  We study the near horizon structure of Extremal Vanishing Horizon (EVH) black
holes, extremal black holes with vanishing horizon area with a vanishing
one-cycle on the horizon. We construct the most general near horizon EVH and
near-EVH ansatz for the metric and other fields, like dilaton and gauge fields
which may be present in the theory. We prove that (1) the near horizon EVH
geometry for generic gravity theory in generic dimension has a three
dimensional maximally symmetric subspace; (2) if the matter fields of the
theory satisfy strong energy condition either this 3d part is AdS$_3$, or the
solution is a direct product of a locally 3d flat space and a $d-3$ dimensional
part; (3) these results extend to the near horizon geometry of near-EVH black
holes, for which the AdS$_3$ part is replaced with BTZ geometry. We present
some specific near horizon EVH geometries in 3, 4 and 5 dimensions for which
there is a classification. We also briefly discuss implications of these
generic results for generic (gauged) supergravity theories and also for the
thermodynamics of near-EVH black holes and the EVH/CFT proposal.
