Morse index and multiplicity of min-max minimal hypersurfaces

The Min-max Theory for the area functional, started by Almgren in the early 1960s and greatly improved by Pitts in 1981, was left incomplete because it gave no Morse index estimate for the min-max minimal hypersurface. We advance the theory further and prove the first general Morse index bounds for minimal hypersurfaces produced by it. We also settle the multiplicity problem for the classical case of one-parameter sweepouts.Comment: Cambridge Journal of Mathematics, 4 (4), 463-511, 201

Min-max theory and the energy of links

Freedman, He, and Wang, conjectured in 1994 that the Mobius energy should be minimized, among the class of all nontrivial links in Euclidean space, by the stereographic projection of the standard Hopf link. We prove this conjecture using the min-max theory of minimal surfaces.Comment: 19 pages. Revised version. To appear in J. Amer. Math. So

Weyl law for the volume spectrum

Given $M$ a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum $\{\omega_p(M)\}_{p\in\mathbb{N}}$ satisfies a Weyl law that was conjectured by Gromov.Comment: Revised version. To appear in Annals of Mathematic

Monitoring 2 REINFFORCE Network arboretums. Health status, growth and phenology

Mestrado em Engenharia Florestal e dos Recursos Naturais / Instituto Superior de Agronomia. Universidade de LisboaClimate change is affecting plant and animal composition in forest ecosystems. Understanding how trees and insects are coping with climate change is most relevant to develop sustainable forest plantations, in the future. Using two REINFFORCE network arboretums, we aimed to analyse how different species and provenances are affected by abiotic and biotic damaging. Additionally, we also aimed to assess phenology influence on herbivory, having as model two Oak species, a native and an exotic, as well as their provenances. In general, there were significant differences in survivorship from different species. However, it did not differ between coniferous and broadleaved species (p = 0.659). It differed at genus level, such as Pinus, where we observed a higher survivorship on natives when compared to exotics. This relation was not verified in oak species. Discolouration was the main damage type verified in some species, in particular on Carob tree, Ceratonia siliqua, which suggests the presence of abiotic stressors, possibly linked to climate conditions and soil. Defoliation values were higher in two broadleaved genera, Quercus and Fagus. Lisbon arboretum had higher damaging and pests than Sintra arboretum, which might be related with the climate differences on both sites and also, the physiological and vegetative status of trees, which presented higher stamina in Sintra. Overall, herbivore damaging was lesser in exotic species than native ones. In herbivory assessment from the two studied Oak species, along Spring season, it was shown that the native species Quercus robur suffered three times more damage, than the exotic species Quercus rubra. Additionally, we showed that herbivore damaging was directly related to phenology development from provenances, thus proving this factor influenceN/