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Nonlinear Analysis

The quaternionic contact Yamabe problem and Bahri-Coron topological argument

Mohameden Ahmedou, Giessen University, Germany

Location:  Hill 705
Date & time: Wednesday, 03 October 2018 at 1:40PM - 3:00PM

ABSTRACT:  A quaternionic contact (qc) structure on a real \(4n+3\)-manifold is a co-dimension 3 distribution, locally given by a \(\R^3\)-valued 1- form \(\eta\). Such a structure generalize to the quaternionic algebra, geometric structures that are boundaries at Infinity of real and complex asymptotically hyperbolic spaces. In the real case these manifolds are simply conformal manifolds and in the complex case, they are Cauchy-Riemann CR manifolds. On such manifolds, O. Biquard defined linear connection preserving the qc structure.  A natural question, arising from the conformal invariance of qc structures is the quaternionic contact Yamabe problem: Find a qc conformally equivalent 1-form with constant qc scalar curvature (i.e. with respect to the Biquard connection).

Through the works of Wang('07), Kunkel('08), Ivanov-Wassilev ('11), Ivanov-Petkov('16), this problem has been solved on non spherical qc 
manifolds.  In this talk, we show how the so called Bahri-Coron topological argument solves the remaining case. Namely on locally spherical qc  manifolds.

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