Isoperimetry and volume preserving stability under symmetry constraint in space forms.

In this talk we will discuss the isoperimetric problem and volume preserving stability under additional restriction of symmetry from the ambient space. The scenarios of interest include the Euclidean ball, the round sphere, and the Gaussian space. Among other partial results, we show an isoperimetric characterization for slabs bounded by two parallel planes for a certain range of the Gaussian volume.