Examples of scalar-flat hypersurfaces in $${\mathbb{R}^{n+1}}$$

Abstract

ABSTRACT. Given a hypersurface M of null scalar curvature in the unit sphere S n , n ≥ 4, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in R n+1 as a normal graph over a truncated cone generated by M . Furthermore, this graph is 1-stable if the cone is strictly 1-stable.MSC 2000: 53C21, 53C42.

Curvature properties of some class of minimal hypersurfaces in Euclidean spaces

“…Precisely, let M the cone over the Clifford hypersurface M m,n−m−1 (c, t) defined in (i). We refer to Section 3 of [45] for precise definition and properties of cones. In particular, from Section 3 of [45] it follows immediately that M has at every point three distinct principal curvatures λ 1 = 0, λ 2 = 1 t ρ 1 and λ 3 = satisfying (1) on U H ⊂ M then the folowing conditions are satisfied on this set: (14) and…”

“…(ii) Let M be a n-dimensional hypersurface in the Euclidean space E n+1 , n ≥ 4. Precisely, let M be the cone over M. We refer to Section 3 of [45] for precise definition and properties of such hypersurfaces. In particular, from Section 3 of [45] it follows immediately that M has at every point three distinct principal curvatures λ 1 = 0, λ 2 = 1 t ρ 1 and λ 3 = 1 t ρ 2 , t ∈ R + , of the multiplicity 1, p and n − p − 1, respectively.…”

“…Precisely, let M be the cone over M. We refer to Section 3 of [45] for precise definition and properties of such hypersurfaces. In particular, from Section 3 of [45] it follows immediately that M has at every point three distinct principal curvatures λ 1 = 0, λ 2 = 1 t ρ 1 and λ 3 = 1 t ρ 2 , t ∈ R + , of the multiplicity 1, p and n − p − 1, respectively. Thus we see that the cone M over the Clifford torus S p (c 1 ) × S n−p−1 (c 2 ) is a hypersurface in E n+1 , n ≥ 4, having exactly three distinct principal curvatures satisfying at every point (2).…”

Enantio- and Diastereoselective Synthesis of N-Acetyl Dihydrotetrafibricin Methyl Ester

“…We initially envisioned that 1 would be accessed by a late-stage fragment-assembly double allylboration reaction of aldehydes 4 and 2 , with the first-generation reagent 3 . As it turned out, several attempts at this coupling with these and related intermediates proceeded in low yield, which we ultimately traced to the instability of 4 . Curran’s group reported similar issues in their attempts to effect a Kociensky–Julia olefination reaction with an analogue of 4 .…”

“…The absolute configuration of the C(13) hydroxyl group of 11 was assigned by using the Mosher ester method . The absolute and relative configurations of the C(17) hydroxyl group were deduced by analogy to the previously reported C(19) TBDPS ether. , The 1,3- syn stereochemistry of diol 23 was assigned by using Rychnovsky’s acetonide analysis …”