Membrane domains known as rafts are rich in cholesterol and sphingolipids, and are thought to be thicker than the surrounding membrane. If so, monolayers should elastically deform so as to avoid exposure of hydrophobic surfaces to water at the raft boundary. We calculated the energy of splay and tilt deformations necessary to avoid such hydrophobic exposure. The derived value of energy per unit length, the line tension gamma, depends on the elastic moduli of the raft and the surrounding membrane; it increases quadratically with the initial difference in thickness between the raft and surround; and it is reduced by differences, either positive or negative, in spontaneous curvature between the two. For zero spontaneous curvature, gamma is approximately 1 pN for a monolayer height mismatch of approximately 0.3 nm, in agreement with experimental measurement. Our model reveals conditions that could prevent rafts from forming, and a mechanism that can cause rafts to remain small. Prevention of raft formation is based on our finding that the calculated line tension is negative if the difference in spontaneous curvature for a raft and the surround is sufficiently large: rafts cannot form if gamma < 0 unless molecular interactions (ignored in the model) are strong enough to make the total line tension positive. Control of size is based on our finding that the height profile from raft to surround does not decrease monotonically, but rather exhibits a damped, oscillatory behavior. As an important consequence, the calculated energy of interaction between rafts also oscillates as it decreases with distance of separation, creating energy barriers between closely apposed rafts. The height of the primary barrier is a complex function of the spontaneous curvatures of the raft and the surround. This barrier can kinetically stabilize the rafts against merger. Our physical theory thus quantifies conditions that allow rafts to form, and further, defines the parameters that control raft merger.