Rigidity for the spectral gap on $RCD(K,\infty )$-spaces

Abstract

We consider a rigidity problem for the spectral gap of the Laplacian on an $RCD(K,\infty )$-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 $RCD(K,\infty )$-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.

References in corpus (8)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• The splitting theorem in non-smooth context
• Well posedness of Lagrangian flows and continuity equations in metric measure spaces
• Obata's rigidity theorem for metric measure spaces
• Sobolev spaces on warped products
• Second order differentiation formula on compact $RC{D}^{\ast }(K,N)$ spaces

Cited by in corpus (4)

• The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces
• Regularity of Lagrangian flows over $RC{D}^{\ast }(K,N)$ spaces
• Equality in the logarithmic Sobolev inequality
• Rigidity of some functional inequalities on RCD spaces