The notion of causal boundary ∂M for a strongly causal spacetime M has been a controversial topic along last decades: on one hand, some attempted definitions were not fully consistent, on the other, there were simple examples where an open conformal embedding i : M → M0 could be defined, but the corresponding conformal boundary ∂iM disagreed drastically with the causal one. Nevertheless, the recent progress in this topic suggests that a final option for ∂M is available in most cases. Our study has two parts:(I) To give general arguments on a boundary in order to ensure that it is admissible as a causal boundary at the three natural levels, i.e., as a point set, as a chronological space and as a topological space. Then, the essential uniqueness of our choice is stressed, and the relatively few admissible alternatives are discussed.(II) To analyze the role of the conformal boundary ∂iM . We show that, in general, ∂iM may present a very undesirable structure. Nevertheless, it is well-behaved under certain general assumptions, and its accessible part ∂ * i M agrees with the causal boundary.This study justifies both boundaries. On one hand, the conformal boundary ∂ * i M , which cannot be defined for a general spacetime but is easily computed in particular examples, appears now as a special case of the causal boundary. On the other, the new redefinition of the causal boundary not only is free of inconsistencies and applicable to any strongly causal spacetime, but also recovers the expected structure in the cases where a natural simple conformal boundary is available. The cases of globally hyperbolic spacetimes and asymptotically conformally flat ends are especially studied.
Recently, the old notion of causal boundary for a spacetime V has been redefined consistently. The computation of this boundary ∂V on any standard conformally stationary spacetime V = R × M , suggests a natural compactification MB associated to any Riemannian metric on M or, more generally, to any Finslerian one. The corresponding boundary ∂BM is constructed in terms of Busemann-type functions. Roughly, ∂BM represents the set of all the directions in M including both, asymptotic and "finite" (or "incomplete") directions.This Busemann boundary ∂BM is related to two classical boundaries: the Cauchy boundary ∂C M and the Gromov boundary ∂GM . In a natural way ∂C M ⊂ ∂BM ⊂ ∂GM , but the topology in ∂BM is coarser than the others. Strict coarseness reveals some remarkable possibilities -in the Riemannian case, either ∂C M is not locally compact or ∂GM contains points which cannot be reached as limits of ray-like curves in M .In the non-reversible Finslerian case, there exists always a second boundary associated to the reverse metric, and many additional subtleties appear. The spacetime viewpoint interprets the asymmetries between the two Busemann boundaries, ∂ + B M (≡ ∂BM ), ∂ − B M , and this yields natural relations between some of their points.Our aims are: (1) to study the subtleties of both, the Cauchy boundary for any generalized (possibly non-symmetric) distance and the Gromov compactification for any (possibly incomplete) Finsler manifold, (2) to introduce the new Busemann compactification MB, relating it with the previous two completions, and (3) to give a full description of the causal boundary ∂V of any standard conformally stationary spacetime.
As known from literature lateral buds from pea (Pisum sativum) plants are released from apical dominance when repeatedly treated with exogenous cytokinins. Little is known, however, about the endogenous role of cytokinins in this process and whether they interact with basipolar transported IAA, generally regarded as the main signal controlling apical dominance. This paper presents evidence that such an interaction exists. The excision of the apex of pea plants resulted in the release of inhibited lateral buds from apical dominance (AD). This could be entirely prevented by applying 1‐naphthylacetic acid (NAA) to the cut end of the shoot. Removal of the apex also resulted in a rapid and rather large increase in the endogenous concentrations of zeatin riboside (ZR), isopentenyladenosine (iAdo) and an as yet unidentified polar zeatin derivative in the node and internode below the point of decapitation. This accumulation of ZR and iAdo, was strongly reduced by the application of NAA. The observed increase in cytokinin concentration preceded the elongation of the lateral buds, suggesting that endogenous cytokinins play a significant role in the release of lateral buds from AD. However, the effect of NAA on the concentration of cytokinins clearly demonstrated the dominant role of the polar basipetally transported auxin in AD. The results suggest a mutual interaction between the basipolar IAA transport system and cytokinins obviously produced in the roots and transported via the xylem into the stem of the pea plants.
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