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

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