Limitations on the smooth confinement of an unstretchable manifold

We prove that an m-dimensional unit ball D^m in the Euclidean space {\mathbb R}^m cannot be isometrically embedded into a higher-dimensional Euclidean ball B_r^d \subset {\mathbb R}^d of radius r < 1/2 unless one of two conditions is met -- (1)The embedding manifold has dimension d >= 2m. (2) The embedding is not smooth. The proof uses differential geometry to show that if d<2m and the embedding is smooth and isometric, we can construct a line from the center of D^m to the boundary that is geodesic in both D^m and in the embedding manifold {\mathbb R}^d. Since such a line has length 1, the diameter of the embedding ball must exceed 1.Comment: 20 Pages, 3 Figure

Polar foliations and isoparametric maps

A singular Riemannian foliation $F$ on a complete Riemannian manifold $M$ is called a polar foliation if, for each regular point $p$, there is an immersed submanifold $\Sigma$, called section, that passes through $p$ and that meets all the leaves and always perpendicularly. A typical example of a polar foliation is the partition of $M$ into the orbits of a polar action, i.e., an isometric action with sections. In this work we prove that the leaves of $F$ coincide with the level sets of a smooth map $H: M\to \Sigma$ if $M$ is simply connected. In particular, we have that the orbits of a polar action on a simply connected space are level sets of an isoparametric map. This result extends previous results due to the author and Gorodski, Heintze, Liu and Olmos, Carter and West, and Terng.Comment: 9 pages; The final publication is available at springerlink.com http://www.springerlink.com/content/c72g4q5350g513n1

A factorization of a super-conformal map

A super-conformal map and a minimal surface are factored into a product of two maps by modeling the Euclidean four-space and the complex Euclidean plane on the set of all quaternions. One of these two maps is a holomorphic map or a meromorphic map. These conformal maps adopt properties of a holomorphic function or a meromorphic function. Analogs of the Liouville theorem, the Schwarz lemma, the Schwarz-Pick theorem, the Weierstrass factorization theorem, the Abel-Jacobi theorem, and a relation between zeros of a minimal surface and branch points of a super-conformal map are obtained.Comment: 21 page

Identifiable Acetylene Features Predicted for Young Earth-like Exoplanets with Reducing Atmospheres Undergoing Heavy Bombardment

The chemical environments of young planets are assumed to be largely influenced by the impacts of bodies lingering on unstable trajectories after the dissolution of the protoplanetary disk. We explore the chemical consequences of impacts within the context of reducing planetary atmospheres dominated by carbon monoxide, methane, and molecular nitrogen. A terawatt high-power laser was selected in order to simulate the airglow plasma and blast wave surrounding the impactor. The chemical results of these experiments are then applied to a theoretical atmospheric model. The impact simulation results in substantial volume mixing ratios within the reactor of 5% hydrogen cyanide (HCN), 8% acetylene (C2H2), 5% cyanoacetylene (HC3N), and 1% ammonia (NH3). These yields are combined with estimated impact rates for the early Earth to predict surface boundary conditions for an atmospheric model. We show that impacts might have served as sources of energy that would have led to steady-state surface quantities of 0.4% C2H2, 400 ppm HCN, and 40 ppm NH3. We provide simulated transit spectra for an Earth-like exoplanet with this reducing atmosphere during and shortly after eras of intense impacts. We predict that acetylene is as observable as other molecular features on exoplanets with reducing atmospheres that have recently gone through their own "heavy bombardments," with prominent features at 3.05 and 10.5 μm

The spectral curve of a quaternionic holomorphic line bundle over a 2-torus

A conformal immersion of a 2-torus into the 4-sphere is characterized by an auxiliary Riemann surface, its spectral curve. This complex curve encodes the monodromies of a certain Dirac type operator on a quaternionic line bundle associated to the immersion. The paper provides a detailed description of the geometry and asymptotic behavior of the spectral curve. If this curve has finite genus the Dirichlet energy of a map from a 2-torus to the 2-sphere or the Willmore energy of an immersion from a 2-torus into the 4-sphere is given by the residue of a specific meromorphic differential on the curve. Also, the kernel bundle of the Dirac type operator evaluated over points on the 2-torus linearizes in the Jacobian of the spectral curve. Those results are presented in a geometric and self contained manner.Comment: 36 page

