in case A then immediately follows from the grand conclusion.

Keywords:   lemmas, conformal invariants, Riemannian invariants

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