A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation

Abstract

Let $(X,\omega_0)$ be a compact K\"ahler manifold of complex dimension $n$, and let $U\Subset X$ have $C^{3,1}$ uniformly strongly pseudoconvex boundary. Consider the signed parallel family $U_t=\{\rho<t\}$, where $\rho$ 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 $\Sigma_t=\partial U_t$, with outward unit normal $\nu_t$. We prove that the normalized Aubin--Mabuchi energy $e(t)=E_{\omega_0}(u_t)$ belongs to $C^{1,1/2}([-\tau,\tau])$ for some $\tau>0$ and satisfies the Hadamard formula \[ e'(t)=\frac{1}{n+1}\int_{\Sigma_t}\left(-\partial_{\nu_t}u_t\right)^2\,d\sigma_{\rho,t}^{\mathrm{tot}},\qquad |t|<\tau. \] Here $d\sigma_{\rho,t}^{\mathrm{tot}}$ is the interior outward Anzellotti trace of \[ \frac{1}{V}\,d^c\rho\wedge \sum_{p=0}^{n-1}(p+1)\,\omega_{u_t}^{p}\wedge\omega_0^{n-1-p},\qquad V=\int_X\omega_0^n,\quad \omega_{u_t}=\omega_0+dd^c u_t. \] 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 $1/2$-H\"older stability of the normal derivatives under the normal-flow identifications. We identify the boundary part of the normalized Monge--Amp\`ere 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.