scholarly journals An uncertainty principle for spectral projections on rank one symmetric spaces of noncompact type

2021 ◽  
Author(s):  
Pritam Ganguly ◽  
Sundaram Thangavelu
Keyword(s):  
Uncertainty Principle ◽  
Symmetric Spaces ◽  
Noncompact Type ◽  
Rank One ◽  
Spectral Projections

HYPERSURFACES IN NONCOMPACT COMPLEX GRASSMANNIANS OF RANK TWO

2012 ◽  
Vol 23 (10) ◽  
pp. 1250103 ◽  
Author(s):  
JÜRGEN BERNDT ◽  
YOUNG JIN SUH
Keyword(s):  
Geometric Structure ◽  
Symmetric Spaces ◽  
Second Fundamental Form ◽  
Kähler Structure ◽  
Noncompact Type ◽  
The Second Fundamental Form ◽  
Special Cases ◽  
Rank One ◽  
Higher Rank ◽  
Riemannian Symmetric Spaces

Consider a Riemannian manifold N equipped with an additional geometric structure, such as a Kähler structure or a quaternionic Kähler structure, and a hypersurface M in N. The geometric structure induces a decomposition of the tangent bundle TM of M into subbundles. A natural problem is to classify all hypersurfaces in N for which the second fundamental form of M preserves these subbundles. This problem is reasonably well understood for Riemannian symmetric spaces of rank one, but not for higher rank symmetric spaces. A general treatment of this problem for higher rank symmetric spaces is out of reach at present, and therefore it is desirable to understand this problem better in a few special cases. Due to some conceptual differences between symmetric spaces of compact type and of noncompact type it appears that one needs to consider these two cases separately. In this paper we investigate this problem for the rank two symmetric space SU 2, m/S(U2Um) of noncompact type.

scholarly journals Maximal convergence groups and rank one symmetric spaces

2007 ◽  
Vol 83 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Ara Basmajian ◽  
Mahmoud Zeinalian
Keyword(s):  
Symmetric Space ◽  
Hyperbolic Plane ◽  
Symmetric Spaces ◽  
Convergence Property ◽  
Noncompact Type ◽  
Hyperbolic Case ◽  
The Real ◽  
Convergence Group ◽  
Rank One ◽  
Möbius Groups

abstractWe show that the group of conformal homeomorphisms of the boundary of a rank one symmetric space (except the hyperbolic plane) of noncompact type acts as a maximal convergence group. Moreover, we show that any family of uniformly quasiconformal homeomorphisms has the convergence property. Our theorems generalize results of Gehring and Martin in the real hyperbolic case for Möbius groups. As a consequence, this shows that the maximal convergence subgroups of the group of self homeomorphisms of the d–sphere are not unique up to conjugacy. Finally, we discuss some implications of maximality.

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