The pepenaactivity is part of the informal sector, characterized by precariousness and social invisibility; however, it is a key element in the recycling production chain, providing an economic income for many people. In the city of Chihuahua, this activity is carried out on a significant scale. Therefore, this study analyzes and dignifies the recycling of urban solid waste from the perspective of scavengers, also explaining their relationship with other social actors at the city's final disposal site. The study starts from the premise that waste picking is, in the current era of global environmental crisis, an effective way to conserve resources and reduce environmental impacts. Methodologically, participatory action research was used as a strategy to raise awareness among scavengers about the importance of their work and to present their community organization as an example to other urban waste picking groups in different locations. Finally, we found issues in these groups such as informality, lack of legal support,specific health risks, and, more broadly, the absence of public policies to recognize this activity as valuable in addressing the global environmental crisis
We develop a loop group (DPW-type) representation for minimal Lagrangian surfaces in the complex quadric $Q_{2}\cong \mathbb S^{2}\times \mathbb S^{2}$, formulated via a flat family of connections $\{ abla^λ\}_{λ\in \mathbb S^{1}}$ on a trivial bundle. We prove that minimality is equivalent to the flatness of $ abla^λ$ for all $λ$, describe the associated isometric $\mathbb S^{1}$-family, and establish a precise correspondence with minimal surfaces in $\mathbb S^{3}$ through their Gauss maps. Our framework unifies and streamlines earlier constructions (e.g., Castro--Urbano) and yields explicit families including $\mathbb R$-equivariant, radially symmetric, and trinoid-type examples.
This is the second of two articles in which we investigate the geometry of free boundary and capillary minimal surfaces in balls $B_R\subset\mathbb{S}^3$. In this article, we find monotonicity formulae which imply that capillary minimal surfaces maximise a certain modified energy in their conformal orbit (preserving $B_R$). In the hemisphere, this energy is precisely the capillary energy. We also prove a partial characterisation by index for capillary minimal surfaces in the hemisphere, analogous to Urbano's characterisation of the Clifford torus.
In this paper, we prove that a closed minimal hypersurface in $\SSS$ with $λ_1<n$ has Morse index at least $n+4$, providing a partial answer to a conjecture of Perdomo. As a corollary, we re-obtain a partial proof of the famous Urbano Theorem for minimal tori in $\mathbb{S}^3$: a minimal torus in $\mathbb{S}^3$ has Morse index at least $5$, with equality holding if and only if it is congruent to the Clifford torus. The proof is based on a comparison theorem between eigenvalues of two elliptic operators, which also provides us simpler new proofs of some known results on index estimates of both minimal and $r$-minimal hypersurfaces in a sphere.
This paper is dedicated to the exploration of the conformal Willmore functional for surfaces within 4-dimensional conformal manifolds. We provide a detailed calculation of both the first and second variations, and present the Euler-Lagrange equation of this functional in a conformally invariant form. Utilizing the second variation formula we derived, we demonstrate that the Clifford torus in $\mathbb{C}P^2$ is strictly Willmore-stable. This finding strongly supports the conjecture proposed by Montiel and Urbano [J. reine angew. Math. 546 2002, 139-154], which posits that the Clifford torus in $\mathbb{C}P^2$ minimizes the Willmore functional among all tori. Moreover, by applying our formula to complex curves in $\mathbb{C}P^2$, we establish that the first nonzero eigenvalue of the Jacobi operator is at least 12. In the context of 4-dimensional locally symmetric spaces, we construct several holomorphic differentials to show that among all minimal 2-spheres, only those super-minimal ones can be Willmore.
In this paper, we classify the hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$ with constant sectional curvature. By applying the so-called Tsinghua principle, which was first discovered by the first three authors in 2013 at Tsinghua University, we prove that the constant sectional curvature can only be $\frac{1}{2}$ and the product angle function $C$ defined by Urbano is identically zero. We show that any such hypersurface is a parallel hypersurface of a minimal hypersurface in $\mathbb{S}^2\times\mathbb{S}^2$ with $C=0$, and we establish a one-to-one correspondence between the involving minimal hypersurface and the famous ``sinh-Gordon equation'' $$ (\frac{\partial^2}{\partial u^2}+\frac{\partial^2}{\partial v^2})h =-\tfrac{1}{\sqrt{2}}\sinh(\sqrt{2}h). $$ As a byproduct, we give a complete classification of the hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$ with constant mean curvature and constant product angle function $C$.
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