Some classifications of biharmonic hypersurfaces with constant scalar curvature

We give some classifications of biharmonic hypersurfaces with constant scalar curvature. These include biharmonic Einstein hypersurfaces in space forms, compact biharmonic hypersurfaces with constant scalar curvature in a sphere, and some complete biharmonic hypersurfaces of constant scalar curvature in space forms and in a non-positively curved Einstein space. Our results provide additional cases (Theorem 2.3 and Proposition 2.8) that supports the conjecture that a biharmonic submanifold in a sphere has constant mean curvature, and two more cases that support Chen's conjecture on biharmonic hypersurfaces (Corollaries 2.2,2.7).Comment: 11 page

Additional file 1 of The core bacterial microbiome of banana (Musa spp.)

Additional file 1: Table S1. The locations and basic properties of the five soils used to grow Musa spp. in the pot experiment of this study. Table S2. Musa spp. genotypes included in our survey of field-grown plants in the Australian Banana Germplasm Collection. Table S3. Studies included in a meta-analysis of the prevalence of core microbes identified in this study in other studies of bacteria associated with Musa spp. Fig. S1. A map showing, in black, countries included in the meta-analysis of the bacterial microbiome of Musa spp. Countries in grey are major producers (FAOSTAT, 2022) for which data were not available and were not included in meta-analysis. Table S4. The impact of soil, genotype and plant compartment on alpha diversity metrics assessed by ANOVA from the bacterial microbiome of Musa spp. These results derive from our pot experiment which included five distinct soils, three Musa spp. genotypes, and eight compartments, each with 10 replicates. Table S5. The impact of soil, genotype and plant compartment on bacterial community composition as represented by Weighted UniFrac distances using PERMANOVA. These results derive from our pot experiment which included five distinct soils, three Musa spp. genotypes, and eight compartments, each with 10 replicates. Table S6. Average percentage similarity plus/minus the standard deviation of various Musa spp. plant compartments. Percentages were produced using a Bayesian approach implemented through SourceTracker. Table S7. The influence of soil on the alpha diversity of bacterial communities within each compartment, as assessed by ANOVA. The results are for the Musa (AAA Group, Cavendish Subgroup) ‘Williams’ plants grown in five distinct soils in our pot experiment. Table S8. The influence of soil on the composition of bacterial communities, as represented by Weighted UniFrac distances, within compartments as assessed using PERMANOVA. The results are for the Musa (AAA Group, Cavendish Subgroup) ‘Williams’ plants grown in five distinct soils in our pot experiment. Fig. S2. Venn diagrams to show the numbers of shared candidate-core OTUs between Musa (AAA Group, Cavendish Subgroup) ‘Williams’ grown in five distinct soils (Pg, Tu, In, Li, and To) within compartments. Candidate-core OTUs are those that were present in ≥ 50% of the ten replicates within each treatment combination at a mean relative abundance of ≥ 0.5%. Core OTUs are those that were shared between all soils within each compartment. These results are from our pot experiment. Fig. S3. Heatmaps showing the number of soils in which each OTUs was considered a ‘candidate-core’ population on a per compartment basis. The results are for the Musa (AAA Group, Cavendish Subgroup) ‘Williams’ plants grown in five distinct soils in our pot experiment. Fig. S4. The ‘candidate’ core bacterial microbiome of Musa spp. The heatmap highlights the relative abundances of core OTUs in each compartment and soil for the Musa (AAA Group, Cavendish Subgroup) ‘Williams’ plants within the pot experiment. Each cell represents the mean of the replicates for that treatment. OTUs in grey indicate ‘candidates’ that were not found in field-grown plants and were subsequently dropped. Those with black text were found in field plants and were elevated to full core status. Table S9. OTUs found as candidate-core associated with Musa (AAA Group, Cavendish Subgroup) ‘Williams’ in plants grown in one to five soils or as key constituents in the field- grown Musa spp. Fig. S5. The mean relative frequencies of bacterial classes associated with each compartment in field-grown Musa spp. Within each phylum, classes represented at < 1% mean relative abundance are grouped as other. Fig. S6. Numbers of observed bacterial OTUs associated with different Musa spp. compartments and settings (pot vs. field). Error bars represent standard errors of the means. The letters indicate treatments that differ across soils according to estimated marginal means post hoc tests with Bejamini-Hochberg corrections. Members of the same groupings share the same letter. Fig. S7. A Principal Coordinate Analysis (PCA) ordination highlighting differences in the composition of bacterial communities (Hellinger transformed OTUs) associated with pot and field-grown Musa spp. in different plant compartments. Points representing field samples are larger than those representing pot samples. The ellipses represent standard deviations of the group centroids. Table S10. SPIEC-EASI network centrality metrics for core OTUs in field-grown Musa spp. Fig. S8. SPIEC-EASI network graphs showing the co-occurrences of core and non-core bacteria in field-grown Musa spp. Nodes are coloured by core status and size is positively associated with a range of centrality metrics. Edge colours represent positive (blue) and negative (red) associations between taxa. Edge width is positively associated with the coefficient for the co-occurrence between the taxa. The node layout in each graph is the same as in Figure 4, which shows the OTU IDs. Fig. S9. The common core bacterial microbiome of Musa spp. Blue tiles highlight which OTUs are core within each plant compartment. Fig. S10. Percentage of total bacterial sequences attributable to core OTUs and non-core OTUs within Musa spp. grown in the field or in pots containing different soils (In, Li, Pg, To, Tu). Fig. S11. A heatmap representing the mean relative abundances of key constituent OTUs in the ectorhizosphere of field-grown Musa spp. (i.e. present at ≥0.5% relative abundance in ≥50% of samples). The green squares indicate taxa that are members of the core microbiome. Fig. S12. A heatmap representing the mean relative abundances of key constituent OTUs in the endorhizosphere of field-grown Musa spp. (i.e. present at ≥ 0.5% relative abundance in ≥ 50% of samples). The green squares indicate taxa that are members of the core microbiome. Fig. S13. A heatmap representing the mean relative abundances of key constituent OTUs in the pseudostem of field-grown Musa spp. (i.e. present at ≥ 0.5% relative abundance in ≥ 50% of samples). The green squares indicate taxa that are members of the core microbiome. Table S11. Representative sequences the 47 candidate-core OTUs identified in the Musa (AAA Group, Cavendish Subgroup) ‘Williams’ plants grown in the five distinct soils in our pot experiment. OTUs that were elevated to full core-status are marked as core in bold text. Fig. S14. A heatmap representing the mean relative abundances of key constituent OTUs in the leaves of field-grown Musa spp. (i.e. present at ≥ 0.5% relative abundance in ≥ 50% of samples). The green squares indicate taxa that are members of the core microbiome. Table S12. Examples from the literature where core and candidate-core taxa identified in this study have been reported to be important to Musa spp. plant fitness or associated with the plant

