Abstract:

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

摘要：

我们考虑正值$ K $在$ {\ rm RCD}（K，\ infty）$空间（满足黎曼曲率维度条件的度量尺度空间）上的拉普拉斯谱隙的刚性问题。对于加权的黎曼流形，Cheng-Zhou表明，只有将$ 1 $维的高斯空间分开，才能实现尖锐的光谱间隙。这可以视为Obata刚度定理的无穷维对应。由于缺乏平滑的结构和加倍的条件，将$ {\ rm RCD}（K，\ infty）$-空间归纳起来并不容易。我们将本征函数的提升应用到Wasserstein空间，并采用了Ambrosio-Trevisan最近为克服这一难题而开发的规则拉格朗日流理论。