TY - JOUR
T1 - Macroscopic scalar curvature and areas of cycles
AU - Alpert, Hannah
AU - Funano, Kei
N1 - Publisher Copyright:
© 2017, Springer International Publishing AG.
PY - 2017/7/1
Y1 - 2017/7/1
N2 - In this paper we prove the following. Let Σ be an n–dimensional closed hyperbolic manifold and let g be a Riemannian metric on Σ × S1. Given an upper bound on the volumes of unit balls in the Riemannian universal cover (Σ × S1~ , g~) , we get a lower bound on the area of the Z2–homology class [ Σ × * ] on Σ × S1, proportional to the hyperbolic area of Σ. The theorem is based on a theorem of Guth and is analogous to a theorem of Kronheimer and Mrowka involving scalar curvature.
AB - In this paper we prove the following. Let Σ be an n–dimensional closed hyperbolic manifold and let g be a Riemannian metric on Σ × S1. Given an upper bound on the volumes of unit balls in the Riemannian universal cover (Σ × S1~ , g~) , we get a lower bound on the area of the Z2–homology class [ Σ × * ] on Σ × S1, proportional to the hyperbolic area of Σ. The theorem is based on a theorem of Guth and is analogous to a theorem of Kronheimer and Mrowka involving scalar curvature.
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U2 - 10.1007/s00039-017-0417-8
DO - 10.1007/s00039-017-0417-8
M3 - Article
AN - SCOPUS:85023747135
SN - 1016-443X
VL - 27
SP - 727
EP - 743
JO - Geometric and Functional Analysis
JF - Geometric and Functional Analysis
IS - 4
ER -