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A discrete version of the Brunn-Minkowski inequality and its stability

Michel Bonnefont (2009)

Annales mathématiques Blaise Pascal

In the first part of the paper, we define an approximated Brunn-Minkowski inequality which generalizes the classical one for metric measure spaces. Our new definition, based only on properties of the distance, allows also us to deal with discrete metric measure spaces. Then we show the stability of our new inequality under convergence of metric measure spaces. This result gives as corollary the stability of the classical Brunn-Minkowski inequality for geodesic spaces. The proof of this stability...

A fully nonlinear version of the Yamabe problem on manifolds with boundary

Aobing Li, Yan Yan Li (2006)

Journal of the European Mathematical Society

We propose to study a fully nonlinear version of the Yamabe problem on manifolds with boundary. The boundary condition for the conformal metric is the mean curvature. We establish some Liouville type theorems and Harnack type inequalities.

A Monge-Ampère equation in conformal geometry

Matthew J. Gursky (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider the Monge-Ampère-type equation det ( A + λ g ) = const . , where A is the Schouten tensor of a conformally related metric and λ > 0 is a suitably chosen constant. When the scalar curvature is non-positive we give necessary and sufficient conditions for the existence of solutions. When the scalar curvature is positive and the first Betti number of the manifold is non-zero we also establish existence. Moreover, by adapting a construction of Schoen, we show that solutions are in general not unique.

A note on conformal vector fields on a Riemannian manifold

Sharief Deshmukh, Falleh Al-Solamy (2014)

Colloquium Mathematicae

We consider an n-dimensional compact Riemannian manifold (M,g) and show that the presence of a non-Killing conformal vector field ξ on M that is also an eigenvector of the Laplacian operator acting on smooth vector fields with eigenvalue λ > 0, together with an upper bound on the energy of the vector field ξ, implies that M is isometric to the n-sphere Sⁿ(λ). We also introduce the notion of φ-analytic conformal vector fields, study their properties, and obtain a characterization of n-spheres...

A property of Wallach's flag manifolds

Teresa Arias-Marco (2007)

Archivum Mathematicum

In this note we study the Ledger conditions on the families of flag manifold ( M 6 = S U ( 3 ) / S U ( 1 ) × S U ( 1 ) × S U ( 1 ) , g ( c 1 , c 2 , c 3 ) ) , ( M 12 = S p ( 3 ) / S U ( 2 ) × S U ( 2 ) × S U ( 2 ) , g ( c 1 , c 2 , c 3 ) ) , constructed by N. R. Wallach in (Wallach, N. R., Compact homogeneous Riemannian manifols with strictly positive curvature, Ann. of Math. 96 (1972), 276–293.). In both cases, we conclude that every member of the both families of Riemannian flag manifolds is a D’Atri space if and only if it is naturally reductive. Therefore, we finish the study of M 6 made by D’Atri and Nickerson in (D’Atri, J. E., Nickerson, H. K., Geodesic...

A Singularity Theorem for Twistor Spinors

Florin Alexandru Belgun, Nicolas Ginoux, Hans-Bert Rademacher (2007)

Annales de l’institut Fourier

We study spin structures on orbifolds. In particular, we show that if the singular set has codimension greater than 2, an orbifold is spin if and only if its smooth part is. On compact orbifolds, we show that any non-trivial twistor spinor admits at most one zero which is singular unless the orbifold is conformally equivalent to a round sphere. We show the sharpness of our results through examples.

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