Global solutions for higher-dimensional stretched small black holes

Small black holes in heterotic string theory have vanishing horizon area at the supergravity level, but the horizon is stretched to the finite radius $AdS_2 \times S^{D-2}$ geometry once higher curvature corrections are turned on. This has been demonstrated to give good agreement with microscopic entropy counting. Previous considerations, however, were based on the classical local solutions valid only in the vicinity of the event horizon. Here we address the question of global existence of extremal black holes in the $D$-dimensional Einstein-Maxwell-Dilaton theory with the Gauss-Bonnet term introducing a variable dilaton coupling $a$ as a parameter. We show that asymptotically flat black holes exist only in a bounded region of the dilaton couplings $0 < a < a_{\rm cr}$ where $a_{\rm cr}$ depends on $D$. For $D \geq 5$ (but not for $D = 4$) the allowed range of $a$ includes the heterotic string values. For $a > a_{\rm cr}$ numerical solutions meet weak naked singularities at finite radii $r = r_{\rm cusp}$ (spherical cusps), where the scalar curvature diverges as $|r - r_{\rm cusp}|^{-1/2}$. For $D \geq 7$ cusps are met in pairs, so that solutions can be formally extended to asymptotically flat infinity choosing a suitable integration variable. We show, however, that radial geodesics cannot be continued through the cusp singularities, so such a continuation is unphysical.