CLOSED MINIMAL HYPERSURFACES IN COMPACT SYMMETRIC SPACES

1992 ◽  
Vol 03 (05) ◽  
pp. 629-651 ◽  
Author(s):  
CLAUDIO GORODSKI
Keyword(s):  
Symmetric Spaces ◽  
Local Geometry ◽  
Minimal Hypersurfaces ◽  
Compact Symmetric Space ◽  
Orbital Geometry ◽  
Rank One ◽  
Minimal Cones ◽  
Area Minimizing ◽  
Simply Connected ◽  
Minimal Embedding

W.Y. Hsiang, W.T. Hsiang and P. Tomter conjectured that every simply-connected, compact symmetric space of dimension ≥4 must contain some minimal hypersurfaces of sphere type. With the aid of equivariant differential geometry, they showed that this is in fact the case for many symmetric spaces of rank one and two. Let M be one of the symmetric spaces: Sn(1)×Sn(1)(n≥4), SU(6)/Sp(3), E6/F4, ℍP2 (quaternionic proj. plane) or CaP2 (Cayley proj. plane). We prove the existence of infmitely many immersed, minimal hypersurfaces of sphere type in M which are invariant under a certain group G of isometries of M. Following Hsiang and the others, the equivariant method is also used here to reduce the problem to an investigation of geodesics in M/G equipped with a metric (with singularities) depending only on the orbital geometry of the transformation group (G, M). However, our constructions are based on area minimizing homogeneous cones, corresponding to a corner singularity of M/G with the local geometry of nodal type; this can be viewed as a variation of some of their constructions which depended on some unstable minimal cones of focal type. We further apply the equivariant method to construct a minimal embedding of S1×Sn−1×Sn−1 into Sn(1)×Sn(1)(n≥2) and a minimal, embedded hypersurface of sphere type in [Formula: see text], ℍPn×ℍPn (n≥2) and CaP2×CaP2.

POSITIVELY CURVED MANIFOLDS WITH LARGE CONJUGATE RADIUS

2013 ◽  
Vol 05 (03) ◽  
pp. 333-344 ◽  
Author(s):  
BENJAMIN SCHMIDT
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Spaces ◽  
Injectivity Radius ◽  
Curved Manifolds ◽  
Rank One ◽  
Sectional Curvatures ◽  
Simply Connected ◽  
Positively Curved

Let M denote a complete simply connected Riemannian manifold with all sectional curvatures ≥1. The purpose of this paper is to prove that when M has conjugate radius at least π/2, its injectivity radius and conjugate radius coincide. Metric characterizations of compact rank one symmetric spaces are given as applications.

scholarly journals On Some Classical andWeighted Estimates for SL4

2020 ◽  
Vol 15 ◽  
Keyword(s):  
Error Term ◽  
Symmetric Spaces ◽  
Compact Riemann Surface ◽  
Real Rank ◽  
Classical Approach ◽  
Locally Symmetric Spaces ◽  
Compact Symmetric Space ◽  
Rank One ◽  
Prime Geodesic Theorem ◽  
Correct Estimate

The purpose of this paper is two-sided. First, we obtain the correct estimate of the error term in the classical prime geodesic theorem for compact symmetric space SL4. As it turns out, the corrected error term depends on the degree of a certain polynomial appearing in the functional equation of the attached zeta function. This is in line with the known result in the case of compact Riemann surface, or more generally, with the corresponding result in the case of compact locally symmetric spaces of real rank one. Second, we derive a weighted form of the theorem. In particular, we prove that the aforementioned error term can be significantly improved when the classical approach is replaced by its higher level analogue.

scholarly journals Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved

2013 ◽  
Vol 65 (4) ◽  
pp. 757-767 ◽  
Author(s):  
Philippe Delanoë ◽  
François Rouvière
Keyword(s):  
Symmetric Space ◽  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Direct Proof ◽  
Locally Symmetric Spaces ◽  
Locally Symmetric Space ◽  
Hopf Fibrations ◽  
Rank One ◽  
Simply Connected ◽  
Positively Curved

AbstractThe squared distance curvature is a kind of two-point curvature the sign of which turned out to be crucial for the smoothness of optimal transportation maps on Riemannian manifolds. Positivity properties of that new curvature have been established recently for all the simply connected compact rank one symmetric spaces, except the Cayley plane. Direct proofs were given for the sphere, and an indirect one (via the Hopf fibrations) for the complex and quaternionic projective spaces. Here, we present a direct proof of a property implying all the preceding ones, valid on every positively curved Riemannian locally symmetric space.

Coupled Vortex Equations on Complete Kähler Manifolds

2008 ◽  
Vol 51 (3) ◽  
pp. 467-480
Author(s):  
Yue Wang
Keyword(s):  
Symmetric Spaces ◽  
Existence Results ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Convex Domains ◽  
Hermitian Symmetric Spaces ◽  
Vortex Equations ◽  
Curved Manifolds ◽  
Pseudo Convex ◽  
Simply Connected

AbstractIn this paper, we first investigate the Dirichlet problem for coupled vortex equations. Secondly, we give existence results for solutions of the coupled vortex equations on a class of complete noncompact Kähler manifolds which include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric.

scholarly journals Polar actions on rank-one symmetric spaces

1999 ◽  
Vol 53 (1) ◽  
pp. 131-175 ◽  
Author(s):  
Fabio Podestà ◽  
Gudlaugur Thorbergsson
Keyword(s):  
Symmetric Spaces ◽  
Polar Actions ◽  
Rank One
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