标题： 夹紧板在RCD(0,N)空间上的主频率：精确性、刚性和稳定性 显示英文标题

标题： Principal frequency of clamped plates on RCD(0,N) spaces: sharpness, rigidity and stability

摘要： 我们研究在满足RCD(0,N)条件的度量测度空间（可能是奇异的）中，夹紧板的主要频率的精细性质，即无限小Hilbert空间，具有非负的Ricci曲率，并且在合成意义上维度不超过N>1。 最初的猜想——夹紧板主要频率的等周不等式——由Lord Rayleigh在1877年在欧几里得情况下提出，并在2和3维中由Ashbaugh和Benguria [Duke Math. J., 1995]以及Nadirashvili [Arch. Rat. Mech. Anal., 1995]正面解决。 本文的主要贡献是在RCD(0,N)空间中，当N足够接近2或3时，夹紧板主要频率的新等周不等式。 该不等式包含所谓的“渐近体积比”，并且在距离函数的次调和性下变得精确，这一条件在度量测度锥中成立。 此外，根据RCD(0,N)空间的锥结构以及由Bessel函数给出的主要频率特征函数的形状，建立了刚性（即等周不等式中的等号）和稳定性结果。 这些结果甚至对于具有非负Ricci曲率的黎曼流形来说也是新的。 我们讨论了光滑和非光滑空间的例子，其中这些结果可以应用。 显示英文摘要

摘要： We study fine properties of the principal frequency of clamped plates in the (possibly singular) setting of metric measure spaces verifying the RCD(0,N) condition, i.e., infinitesimally Hilbertian spaces with non-negative Ricci curvature and dimension bounded above by N>1 in the synthetic sense. The initial conjecture -- an isoperimetric inequality for the principal frequency of clamped plates -- has been formulated in 1877 by Lord Rayleigh in the Euclidean case and solved affirmatively in dimensions 2 and 3 by Ashbaugh and Benguria [Duke Math. J., 1995] and Nadirashvili [Arch. Rat. Mech. Anal., 1995]. The main contribution of the present work is a new isoperimetric inequality for the principal frequency of clamped plates in RCD(0,N) spaces whenever N is close enough to 2 or 3. The inequality contains the so-called ``asymptotic volume ratio", and turns out to be sharp under the subharmonicity of the distance function, a condition satisfied in metric measure cones. In addition, rigidity (i.e., equality in the isoperimetric inequality) and stability results are established in terms of the cone structure of the RCD(0,N) space as well as the shape of the eigenfunction for the principal frequency, given by means of Bessel functions. These results are new even for Riemannian manifolds with non-negative Ricci curvature. We discuss examples of both smooth and non-smooth spaces where the results can be applied.