  In multicomponent membranes, internal scalar fields may couple to membrane
curvature, thus renormalizing the membrane elastic constants and destabilizing
the flat membranes. Here, a general elasticity theory of membranes is
considered that employs a quartic curvature expansion. The shape of the
membrane and its deformation energy near a long rod-like inclusion are studied
analytically. In the limit where one can neglect the end-effects, the nonlinear
response of the membrane to such inclusions is found in exact form. Notably,
new shape solutions are found when the membrane is curvature unstable,
manifested by a negative rigidity. Near the instability point (i.e. at
vanishing rigidity), the membrane is stabilized by the quartic term, giving
rise to a new length scale and new scale exponents for the shape and its energy
profile. The contact angle induced by an applied force at the inclusion
provides a method to experimentally determine the quartic curvature modulus.
