A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation

Ziyu Li

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arXiv · 2607.17187

A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation

Ziyu Li

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Abstract

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 $U_t=\{ρ 0$ 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.

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Ziyu Li. 2026-09-15. A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation. https://arxiv.org/abs/2607.17187

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