A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation
Abstract
Let (X,ω0) be a compact Kähler manifold of complex dimension n, and let USubset X have C3,1 uniformly strongly pseudoconvex boundary. Consider the signed parallel family Ut={ρ<t}, where ρ is a defining function agreeing near ∂ U with the signed distance, negative in U. Let ut be the equilibrium envelope with zero obstacle on X∖ Ut, and write Σt=∂ Ut, with outward unit normal νt. We prove that the normalized Aubin--Mabuchi energy e(t)=Eω0(ut) belongs to C1,1/2([−τ,τ]) for some τ>0 and satisfies the Hadamard formula e'(t)=1/(n+1)∫Σt(−∂νtut)2 dσρ,ttot, |t|<τ. Here dσρ,ttot is the interior outward Anzellotti trace of 1/V dcρ∧ ∑p=0n−1(p+1) ωutp∧ω0n−1−p, V=∫Xω0n, ωut=ω0+ddc ut. This measure is positive and uniformly comparable to the induced surface measure. The proof combines uniform C1,1 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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BibTeX
@misc{li2026hadamard,
title = {{A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation}},
author = {Ziyu Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.17187},
eprint = {2607.17187},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.17187},
}
RIS
TY - UNPB
TI - A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation
AU - Ziyu Li
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2607.17187
UR - https://arxiv.org/abs/2607.17187
AB - Let (X,ω_0) be a compact Kähler manifold of complex dimension n, and let USubset 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 ∂ U with the signed distance, negative in U. Let u_t be the equilibrium envelope with zero obstacle on X∖ U_t, and write Σ_t=∂ 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)=1/(n+1)∫_Σ_t(−∂_ν_tu_t)^2 dσ_ρ,t^tot, |t|<τ. Here dσ_ρ,t^tot is the interior outward Anzellotti trace of 1/V d^cρ∧ ∑_p=0^n−1(p+1) ω_u_t^p∧ω_0^n−1−p, V=∫_Xω_0^n, 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.
ER -