The emergence of SARS-CoV-2 P.1 lineage coincided with a surge in hospitalisations in the North region of Brazil. In the South region’s Rio Grande do Sul state, severe COVID-19 case numbers rose 3.8 fold in February 2021. During that month, at a COVID-19 referral hospital in this state, whole-genome sequencing of a subset of cases’ specimens (n = 27) revealed P.1 lineage SARS-CoV-2 in most (n = 24). Findings raise concerns regarding a possible association between lineage P.1 and rapid case and hospitalisation increases.

The main purpose of the paper is twofold: First, to extend a well known theorem of Ruh-Vilms [RV] in the Euclidean space to symmetric spaces and, secondly, to apply this result to extend HoffmanOsserman-Schoen Theorem ([HOS]) (HOS Theorem) to 3−dimensional symmetric spaces. Precisely, it is defined a Gauss map of a hypersurface M n−1 immersed in a symmetric space N n taking values in the unit pseudo sphere S m of the Lie algebra g of the isometry group of N , dim g = m + 1, and it is proved that M has CMC if and only if its Gauss map is harmonic. As an application, it is proved that if dim N = 3 and the image of the Gauss map of a CMC surface S immersed in N is contained in a hemisphere of S m determined by a vector X, then S is invariant by the one parameter subgroup of isometries of N of the Killing field determined by X. In particular, it is obtained an extension of HOS Theorem to the 3−dimensional hyperbolic space, which, as the authors know, had not been done so far.In the paper it is also shown that the holomorphic quadratic form induced by the Gauss map coincides (up to a sign) with the Hopf quadratic form when the ambient space is H 3 , R 3 and S 3 and with the Abresch-Rosenberg quadratic form when the ambient space is H 2 × R and S 2 × R providing, then, an unified way of relating Hopf's and Abresch-Rosenberg's quadratic form with the quadratic form induced by a harmonic Gauss map of a CMC surface in these 5 spaces.

We study complete finite topology immersed surfaces Σ in complete Riemannian 3-manifolds N with sectional curvature K N ≤ −a 2 ≤ 0, such that the absolute mean curvature function of Σ is bounded from above by a and its injectivity radius function is not bounded away from zero on each of its annular end representatives. We prove that such a surface Σ must be proper in N and its total curvature must be equal to 2πχ(Σ). If N is a hyperbolic 3-manifold of finite volume and Σ is a properly immersed surface of finite topology with nonnegative constant mean curvature less than 1, then we prove that each end of Σ is asymptotic (with finite positive multiplicity) to a totally umbilic annulus, properly embedded in N .Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42. Key words and phrases: Calabi-Yau problem, hyperbolic 3-manifolds, asymptotic injectivity radius, bounded mean curvature, isoperimetric inequality. Definition 1.1. Let e be an end of a complete Riemannian surface Σ whose injectivity radius function we denote by I Σ : Σ → (0, ∞). Let E(e) be the collection of proper subdomains E ⊂ Σ, with compact boundary, that represents e. We define the (lower) asymptotic injectivity radius of e byNote that if Σ has an end e which admits a one-ended representative E, then I ∞ Σ (e) = lim inf E I Σ | E . If Σ has finite topology, then every end of Σ has a one-ended representative which is an annulus (i.e., a surface with the topology of S 1 × [0, ∞)), hence this simpler definition can be used. Moreover, Lemma 5.1 in the Appendix shows that if Σ has nonpositive Gaussian curvature and an end e has a representative E which is an annulus, then for every divergent sequence of points {p n } n∈N on E,We define the mean curvature function of an immersed two-sided surface Σ with a given unit normal field in a Riemannian 3-manifold to be the pointwise average of its principal curvatures; note that if Σ does not have a unit normal field, then the absolute value |H Σ | of the mean curvature function of Σ still makes sense because a unit normal field locally exists on Σ and under a change of this local choice, the principal curvatures change sign.Theorem 1.2. Let N be a complete 3-manifold with sectional curvature K N ≤ −a 2 ≤ 0, for some a ≥ 0. Let ϕ : Σ → N be an isometric immersion of a complete surface Σ with finite topology, whose mean curvature function satisfies |H ϕ | ≤ a. Then Σ has nonpositive Gaussian curvature and the following hold:A. If N is simply connected, then I ∞ Σ (e) = ∞ for every end e of Σ.B. If N has positive injectivity radius Inj(N ) = δ > 0, then every end e of Σ satisfies I ∞ Σ (e) ≥ δ. In particular, Σ has positive injectivity radius.C. If I Σ is bounded, then Σ has finite total curvature Σ K Σ = 2πχ(Σ), where χ(Σ) is the Euler characteristic of Σ. Furthermore, for each annular end representative E of Σ, the induced map ϕ * : π 1 (E) → π 1 (N ) on fundamental groups is injective.

Resumen: El objeto del presente artículo es definir consistentemente y con claridad qué características son propias definen a la "Agricultura familiar". De esta forma se realizó una recopilación de las definiciones y metodologías comúnmente más aceptadas y utilizadas a nivel de los países, así como a nivel regional e internacional para definir y caracterizar a la agricultura familiar, tomando en cuenta su diversidad. Además o estudio intenta hacer definiciones sobre la AF como categoría socioeconómica de análisis y como las dichas definiciones se sustenten en criterios y luego en parámetros e indicadores, que puedan hacer comparables situaciones muy diferentes. El desafío ahora es reconocer la heterogenidad de la AF y comenzar a hablar de "matrices de políticas públicas" que permitan atender la complejidad de la finca familiar, la necesidad de asociar las mismas para asegurar el autoconsumo familiar y competir, así como la necesidad de asegurar los parámetros de la SAN. Finalmente, encarar una análisis de la AF como categoría socio / económica no es atemporal ni ageográfica, está fuertemente vinculada a cada contexto y cada contexto tiene que ver con estos factores: (a) los derivados de la situación geográfica y agroecológica y distancia relativa a los mercados; (b) los derivados de los procesos económicos; (c) los derivados de la historia, la cultura, la relación con la tierra y el territorio. El Concepto de AF es una construcción multidimensional. Dentro de ella los criterios fundamentales son: (i) organización del trabajo familiar en la finca y contratación eventual de mano de obra asalariada; (ii) administración de la finca o empresa familiar, o sea quién toma las decisiones productivas y/o técnicas y/o comerciales; (c) situación relativa de autonomía o dependencia respecto de los distintos mercados (insumos, factores, productos). Palabras-llave: El desarrollo rural. Definición teórica. Agricultura.Resumo: O objetivo deste artigo é definir, claramente, quais recursos são característicos do conceito de agricultura familiar -AF. Assim, realizou-se uma compilação das mais comuns, aceitas e utilizadas definições em nível de país. A metodologia utilizada considerou os níveis regional e internacional para definir e caracterizar a agricultura familiar, levando em conta a sua diversidade. Além disso, o estudo busca definições de AF como uma categoria socioeconômica de análise nas quais as definições são sustentadas por critérios e, seguidas de parâmetros e de indicadores que permitam a utilização de forma comparável em diferentes situações. O desafio é reconhecer a heterogeneidade da AF e começar a utilizar "matrizes de ordem pública" que abordam a complexidade da propriedade familiar, a necessidade de associá-las para garantir o consumo familiar e a concorrência, bem como a necessidade de garantir parâmetros de SAN. Finalmente, enfrenta-se uma análise da AF como uma categoria econômica não atemporal ou geográfica, mas fortemente ligada a cada contexto que deve derivar dos seguintes fatores: (a) situação geográfica e...

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