Dual quadratic differentials and entire minimal graphs in Heisenberg space

We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces $\mathbb{L}(\kappa,\tau)$ with isometry group of dimension 4, which are dual to the Abresch-Rosenberg differentials in the Riemannian counterparts $\mathbb{E}(\kappa,\tau)$, and obtain some consequences. On the one hand, we give a very short proof of the Bernstein problem in Heisenberg space, and provide a geometric description of the family of entire graphs sharing the same differential in terms of a 2-parameter conformal deformation. On the other hand, we prove that entire minimal graphs in Heisenberg space have negative Gauss curvature.Comment: 19 page

L'Athlète : journal hebdomadaire de tous les sports

23 juillet 19301930/07/23 (N688)-1930/07/23.Appartient à l’ensemble documentaire : Aquit

Additional file 4: Figure S3. of Sugary Endosperm is Modulated by Starch Branching Enzyme IIa in Rice (Oryza sativa L.)

OsISA1 expression patterns in different organs and at different stages of seed development using qRT-PCR analysis. (a) Transcript levels decreased in seed (10 DAF) of the sug-h mutant. (b) OsISA1 expression in 10 and 20 DAF seeds decreased in the sug-h mutant. All data are mean ± SD (n = 3). Statistical significance was determined using Student’s t-test (*P < 0.05, **P < 0.01). WT, wild-type rice (Hwacheong); DAF, days after flowering. (PDF 51 kb