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In the ¯rst Heisenberg group H1 with its sub-Riemannian structure generated by the horizontal subbundle, we single out a class of C2 non-characteristic entire intrinsic graphs which we call strict graphical strips. We prove that such strict graphical strips have vanishing horizontal mean curvature (i.e., they are H-minimal) and are unstable (i.e., there exist compactly supported deformations for which the second variation of the horizontal perimeter is strictly negative). We then show that, modulo left-translations and rotations about the center of the group, every C2 entire Hminimal graph with empty characteristic locus and which is not a vertical plane contains a strict graphical strip. Combining these results we prove the conjecture that in H1 the only stable C2 Hminimal entire graphs, with empty characteristic locus, are thevertical planes.

No abstract uploaded!

No abstract uploaded!

This paper considers the problem of solving systems of quadratic equations, namely, recovering an object of interest \mathbf {x}^{atural}\in\mathbb {R}^{n} from \mathbf {x}^{atural}\in\mathbb {R}^{n} quadratic equations/samples \mathbf {x}^{atural}\in\mathbb {R}^{n} , \mathbf {x}^{atural}\in\mathbb {R}^{n} . This problem, also dubbed as phase retrieval, spans multiple domains including physical sciences and machine learning.