Neumann谱和Aharonov--Bohm谱的锐调和平均不等式

中文摘要 AI 辅助

我们证明了曲面上Neumann谱和Aharonov--Bohm Neumann谱的锐双特征值等周不等式。若$\Omega\subset\mathbb S^2$是光滑、单连通且真包含的，则$\mu_2(\Omega)$和$\mu_3(\Omega)$的调和平均被等面积测地圆盘的第一正Neumann特征值所上界控制，且等号仅在圆盘时成立。对于Gauss曲率有上界的单连通曲面，我们获得了前两个Aharonov--Bohm特征值的磁类比。证明结合了二维互反Rayleigh--Ritz原理与Green水平比较。一个关键附加要素是磁球冠的谱序定理：对于$0<\nu<1/2$，前两个特征值位于有效阶$\nu$和$1-\nu$的角扇区中。我们通过分解$L_0=T^*T$，$L_1=TT^*$和精确的Neumann--Dirichlet谱移位来证明这一点。我们还获得了锐的整球和闭曲面界，以及关于共形模量和磁通量的环形不等式。

英文摘要

We prove sharp two-eigenvalue isoperimetric inequalities for Neumann and Aharonov--Bohm Neumann spectra on surfaces. If $Ω\subset\mathbb S^2$ is smooth, simply connected and proper, then the harmonic mean of $μ_2(Ω)$ and $μ_3(Ω)$ is bounded above by the first positive Neumann eigenvalue of the equal-area geodesic disk, with equality only for disks. For simply connected surfaces with Gaussian curvature bounded above, we obtain the magnetic analogue for the first two Aharonov--Bohm eigenvalues. The proof combines a two-dimensional reciprocal Rayleigh--Ritz principle with Green-level comparison. A key additional ingredient is a spectral ordering theorem for magnetic spherical caps: for $0<ν<1/2$, the first two eigenvalues lie in the angular sectors of effective orders $ν$ and $1-ν$. We prove this by the factorization $L_0=T^*T$, $L_1=TT^*$ and an exact Neumann--Dirichlet spectral shift. We also obtain sharp full-sphere and closed-surface bounds, and an annular inequality in terms of conformal modulus and flux.

发表机构

• Tsinghua University（清华大学）

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