Vol. 7, No. 1, 2014

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Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications

Paul Laurain and Tristan Rivière

Vol. 7 (2014), No. 1, 1–41

We establish a quantization result for the angular part of the energy of solutions to elliptic linear systems of Schrödinger type with antisymmetric potentials in two dimensions. This quantization is a consequence of uniform Lorentz–Wente type estimates in degenerating annuli. Moreover this result is optimal in the sense that we exhibit a sequence of functions satisfying our hypothesis whose radial part of the energy is not quantized. We derive from this angular quantization the full energy quantization for general critical points to functionals which are conformally invariant or also for pseudoholomorphic curves on degenerating Riemann surfaces.

analysis of PDEs, differential geometry
Mathematical Subject Classification 2010
Primary: 35J20, 35J60, 53C42, 58E20, 35J47
Secondary: 49Q05, 53C21, 32Q65
Received: 1 December 2011
Revised: 7 February 2013
Accepted: 3 April 2013
Published: 7 May 2014
Paul Laurain
Department of Mathematics
IMJ-Université Paris 7
75013 Paris
Tristan Rivière
Department of Mathematics
ETH Zentrum
CH-8093 Zürich