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Award ID contains: 2247544

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  1. ; (, Journal für die reine und angewandte Mathematik (Crelles Journal))
    Abstract We prove the rigidity ofrectifiableboundaries with constantdistributionalmean curvature in the Brendle class of warped product manifolds (which includes important models in general relativity, like the de Sitter–Schwarzschild and Reissner–Nordstrom manifolds).As a corollary, we characterize limits of rectifiable boundaries whose mean curvatures converge, as distributions, to a constant.The latter result is new, and requires the full strength of distributional CMC-rigidity, even when one considers smooth boundaries whose mean curvature oscillations vanish in arbitrarily strong C k C^{k}-norms.Our method also establishes that rectifiable boundaries of sets of finite perimeter in the hyperbolic space with constant distributional mean curvature are finite unions of possibly mutually tangent geodesic spheres. 
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    Free, publicly-accessible full text available October 2, 2026
  2. ; ; (, Journal of Functional Analysis)
    Free, publicly-accessible full text available January 1, 2027
  3. ; ; (, arxiv)
    Several commonly observed physical and biological systems are arranged in shapes that closely resemble an honeycomb cluster, that is, a tessellation of the plane by regular hexagons. Although these shapes are not always the direct product of energy minimization, they can still be understood, at least phenomenologically, as low-energy configurations. In this paper, explicit quantitative estimates on the geometry of such low-energy configurations are provided, showing in particular that the vast majority of the chambers must be generalized polygons with six edges, and be closely resembling regular hexagons. Part of our arguments is a detailed revision of the estimates behind the global isoperimetric principle for honeycomb clusters due to Hales (T. C. Hales. The honeycomb conjecture. Discrete Comput. Geom., 25(1):1-22, 2001). 
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    Accepted Manuscript Full Text Available
  4. ; ; ; (, Advances in Mathematics)
    Full Text Available 
  5. ; ; (, arxiv)
    We consider a diffused interface version of the volume-preserving mean curvature flow in the Euclidean space, and prove, in every dimension and under natural assumptions on the initial datum, exponential convergence towards single "diffused balls". 
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    Accepted Manuscript Full Text Available
  6. ; ; (, arxiv)
    We introduce a diffused interface formulation of the Plateau problem, where the Allen--Cahn energy is minimized under a volume constraint and a spanning condition on the level sets of the densities. We discuss two singular limits of these Allen--Cahn Plateau problems: when , we prove convergence to the Gauss' capillarity formulation of the Plateau problem with positive volume ; and when , and , we prove convergence to the classical Plateau problem (in the homotopic spanning formulation of Harrison and Pugh). As a corollary of our analysis we resolve the incompatibility between Plateau's laws and the Allen--Cahn equation implied by a regularity theorem of Tonegawa and Wickramasekera. In particular, we show that Plateau-type singularities can be approximated by energy minimizing solutions of the Allen--Cahn equation with a volume Lagrange multiplier and a transmission condition on a spanning free boundary. 
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    Accepted Manuscript Full Text Available
  7. ; ; (, arxiv)
    We provide, in the setting of Gauss' capillarity theory, a rigorous derivation of the equilibrium law for the three dimensional structures known as Plateau borders which arise in "wet" soap films and foams. A key step in our analysis is a complete measure-theoretic overhaul of the homotopic spanning condition introduced by Harrison and Pugh in the study of Plateau's laws for two-dimensional area minimizing surfaces ("dry" soap films). This new point of view allows us to obtain effective compactness theorems and energy representation formulae for the homotopic spanning relaxation of Gauss' capillarity theory which, in turn, lead to prove sharp regularity properties of energy minimizers. The equilibrium law for Plateau borders in wet foams is also addressed as a (simpler) variant of the theory for wet soap films. 
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    Accepted Manuscript Full Text Available