  Active solids consume energy to allow for actuation, shape change, and wave
propagation not possible in equilibrium. For two-dimensional active surfaces,
powerful design principles exist that realise this phenomenology across systems
and length scales. However, control of three-dimensional bulk solids remains a
challenge. Here, we develop both a continuum theory and microscopic simulations
that describe an active surface wrapped around a passive soft solid. The
competition between active surface stresses and bulk elasticity leads to a
broad range of previously unexplored phenomena, which we dub active
elastocapillarity. In passive materials, positive surface tension rounds out
corners and drives every shape towards a sphere. By contrast, activity can send
the surface tension negative, which results in a diversity of stable shapes
selected by elasticity. We discover that in these reconfigurable objects,
material nonlinearity controls reversible switching and snap-through
transitions between anisotropic shapes, as confirmed by a particle-based
numerical model. These transition lines meet at a critical point, which allows
for a classification of shapes based on universality. Even for stable surfaces,
a signature of activity arises in the negative group velocity of surface
Rayleigh waves. These phenomena offer insights into living cellular membranes
and underpin universal design principles across scales from robotic
metamaterials down to shape-shifting nanoparticles.
