We consider a rigidity problem for the spectral gap of the Laplacian on an RCD(K, ∞)-space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive K. For a weighted Riemannian manifold, Cheng-Zhou showed that the sharp spectral gap is achieved only when a 1-dimensional Gaussian space is split off. This can be regarded as an infinite-dimensional counterpart to Obata’s rigidity theorem. Generalizing to RCD(K, ∞)-spaces is not straightforward due to the lack of smooth structure and doubling condition. We employ the lift of an eigenfunction to the Wasserstein space and the theory of regular Lagrangian flows recently developed by Ambrosio-Trevisan to overcome this difficulty.

Rigidity for the spectral gap on rcd(K, ∞)-spaces / Gigli, N.; Ketterer, C.; Kuwada, K.; Ohta, S. -I.. - In: AMERICAN JOURNAL OF MATHEMATICS. - ISSN 0002-9327. - 142:5(2020), pp. 1559-1594. [10.1353/ajm.2020.0039]

Rigidity for the spectral gap on rcd(K, ∞)-spaces

Gigli N.
;
2020-01-01

Abstract

We consider a rigidity problem for the spectral gap of the Laplacian on an RCD(K, ∞)-space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive K. For a weighted Riemannian manifold, Cheng-Zhou showed that the sharp spectral gap is achieved only when a 1-dimensional Gaussian space is split off. This can be regarded as an infinite-dimensional counterpart to Obata’s rigidity theorem. Generalizing to RCD(K, ∞)-spaces is not straightforward due to the lack of smooth structure and doubling condition. We employ the lift of an eigenfunction to the Wasserstein space and the theory of regular Lagrangian flows recently developed by Ambrosio-Trevisan to overcome this difficulty.
2020
142
5
1559
1594
Gigli, N.; Ketterer, C.; Kuwada, K.; Ohta, S. -I.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/118317
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