  We discuss relevant scalar deformations of a holographic theory with a
compact boundary. An example of such a theory would be the global AdS$_4$ with
its spatially compact boundary $S^2$. To introduce a relevant deformation, we
choose to turn on a time-independent and spatially homogeneous non-normalizable
scalar operator with $m^2 = -2$. The finite size of a compact boundary cuts
down the RG flow at a finite length scale leading to an incomplete RG flow to
IR. We discuss a version of {\it incomplete} C-theorem and an {\it incomplete}
attractor like mechanism. We discuss the implication of our results for
entanglement entropy and geometric quantities like scalar curvature, volume and
mass scale of fundamental excitation of the how these quantities increase or
decrease (often monotonically) with the strength of the deformation. Thermal
physics of a holographic theory defined on a compact boundary is more
interesting than its non-compact counterpart. It is well known that with a
compact boundary, there is a possibility of a first order Hawking-Page
transition dual to a de-confinement phase transition. From a gravity
perspective, a relevant deformation dumps negative energy inside the bulk,
increasing the effective cosmological constant ($\Lambda$) of the AdS. Dumping
more negative energy in the bulk would make the HP transition harder and the
corresponding HP transition temperature would increase. However, we have found
the size of the BH at the transition temperature decreases.
