Improved generic regularity of codimension-1 minimizing integral currents (2306.13191v2)
Abstract: Let $\Gamma$ be a smooth, closed, oriented, $(n-1)$-dimensional submanifold of $\mathbb{R}{n+1}$. We show that there exist arbitrarily small perturbations $\Gamma'$ of $\Gamma$ with the property that minimizing integral $n$-currents with boundary $\Gamma'$ are smooth away from a set of Hausdorff dimension $\leq n-9-\varepsilon_n$, where $\varepsilon_n \in (0, 1]$ is a dimensional constant. This improves on our previous result (where we proved generic smoothness of minimizers in $9$ and $10$ ambient dimensions). The key ingredients developed here are a new method to estimate the full singular set of the foliation by minimizers and a proof of superlinear decay of closeness (near singular points) that holds even across non-conical scales.
- William K. Allard. On the first variation of a varifold: boundary behavior. Ann. of Math. (2), 101:418–446, 1975.
- Generic regularity for minimizing hypersurfaces in dimensions 9 and 10, Preprint, 2023.
- Quantitative stratification and the regularity of harmonic maps and minimal currents. Comm. Pure Appl. Math., 66(6):965–990, 2013.
- Ennio De Giorgi. Frontiere orientate di misura minima. Editrice Tecnico Scientifica, Pisa,, 1961. Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61.
- Generic regularity of free boundaries for the obstacle problem. Publ. Math. Inst. Hautes Études Sci., 132:181–292, 2020.
- Boundary regularity and embedded solutions for the oriented Plateau problem. Ann. of Math. (2), 110(3):439–486, 1979.
- Area minimizing hypersurfaces with isolated singularities. J. Reine Angew. Math., 362:102–129, 1985.
- The singular structure and regularity of stationary varifolds. J. Eur. Math. Soc. (JEMS), 22(10):3305–3382, 2020.
- Leon Simon. Lectures on geometric measure theory, volume 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University. Australian National University, Centre for Mathematical Analysis, Canberra, 1983.
- Leon Simon. A strict maximum principle for area minimizing hypersurfaces. J. Differential Geom., 26(2):327–335, 1987.
- Leon Simon. Cylindrical tangent cones and the singular set of minimal submanifolds. J. Differential Geom., 38(3):585–652, 1993.
- Leon Simon. A general asymptotic decay lemma for elliptic problems. In Handbook of geometric analysis. No. 1, volume 7 of Adv. Lect. Math. (ALM), pages 381–411. Int. Press, Somerville, MA, 2008.
- Leon Simon. Stable minimal hypersurfaces in \varmathbbRN+1+ℓ\varmathbbsuperscript𝑅𝑁1ℓ\varmathbb{R}^{N+1+\ell}italic_R start_POSTSUPERSCRIPT italic_N + 1 + roman_ℓ end_POSTSUPERSCRIPT with singular set an arbitrary closed K⊂{0}×\varmathbbRℓ𝐾0\varmathbbsuperscript𝑅ℓK\subset\{0\}\times\varmathbb{R}^{\ell}italic_K ⊂ { 0 } × italic_R start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT. Ann. of Math. (2), 197(3):1205–1234, 2023.
- Zhihan Wang. Mean Convex Smoothing of Mean Convex Cones. Geom. Funct. Anal., 34:263–301, 2024.
- Brian White. Regularity of area-minimizing hypersurfaces at boundaries with multiplicity. In Seminar on minimal submanifolds, volume 103 of Ann. of Math. Stud., pages 293–301. Princeton Univ. Press, Princeton, NJ, 1983.
- Brian White. Stratification of minimal surfaces, mean curvature flows, and harmonic maps. J. Reine Angew. Math., 488:1–35, 1997.