A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation

[2607.17187] A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Complex Variables arXiv:2607.17187 (math) [Submitted on 19 Jul 2026 ( v1 ), last revised 15 Sep 2026 (this version, v3)] Title: A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation Authors: Ziyu Li View a PDF of the paper titled A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation, by Ziyu Li View PDF HTML (experimental) Abstract: Let $(X,\omega_0)$ 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 $U_t=\{\rho<t\}$, 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 $1/2$-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. Comments: 35 pages, 1 figure. Substantially revised and expanded: the main result is stated for a two-sided parallel family, and detailed uniform-regularity proofs are included in an appendix. Comments are welcome! Subjects: Complex Variables (math.CV) ; Analysis of PDEs (math.AP); Differential Geometry (math.DG) MSC classes: 32W20 (Primary) 32U15, 32Q15 (Secondary) Cite as: arXiv:2607.17187 [math.CV] (or arXiv:2607.17187v3 [math.CV] for this version) https://doi.org/10.48550/arXiv.2607.17187 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Ziyu Li [ view email ] [v1] Sun, 19 Jul 2026 10:59:02 UTC (33 KB) [v2] Sat, 15 Aug 2026 09:55:46 UTC (34 KB) [v3] Tue, 15 Sep 2026 12:55:33 UTC (52 KB) Full-text links: Access Paper: View a PDF of the paper titled A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation, by Ziyu Li View PDF HTML (experimental) TeX Source view license Current browse context: math.CV < prev | next > new | recent | 2026-07 Change to browse by: math math.AP math.DG References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

Related documents

• A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation#609996
• Geometric Folding Algorithms: Linkages, Origami, Polyhedra#3215
• Projected Gradient Method on Hadamard Manifolds#819979
• A variational principle of minimum for Navier–Stokes equation and Bingham fluids based on the symplectic formalism#194448
• Frictional sliding strength of knotted and capstan configurations along the axis of a cylinder#147520