Acknowledgements ................................................ 5
Chapter 1 Introduction ......................................... 7
1.1 Preliminaries ............................................. 15
1.1.1 Vector-valued forms ................................ 16
1.1.2 LP spaces of differential forms .................... 19
1.1.3 Bundle-valued forms ................................ 21
Chapter 2 The curvature of non-smooth connections ............. 25
2.1 Smooth connections and curvature .......................... 27
2.2 Non-smooth connections .................................... 30
2.3 Holonomy bounds for smooth connections .................... 31
2.4 Smooth approximation of non-smooth connections ............ 40
2.5 Holonomy bounds for non-smooth connections ................ 46
2.5.1 Almost every plane is typical ...................... 50
2.5.2 The proof of Theorem 1.1 ........................... 54
2.6 A Frobenius theorem for non-smooth connections ............ 56
2.6.1 Frobenius' theorem for Lipschitz distributions ..... 59
Chapter 3 Quasiconformal co-frames and p-harmonic maps to
SO(n) .......................................................... 63
3.1 p-harmonic maps and SO(n) ................................. 65
3.2 The Euler Lagrange equations .............................. 67
3.2.1  -harmonic maps to SO(n) ........................... 71
3.2.2 Minimisers in the class of an exact frame .......... 73
3.3 Minimisers of exterior energy ............................. 75
3.4 Another exterior energy ................................... 83
References ..................................................... 89
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