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We construct solutions of higher-dimensional Einstein gravity coupled tononlinear $\sigma$-model with cosmological constant. The $\sigma$-model can beperceived as exterior configuration of a spontaneously-broken $SO(D-1)$ globalhigher-codimensional "monopole". Here we allow the kinetic term of the$\sigma$-model to be noncanonical; in particular we specifically study aquadratic-power-law type. This is some possible higher-dimensionalgeneralization of the Bariola-Vilenkin (BV) solutions with $k$-global monopolestudied recently. The solutions can be perceived as the exterior solution of ablack hole swallowing up noncanonical global defects. Even in the absence ofcomological constant its surrounding spacetime is asymptotically non-flat; itsuffers from deficit solid angle. We discuss the corresponding horizons. For$\Lambda>0$ in $4d$ there can exist three extremal conditions (the cold,ultracold, and Nariai black holes), while in higher-than-four dimensions theextremal black hole is only Nariai. For $\Lambda<0$ we only have black holesolutions with one horizon, save for the $4d$ case where there can exist twohorizons. We give constraints on the mass and the symmetry-breaking scale forthe existence of all the extremal cases. In addition, we also obtain factorizedsolutions, whose topology is the direct product of two-dimensional spaces ofconstant curvature ($M_2$, $dS_2$, or $AdS_2$) with (D-2)-sphere. We study allpossible factorized channels.

Classical defects in higher-dimensional Einstein gravity coupled to nonlinear $\sigma$-models