  There are several examples in which algebraic properties of Jacobian algebras
from (unpunctured) Riemann surfaces can be computed from the geometry of the
Riemann surface.
  In this work, we compute the dimension of the Hochschild cohomology groups of
any Jacobian algebra from unpunctured Riemann surfaces. In those expressions
appear geometric objects of the triangulated surface, namely: the number of
internal triangles and certain types of boundaries. Moreover, we give geometric
conditions on the triangulated surface $(S,M,\mathbb T)$ such that the
Gerstenhaber algebra $\operatorname{HH}^*(A_{\mathbb T})$ has non-trivial
multiplicative structures.
  We also show that the derived class of Jacobian algebras from an unpunctured
surface $(S,M)$ is not always completely determined by the Hochschild
cohomology.
