Inverse mean curvature flow with outer obstacle

Xu, Kai

doi:10.2140/gt.2026.30.3069

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ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

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Inverse mean curvature flow with outer obstacle

Kai Xu

Geometry & Topology 30 (2026) 3069–3144

DOI: 10.2140/gt.2026.30.3069

Abstract

We develop a new boundary condition for the weak inverse mean curvature flow, which gives canonical and nontrivial solutions in bounded domains. Roughly speaking, the boundary of the domain serves as an outer obstacle, and the evolving hypersurfaces are assumed to stick tangentially to the boundary upon contact. In smooth bounded domains, we prove an existence and uniqueness theorem for weak solutions, and establish $C^{1, α}$ regularity of the level sets up to the obstacle. The proof combines various techniques, including elliptic regularization, blow-up analysis, and certain parabolic estimates. As an analytic application, we address the well-posedness problem for the usual weak inverse mean curvature flow, showing that the initial value problem always admits a unique maximal (or innermost) weak solution.

Keywords

inverse mean curvature flow, outer obstacle

Mathematical Subject Classification

Primary: 49Q20, 53E10

References

Publication

Received: 14 February 2025

Revised: 15 February 2026

Accepted: 30 March 2026

Published: 30 August 2026

Proposed: Tobias H Colding

Seconded: Gang Tian, Bruce Kleiner

Authors

Kai Xu
Department of Mathematics Duke University Durham, NC United States	Department of Mathematics Univeristy of California, Berkeley Berkeley, CA United States

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Volume 30, issue 8 (2026)