The distribution of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> (Foc) vegetative compatibility groups (VCGs) in Vietnam.

<p>VCG 0123 is shown in light green, VCG 0124/5 is shown in dark green, VCG 0128 is shown in blue, VCG 01221 is shown in white and VCG 0124/22 is shown in purple.</p

Distribution of vegetative compatibility groups of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> found in Asian countries.

<p>The y-axis shows the number of isolates, while the x-axis shows countries represented. The legend corresponds each of the VCGs to a specific colour: VCG 0120/15 (light orange), 0121 (dark orange), 0122 (burgundy), 0123 (light green), 0124/5 (dark green), 0126 (light purple), 0128 (blue), 01213/16 (red), 01217 (dark grey), 01218 (black), 01219 (yellow), 01220 (light grey), 0124/22 (dark purple) and self-incompatible and isolates incompatible to known VCGs (pink).</p

Vegetative compatibility group (VCG) and lineage distribution of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> isolates in Asia.

<p>Vegetative compatibility group (VCG) and lineage distribution of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> isolates in Asia.</p

The distribution of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> (Foc) vegetative compatibility groups (VCGs) in Indonesia.

<p>VCG 0120/15 is shown in dark yellow, VCG 0121 is shown in light orange, VCG 0123 is shown in light green, VCG 0124/5 is shown in dark green, VCG 01213/16 is shown in red, VCG 01218 is shown in black and VCG 01219 is shown in light yellow.</p

The distribution of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> (Foc) vegetative compatibility groups (VCGs) in Bangladesh, India and Sri Lanka.

<p>VCG 0123 is shown in light green, 0124/5 is shown in dark green, VCG 0128 is shown in blue, VCG 01217 is shown in dark grey, VCG 01220 is shown in light grey and VCG 0124/22 is shown in purple.</p

The group, subgroup and common synonyms of <i>Musa</i> cultivars and their relationship with vegetative compatibility groups (VCGs) of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i>.

<p>The group, subgroup and common synonyms of <i>Musa</i> cultivars and their relationship with vegetative compatibility groups (VCGs) of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i>.</p