Here are two relatively recent papers. Ivan visits MO, so he may answer more definitively.

Izmestiev, Ivan. "Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional II: Smooth case." arXiv:1105.5067. (2011).

"The paper is centered around a new proof of the infinitesimal rigidity of smooth closed surfaces with everywhere positive Gauss curvature."

Martinez-Maure, Yves. "Rigidity and Bellows-type Theorem for hedgehogs." (2011). Author's link.

OP: "I'm curious about more recent research in this area." Here are two relatively recent papers. Ivan visits MO, so he may answer more definitively.

Izmestiev, Ivan. "Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional II: Smooth case." arXiv:1105.5067. (2011).

"The paper is centered around a new proof of the infinitesimal rigidity of smooth closed surfaces with everywhere positive Gauss curvature."

Martinez-Maure, Yves. "Rigidity and Bellows-type Theorem for hedgehogs." (2011). Author's link.