For the two-dimensional Liouville equation arising in embedding theory, the authors prove that decay at infinity alone forces rotational symmetry, and for the three-dimensional case they prove uniqueness of smooth spherically symmetric solutions.
Fundamental properties of the dark and the luminous matter from Low Surface Brightness discs
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abstract
Dark matter (DM) is one of the biggest mystery in the Universe. In this review, after a brief discussion of the DM evidences and the main proposed candidates and scenarios for the DM phenomenon, we focus on recent results on rotating disc galaxies giving a special attention to the Low Surface Brightness (LSB) galaxies. The main observational properties related to the baryonic matter in LSBs, investigated over the last decades, are briefly recalled. Next, the LSBs are analysed by means of the mass modelling of their rotation curves both individually and stacked. The latter analysis, via the Universal Rotation Curve (URC) method, results really powerful in giving a global/universal description of the disc galaxies properties. We show the presence in LSBs of scaling relations between the galactic structural properties and we compare them with those of galaxies of different morphologies. The findings confirm, for all disc systems, a strong entanglement between the luminous matter (LM) and the DM. Moreover, we report how in LSBs the tight relationship between their radial gravitational acceleration $g$ and their baryonic component $g_b$ results to also depend on the galactic radius at which the former have been measured. Finally, LSB galaxies strongly challenge the $\Lambda$CDM scenario with the relative collisionless dark particle and, alongside with the non-detection of the latter, contribute to guide us towards a new scenario for the DM phenomenon.
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Analysis of the equations of motion of fictitious matter in embedding theory
For the two-dimensional Liouville equation arising in embedding theory, the authors prove that decay at infinity alone forces rotational symmetry, and for the three-dimensional case they prove uniqueness of smooth spherically symmetric solutions.