From Grushin to Heisenberg via an isoperimetric problem ∗ Nicola Arcozzi and Annalisa Baldi † Abstract The Grushin plane is a right quotient of the Heisenberg group. Heisenberg geodesics’ projections are solutions of an isoperimetric problem in the Grushin plane. 1 Introduction It is a known fact that there is a correspondence between isoperi- metric problems in Riemannian surfaces and sub-Riemannian geome- tries in three-dimensional manifolds. The most significant example is the isoperimetric problem in the plane, corresponding to the sub- Riemannian geometry of the Heisenberg group H. We briefly recall this connection following the exposition in [Mont]. Consider, on the Euclidean plane, the one-form α = 1 2 (xdy − ydx), which satisfies dα = dx ∧ dy and which vanishes on straight lines through the origin. By Stokes’ Theorem, the signed area enclosed by a curve γ is  γ α. Let c :[a, b] → R 2 be a curve. For each s in [a, b], let γ s be the union of the curve c restricted to [a, s], of the segment of straight line joining c(s) with the origin O and of the segment of straight line joining O with c(a). Let C :[a, b] → R 3 be the curve * MSC (2000): 43A80, 58E10, 49Q20. Keywords: Heisenberg group, Grushin plane, Isoperimetric problem † Investigation supported by University of Bologna, funds for selected research topics. The first author was supported by Italian Minister of Research, COFIN project ”Analisi Armonica”. The second author was supported by GNAMPA of INdAM, Italy, project “Analysis in metric spaces and subelliptic equations” and by MURST, Italy. 1