平行变形下平衡包络的哈达玛公式
A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation
- Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
在紧致凯勒流形中，针对具有特定边界的区域，研究平衡包络相关问题。通过识别边界分量为散度测度通量流的负向外安泽洛蒂迹，证明了哈达玛公式，给出相关边界迹，为正则界面问题提供局部弱形式。
AI中文摘要：
设$(X,\omega_0)$为紧致凯勒流形，$U\Subset X$具有$C^{3,1}$一致强伪凸边界。对于与$X\setminus U$相关的平衡包络$u_0$，归一化蒙日 - 安培测度在$U$上消失，在$X\setminus\overline U$上等于背景测度$V^{-1}\omega_0^n$，其剩余部分是支撑在$\partial U$上的奇异测度。我们将此边界分量识别为散度测度通量流的负向外安泽洛蒂迹。然后证明了沿向外平行族$U_\varepsilon=\{\rho<\varepsilon\} $的归一化蒙日 - 安培能量的单侧哈达玛公式。非线性伸缩恒等式给出了混合贝德福德 - 泰勒边界迹，它们加起来等于流$\mathcal J_{\mathrm{tot}} =\frac1V\mathrm{d}^c u_0\wedge\sum_{p=0}^{n - 1}(p + 1)\omega_{u_0}^p\wedge\omega_0^{n - 1 - p}$的迹，其中$\omega_{u_0}=\omega_0+\mathrm{d}\mathrm{d}^c u_0$。这些结果为与达西/赫勒 - 肖型问题和蒙日 - 安培增长相关的正则界面问题提供了边界通量和法向变化的局部弱形式。
英文摘要：
Let $(X,ω_0)$ be a compact Kähler manifold of complex dimension $n$, and let $U\Subset X$ have $C^{3,1}$ uniformly strongly pseudoconvex boundary. Consider the signed parallel family $U_t=\{ρ<t\}$, where $ρ$ is a defining function agreeing near $\partial U$ with the signed distance, negative in $U$. Let $u_t$ be the equilibrium envelope with zero obstacle on $X\setminus U_t$, and write $Σ_t=\partial U_t$, with outward unit normal $ν_t$. We prove that the normalized Aubin--Mabuchi energy $e(t)=E_{ω_0}(u_t)$ belongs to $C^{1,1/2}([-τ,τ])$ for some $τ>0$ and satisfies the Hadamard formula \[ e'(t)=\frac{1}{n+1}\int_{Σ_t}\left(-\partial_{ν_t}u_t\right)^2\,dσ_{ρ,t}^{\mathrm{tot}},\qquad |t|<τ. \] Here $dσ_{ρ,t}^{\mathrm{tot}}$ is the interior outward Anzellotti trace of \[ \frac{1}{V}\,d^cρ\wedge \sum_{p=0}^{n-1}(p+1)\,ω_{u_t}^{p}\wedgeω_0^{n-1-p},\qquad V=\int_Xω_0^n,\quad ω_{u_t}=ω_0+dd^c u_t. \] This measure is positive and uniformly comparable to the induced surface measure. The proof combines uniform $C^{1,1}$ estimates up to the boundary from the interior with $1/2$-Hölder stability of the normal derivatives under the normal-flow identifications. We identify the boundary part of the normalized Monge--Ampère measure with the negative outward trace of a divergence-measure flux current. The one-sided weak-* limits of the mixed Bedford--Taylor boundary measures yield the weighted total flux appearing in the variation formula.