Risks, benefits, and knowledge gaps of non-native tree species in Europe

Changing ecosystem conditions and diverse socio-economical events have contributed to an ingrained presence of non-native tree species (NNTs) in the natural and cultural European landscapes. Recent research endeavors have focused on different aspects of NNTs such as legislation, benefits, and risks for forestry, emphasizing that large knowledge gaps remain. As an attempt to fulfill part of these gaps, within the PEN-CAFoRR COST Action (CA19128) network, we established an open-access questionnaire that allows both academic experts and practitioners to provide information regarding NNTs from 20 European countries. Then, we integrated the data originating from the questionnaire, related to the country-based assessment of both peer-reviewed and grey literature, with information from available datasets (EUFORGEN and EU-Forest), which gave the main structure to the study and led to a mixed approach review. Finally, our study provided important insights into the current state of knowledge regarding NNTs. In particular, we highlighted NNTs that have shown to be less commonly addressed in research, raising caution about those characterized by an invasive behavior and used for specific purposes (e.g., wood production, soil recultivation, afforestation, and reforestation). NNTs were especially explored in the context of resilient and adaptive forest management. Moreover, we emphasized the assisted and natural northward migration of NNTs as another underscored pressing issue, which needs to be addressed by joint efforts, especially in the context of the hybridization potential. This study represents an additional effort toward the knowledge enhancement of the NNTs situation in Europe, aiming for a continuously active common source deriving from interprofessional collaboration. Copyright © 2022 Dimitrova, Csilléry, Klisz, Lévesque, Heinrichs, Cailleret, Andivia, Madsen, Böhenius, Cvjetkovic, De Cuyper, de Dato, Ferus, Heinze, Ivetić, Köbölkuti, Lazarević, Lazdina, Maaten, Makovskis, Milovanović, Monteiro, Nonić, Place, Puchalka and Montagnoli

Ariel – a window to the origin of life on early earth?

Is there life beyond Earth? An ideal research program would first ascertain how life on Earth began and then use this as a blueprint for its existence elsewhere. But the origin of life on Earth is still not understood, what then could be the way forward? Upcoming observations of terrestrial exoplanets provide a unique opportunity for answering this fundamental question through the study of other planetary systems. If we are able to see how physical and chemical environments similar to the early Earth evolve we open a window into our own Hadean eon, despite all information from this time being long lost from our planet’s geological record. A careful investigation of the chemistry expected on young exoplanets is therefore necessary, and the preparation of reference materials for spectroscopic observations is of paramount importance. In particular, the deduction of chemical markers identifying specific processes and features in exoplanetary environments, ideally “uniquely”. For instance, prebiotic feedstock molecules, in the form of aerosols and vapours, could be observed in transmission spectra in the near future whilst their surface deposits could be observed from reflectance spectra. The same detection methods also promise to identify particular intermediates of chemical and physical processes known to be prebiotically plausible. Is Ariel truly able to open a window to the past and answer questions concerning the origin of life on our planet and the universe? In this paper, we discuss aspects of prebiotic chemistry that will help in formulating future observational and data interpretation strategies for the Ariel mission. This paper is intended to open a discussion and motivate future detailed laboratory studies of prebiotic processes on young exoplanets and their chemical signatures

A chemical survey of exoplanets with ARIEL

Thousands of exoplanets have now been discovered with a huge range of masses, sizes and orbits: from rocky Earth-like planets to large gas giants grazing the surface of their host star. However, the essential nature of these exoplanets remains largely mysterious: there is no known, discernible pattern linking the presence, size, or orbital parameters of a planet to the nature of its parent star. We have little idea whether the chemistry of a planet is linked to its formation environment, or whether the type of host star drives the physics and chemistry of the planet’s birth, and evolution. ARIEL was conceived to observe a large number (~1000) of transiting planets for statistical understanding, including gas giants, Neptunes, super-Earths and Earth-size planets around a range of host star types using transit spectroscopy in the 1.25–7.8 μm spectral range and multiple narrow-band photometry in the optical. ARIEL will focus on warm and hot planets to take advantage of their well-mixed atmospheres which should show minimal condensation and sequestration of high-Z materials compared to their colder Solar System siblings. Said warm and hot atmospheres are expected to be more representative of the planetary bulk composition. Observations of these warm/hot exoplanets, and in particular of their elemental composition (especially C, O, N, S, Si), will allow the understanding of the early stages of planetary and atmospheric formation during the nebular phase and the following few million years. ARIEL will thus provide a representative picture of the chemical nature of the exoplanets and relate this directly to the type and chemical environment of the host star. ARIEL is designed as a dedicated survey mission for combined-light spectroscopy, capable of observing a large and well-defined planet sample within its 4-year mission lifetime. Transit, eclipse and phase-curve spectroscopy methods, whereby the signal from the star and planet are differentiated using knowledge of the planetary ephemerides, allow us to measure atmospheric signals from the planet at levels of 10–100 part per million (ppm) relative to the star and, given the bright nature of targets, also allows more sophisticated techniques, such as eclipse mapping, to give a deeper insight into the nature of the atmosphere. These types of observations require a stable payload and satellite platform with broad, instantaneous wavelength coverage to detect many molecular species, probe the thermal structure, identify clouds and monitor the stellar activity. The wavelength range proposed covers all the expected major atmospheric gases from e.g. H2O, CO2, CH4 NH3, HCN, H2S through to the more exotic metallic compounds, such as TiO, VO, and condensed species. Simulations of ARIEL performance in conducting exoplanet surveys have been performed – using conservative estimates of mission performance and a full model of all significant noise sources in the measurement – using a list of potential ARIEL targets that incorporates the latest available exoplanet statistics. The conclusion at the end of the Phase A study, is that ARIEL – in line with the stated mission objectives – will be able to observe about 1000 exoplanets depending on the details of the adopted survey strategy, thus confirming the feasibility of the main science objectives.Peer reviewedFinal Published versio