TY - JOUR
T1 - A polyhedron comparison theorem for 3-manifolds with positive scalar curvature
AU - Li, Chao
N1 - Funding Information:
The author wishes to thank Rick Schoen, Brian White, Leon Simon, Rafe Mazzeo, Or Hershkovits and Christos Mantoulidis for stimulating conversations. He also wishes to thanks the referee for greatly improving the exposition. Part of this work was carried out when the author was visiting the University of California, Irvine. He wants to thank Department of Mathematics, UCI, for their hospitality.
Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2020/1/1
Y1 - 2020/1/1
N2 - The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is isometric to a flat polyhedron.
AB - The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is isometric to a flat polyhedron.
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U2 - 10.1007/s00222-019-00895-0
DO - 10.1007/s00222-019-00895-0
M3 - Article
AN - SCOPUS:85068875664
SN - 0020-9910
VL - 219
SP - 1
EP - 37
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
IS - 1
ER -