On Lipschitz rigidity of complex analytic sets

We prove that any complex analytic set in $\mathbb{C}^n$ which is Lipschitz normally embedded at infinity and has tangent cone at infinity that is a linear subspace of $\mathbb{C}^n$ must be an affine linear subspace of $\mathbb{C}^n$ itself. No restrictions on the singular set, dimension nor codimension are required. In particular, any complex algebraic set in $\mathbb{C}^n$ which is Lipschitz regular at infinity is an affine linear subspace.The first named author was partially supported by CNPq-Brazil grant 302764/2014-

Global bi-Lipschitz classification of semi-algebraic surfaces

We classify semi-algebraic surfaces in $\mathbb{R}^n$ with isolated singularities up to bi-Lipschitz homeomorphisms with respect to the inner distance. In particular, we obtain complete classifications for the Nash surfaces and the complex algebraic curves. We also address the minimal surfaces with finite total curvature.Comment: 20 pages and 2 figure

Characteristics of the dairy goat primary sector at the Rio de Janeiro State, Brazil.

A survey was done based on 19 goat shepherds at counties of Centre-highlands, Northern and North-western regions of the Rio de Janeiro State and at the county of Pedra Dourada, Zona da Mata region, State of Minas Gerais. We aimed to characterise the primary sector of the goat milk production chain settled at those regions. Therefore, questionnaires were applied in order to depict profiles of the shepherds, their families, the role of the wife in the activity, the resources available, dependence on income generated by the activity, and how producers administrate their business. Farms were distributed in five strata according to the following daily milk production averages and standard deviations: 8.8 ± 0.9, 15.7 ± 3.9, 22.6 ± 2.7, 34.4 ± 3.4, and 183.8 ± 54.2 L/d. Approximately 42% of the interviewed producers conducted their activities according to a household production model and the income earned was exclusively from the dairy goat husbandry. Sons and daughters performed an important role in the business (27.80%), but most of them (62.73%) worked out at non farm activities. The percentage of wives that worked directly in the activity (@47%) indicated that it could contribute to gender equity in the rural environment. Most of the production systems (63.16%) presented positive gross margins. We have noticed, however, that shepherds perceived only the business gross margin and that the most accurate registries taken were those related to revenues. In general, producers of the higher strata were favoured by their larger production scale, but asymptotic behaviours for costs and amounts invested in animals, equipments and buildings were observed. These characteristics should be considered when policies related to the dairy goat primary sector have to be planned

Bi-Lipschitz equivalent cones with different degrees

We show that for every $k\ge 3$ there exist complex algebraic cones of dimension $k$ with isolated singularities, which are bi-Lipschitz and semi-algebraically equivalent but they have different degrees. We also prove that homeomorphic projective hypersurfaces with dimension greater than 2 have the same degree. In the final part of the paper, we classify links of real cones with base $\mathbb{P}^1\times \mathbb{P}^2.$ As an application we give an example of three four dimensional real algebraic cones in $\mathbb{R}^8$ with isolated singularity which are semi-algebraically and bi-Lipschitz equivalent but they have non-homeomorphic bases.Comment: 13 pages. arXiv admin note: text overlap with arXiv:2302.0538