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

Given a hypersurface $M$ of null scalar curvature in the unit sphere $\mathbb{S}^n$, $n\ge 4$, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in \Rr^{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.Comment: Paper accepted to publication in Manuscripta Mathematic

The Dirichlet Problem for Curvature Equations in Riemannian Manifolds

We prove the existence of classical solutions to the Dirichlet problem for a class of fully nonlinear elliptic equations of curvature type on Riemannian manifolds. We also derive new second derivative boundary estimates which allows us to extend some of the existence theorems of Caffarelli, Nirenberg and Spruck [4] and Ivochkina, Trundinger and Lin [19] to more general curvature functions and less convex domains.Comment: 32 pages, no figures. Final version. Paper accepted to publication in Indiana University Mathematics Journa

Height estimates for Killing graphs

The paper aims at proving global height estimates for Killing graphs defined over a complete manifold with nonempty boundary. To this end, we first point out how the geometric analysis on a Killing graph is naturally related to a weighted manifold structure, where the weight is defined in terms of the length of the Killing vector field. According to this viewpoint, we introduce some potential theory on weighted manifolds with boundary and we prove a weighted volume estimate for intrinsic balls on the Killing graph. Finally, using these tools, we provide the desired estimate for the weighted height in the assumption that the Killing graph has constant weighted mean curvature and the weighted geometry of the ambient space is suitably controlled.Comment: 26 pages. Final version. To appear on Journal of Geometric Analysi

NEOTROPICAL XENARTHRANS: a data set of occurrence of xenarthran species in the Neotropics

Xenarthrans – anteaters, sloths, and armadillos – have essential functions for ecosystem maintenance, such as insect control and nutrient cycling, playing key roles as ecosystem engineers. Because of habitat loss and fragmentation, hunting pressure, and conflicts with 24 domestic dogs, these species have been threatened locally, regionally, or even across their full distribution ranges. The Neotropics harbor 21 species of armadillos, ten anteaters, and six sloths. Our dataset includes the families Chlamyphoridae (13), Dasypodidae (7), Myrmecophagidae (3), Bradypodidae (4), and Megalonychidae (2). We have no occurrence data on Dasypus pilosus (Dasypodidae). Regarding Cyclopedidae, until recently, only one species was recognized, but new genetic studies have revealed that the group is represented by seven species. In this data-paper, we compiled a total of 42,528 records of 31 species, represented by occurrence and quantitative data, totaling 24,847 unique georeferenced records. The geographic range is from the south of the USA, Mexico, and Caribbean countries at the northern portion of the Neotropics, to its austral distribution in Argentina, Paraguay, Chile, and Uruguay. Regarding anteaters, Myrmecophaga tridactyla has the most records (n=5,941), and Cyclopes sp. has the fewest (n=240). The armadillo species with the most data is Dasypus novemcinctus (n=11,588), and the least recorded for Calyptophractus retusus (n=33). With regards to sloth species, Bradypus variegatus has the most records (n=962), and Bradypus pygmaeus has the fewest (n=12). Our main objective with Neotropical Xenarthrans is to make occurrence and quantitative data available to facilitate more ecological research, particularly if we integrate the xenarthran data with other datasets of Neotropical Series which will become available very soon (i.e. Neotropical Carnivores, Neotropical Invasive Mammals, and Neotropical Hunters and Dogs). Therefore, studies on trophic cascades, hunting pressure, habitat loss, fragmentation effects, species invasion, and climate change effects will be possible with the Neotropical Xenarthrans dataset