RCD(0,N) 空间上夹紧板的主频率：锐度、刚性与稳定性
Principal frequency of clamped plates on RCD(0,N) spaces: sharpness, rigidity and stability
AI总结：
本文研究了满足 RCD(0,N) 条件的度量测度空间中夹紧板主频率的精细性质，在 N 接近 2 或 3 时提出了包含渐近体积比的新等周不等式，并证明了其锐度、基于锥结构的刚性及稳定性，结果对黎曼流形亦属全新。
摘要：
我们研究了在满足 RCD(0,N) 条件（即在综合意义上，具有非负 Ricci 曲率且维数有上界 N>1 的无穷小 Hilbert 空间）的度量测度空间（可能是奇异的）中，夹紧板主频率的精细性质。最初的猜想——关于夹紧板主频率的等周不等式——由 Lord Rayleigh 于 1877 年在欧几里得情形下提出，并由 Ashbaugh 和 Benguria [Duke Math. J., 1995] 以及 Nadirashvili [Arch. Rat. Mech. Anal., 1995] 在 2 维和 3 维中给予了肯定解决。本工作的主要贡献是：当 N 足够接近 2 或 3 时，针对 RCD(0,N) 空间上夹紧板主频率提出了一项新的等周不等式。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（度量测度锥满足此条件）下被证明是锐度的。此外，基于 RCD(0,N) 空间的锥结构以及由 Bessel 函数给出的主频率特征函数形状，建立了刚性（即等周不等式中的等号成立）和稳定性结果。这些结果即使对于具有非负 Ricci 曲率的黎曼流形而言也是全新的。我们讨论了结果可应用的平滑与非平滑空间示例。
英文摘要：
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.
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