O(m) � O(n)-Invariant Minimal Hypersurfaces in $\mathbb{R}$ m+n

Abstract

A b s tra c t Complex common names such as Indian elephant or green tea denote a certain type o f entity, viz. kinds. Moreover, those kinds are always subkinds o f the kind denoted by their head noun. Establishing such subkinds is essentially the task o f classifying modifiers that are a defining trait of endocentrically structured complex common names. Examining complex common names of different lexico-syntactic types (N N compounds, N + N syntagmas, N P /P P syntagmas, A + N syntagmas) and from different languages (particularly English, G erm an and French) it can be shown that complex common names are subject to language-independent formal and semantic constraints. In particular, complex common names qualify as name-like expressions in that they tend to be deficient in term s of formal complexity and semantic compositionality.

Free boundary minimal surfaces in the unit ball with low cohomogeneity

“…Note: this sign convention is consistent with [1] but has the opposite sign of the vector field X in [2]. On the domain …”

“…By Theorem 1.1 in [1] and Theorem 1.1, part (3) in [2], the profile curve γ is asymptotic to the line m,n . Since clearly lim k→∞ |γ(t k )| = ∞, it follows that the scalings γ k converge in C 0 to m,n on Q and moreover converge in C ∞ to m,n on compact subsets of Q \ {0}.…”

“…When m + n ≥ 8, the vector field V has nodal singularities. It is easy to check that in this case, the integral curve of V defined in an analogous way to ψ from the proof of Theorem 1 does not intersect the line {ϕ = θ} (see also [2], Proposition 4.4, parts (1) and (3)). Therefore, there are no free boundary minimal surfaces in B m+n (1) of the type from Theorem 1, part (1) when m + n ≥ 8.…”

“…al. in [2]. In section 3, we review relevant properties of such minimal surfaces and prove Theorem 1 by finding surfaces which "fit" inside B m+n (1) to satisfy the free boundary condition.…”

O(p + 1) × O(q + 1)-invariant (r – 1)-minimal hypersurfaces in Euclidean space

“…Years later Alencar et al [2] presented a study of minimal hypersurfaces of R p+q+2 invariant by the action of O(p + 1) × O(q + 1) with p, q > 1. The study of O(p + 1) × O(q + 1)-invariant hypersurfaces in R p+q+2 with zero scalar curvature began with the work due to Palmas [14] when p = q = 1 whereas the case p = q > 1 was generalized by Sato [16].…”