Journal
MATHEMATISCHE ANNALEN
Volume -, Issue -, Pages -Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s00208-022-02537-y
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Funding
- National Key R and D Program of China [2020YFA0713100]
- National Natural Science Foundation of China [12031017]
- German Research Foundation (DFG)
- German Academic Scholarship Foundation
- EPSRC [EP/K016687/1]
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In this study, we provide rigidity results for the discrete Bonnet-Myers diameter bound and the Lichnerowicz eigenvalue estimate. We prove that both inequalities are sharp only when the underlying graph is a hypercube. Our approach combines well-known semigroup methods with new direct methods that translate curvature to combinatorial properties. These results can be regarded as the first known discrete analogues of Cheng's and Obata's rigidity theorems.
We give rigidity results for the discrete Bonnet-Myers diameter bound and the Lichnerowicz eigenvalue estimate. Both inequalities are sharp if and only if the underlying graph is a hypercube. The proofs use well-known semigroup methods as well as new direct methods which translate curvature to combinatorial properties. Our results can be seen as first known discrete analogues of Cheng's and Obata's rigidity theorems.
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