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

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

Gross morphology of the overproducing mutant <i>D-h</i> protein transgenic plant (right) and WT (left).

<p>Bar = 20 cm.</p

Positional cloning of the <i>D-h</i> gene.

<p>(a) Genetic mapping of the <i>D-h</i> locus with STS markers. (b) Fine mapping of the <i>D-h</i> locus with additional STS markers. (c) Candidate gene in the 48 kb genomic DNA region identified by fine mapping. (d) Co-segregation analysis in F₂ plants of the HD1×Hwacheongbyeo cross using STS marker S10460. A 583 bp PCR product was observed in the tall homozygotes, whereas a shorter 520 bp PCR product was observed in the dwarf homozygotes. In dwarf heterozygotes, both fragments were observed. Lane 1, Hwacheong; Lane 2, HD1; Lane 3, HD1×HwacheongF₁; Lanes 4–19, dwarf phenotype, Lanes 20 to 34, tall phenotype. (e) Genotype of the STS marker S10460 among the HD1 and 33 rice cultivars having normal clum length. Lane 1, HD1; Lanes 2–34, rice cultivars.</p

Comparison of the cDNA and predicted amino acid sequence alignments of <i>d-h</i>(WT)and<i>D-h</i>(MT).

<p>(a) Comparison of the cDNA sequences <i>d-h</i>(WT)and<i>D-h</i>(MT). Asterisks indicate single nucleotide substitution, <i>d-h</i> start codon, <i>D-h</i> start codon, and stop codon. (b) Sequence ekectrophoregrams of the RT-PCR products of <i>d-h</i>(WT)and<i>D-h</i>(MT). (c) Alignment of the predicted <i>d-h</i>protein with hypothetical proteins from <i>Zea mays</i> (NP_001147534), <i>Sorghum bicolor</i> (XP_002454989), and <i>Hordeumvulgare</i> gene (BAJ91554, AK360345).</p

Subcellular localization of the <i>D-h</i>-green fluorescent protein (GFP) fusion protein in transformed onion epidermal cells.

<p>(a, b) Expression of the control CaMv35S::GFP construct and corresponding DIC image. (c, d) Expression of the CaMv35S::<i>d-h</i>-GFP construct and corresponding DIC image. (e, f) Expression of the CaMv35S::<i>D-h</i>-GFP construct and corresponding DIC image. Bars = 50 µm.</p

Characterization of the HD1.

<p>(a) Seedling phenotype of the HD1 and WT. Bar = 5 cm. (b) The HD1 and WT at 3 weeks after the heading stage. Bar = 20 cm. (c) Mature panicle and seeds of the HD1 and WT. Bottom bar = 5 cm (panicle), top bar = 5 mm (seeds). (d) Internode lengths of the HD1 and WT. The average values were calculated from measurements of at least 10 plants. (e) Parenchyma cells in the first internode in the HD1 and WT. Bars = 50 µm. (f) Quantitative measurements of the cell length and cell width of the HD1 and WT (n = 20). Data are mean±SD. Asterisks indicate a significant difference at P≤0.01 compared with the WT by Student's t test. (g) Induction of α-amylase activity by gibberellic acid (GA₃) in the HD1 and WT. (h) Elongation of the second leaf sheath in the HD1 and the WT in response to GA₃. Data are mean±SD (n = 10). Asterisks indicate a significant difference at P≤0.01 compared with the WT by Student's t test.</p

Expression patterns of the <i>D-h</i> gene.

<p>(a) QuantitativeRT-PCR analysis of the <i>D-h</i> gene in organs. (b) Semi-quantitative RT-PCR analysis of the <i>D-h</i> gene in organs. (c)–(i) GUS activity detected in the <i>D-h</i> promoter::GUS transgenic plants.</p

Inactivation of the gene causes genic male sterility and endosperm chalkiness in rice ( L.)-3

<p><b>Copyright information:</b></p><p>Taken from "Inactivation of the gene causes genic male sterility and endosperm chalkiness in rice ( L.)"</p><p></p><p>The Plant Journal 2008;54(2):190-204.</p><p>Published online 01 Apr 2008</p><p>PMCID:PMC2327258.</p><p>© 2008 The Authors Journal compilation © 2008 Blackwell Publishing Ltd</p

Saturated map of the region containing the ‘’ locus and candidate genes

<p><b>Copyright information:</b></p><p>Taken from "Inactivation of the gene causes genic male sterility and endosperm chalkiness in rice ( L.)"</p><p></p><p>The Plant Journal 2008;54(2):190-204.</p><p>Published online 01 Apr 2008</p><p>PMCID:PMC2327258.</p><p>© 2008 The Authors Journal compilation © 2008 Blackwell Publishing Ltd</p