  Constraints on the geometries of static spherically symmetric black holes are
obtained by requiring that the spacetime curvature be analytic at the event
horizon. Further constraints are obtained by requiring that the semiclassical
trace equation be satisfied in the case that only conformally invariant fields
are present. It is found that there exists a range of sizes for which zero
temperature black holes do not exist. The range depends on the number and types
of quantized fields present.
