  This paper treats boundary conditions on black hole horizons for the full
3+1D Einstein equations. Following a number of authors, the apparent horizon is
employed as the inner boundary on a space slice. It is emphasized that a
further condition is necessary for the system to be well posed; the
``prescribed curvature conditions" are therefore proposed to complete the
coordinate conditions at the black hole. These conditions lead to a system of
two 2D elliptic differential equations on the inner boundary surface, which
coexist nicely to the 3D equation for maximal slicing (or related slicing
conditions). The overall 2D/3D system is argued to be well posed and globally
well behaved. The importance of ``boundary conditions without boundary values"
is emphasized. This paper is the first of a series. This revised version makes
minor additions and corrections to the previous version.
