Einstein solvmanifolds: existence and non-existence questions

The general aim of this paper is to study which are the solvable Lie groups admitting an Einstein left invariant metric. The space N of all nilpotent Lie brackets on R^n parametrizes a set of (n+1)-dimensional rank-one solvmanifolds, containing the set of all those which are Einstein in that dimension. The moment map for the natural GL(n)-action on N evaluated in a point of N encodes geometric information on the corresponding solvmanifold, allowing us to use strong and well-known results from geometric invariant theory. For instance, the functional on N whose critical points are precisely the Einstein solvmanifolds is the square norm of this moment map. We also use a GL(n)-invariant stratification for the space N following essentially a construction given by F. Kirwan and show that there is a strong interplay between the strata and the Einstein condition on the solvmanifolds. As applications, we obtain several examples of graded (even 2-step) nilpotent Lie algebras which are not the nilradicals of any standard Einstein solvmanifold, as well as a classification in the 7-dimensional 6-step case and an existence result for certain 2-step algebras associated to graphs.Comment: Final version to appear in Math. Annale

Dynamic Existence

I am an individual. Nothing and nobody else occupies my standpoint. Otherwise, he would be I. Thus, all what I perceive is individual, perspective of an individual, part of me. The computer screen should be a part of me

Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation

Published versio

Existence and non existence results for the singular Nirenberg problem

In this paper we study the problem, posed by Troyanov (Trans AMS 324: 793–821, 1991), of prescribing the Gaussian curvature under a conformal change of the metric on surfaces with conical singularities. Such geometrical problem can be reduced to the solvability of a nonlinear PDE with exponential type non-linearity admitting a variational structure. In particular, we are concerned with the case where the prescribed function K changes sign. When the surface is the standard sphere, namely for the singular Nirenberg problem, we give sufficient conditions on K, concerning mainly the regularity of its nodal line and the topology of its positive nodal region, to be the Gaussian curvature of a conformal metric with assigned conical singularities. Besides, we find a class of functions on S^2 which do not verify our conditions and which can not be realized as the Gaussian curvature of any conformal metric with one conical singularity. This shows that our result is somehow sharp

The Existence of Mèkol Nyo'on Formula in Tradition of Inheritance Division in Madura

The tradition of the inheritance division in Madura knows a term namely mèkol nyo'on formula or formula 2:1. However, nowaday, the existence of that formula is very difficult to find in its implementation, although it still very sticks in each individual mind until now. This research describes what kind of traditions that have replaced the mekol nyo'on formula and why it happens in Madurese society whom are well known as very religious society. This research takes place in Pamekasan regency where most of the populations are moslem. By implementing sociological-descriptive approach, this research concludes that the tradition of mèkol-nyo'on formula has been replaced by the different formula, which emphasizes more on the goodness and benefits to all the heirs regardless of gender. with the main standard of equality in economic prosperity and humanity among fellow brothers in the family, where men and women have the same opportunity. Copyright (c) 2016 by KARSA. All right reserved DOI: 10.19105/karsa.v24i1.100