Papers
Topics
Authors
Recent
Search
2000 character limit reached

Stable free boundary minimal hypersurfaces in locally wedge-shaped manifolds

Published 12 Mar 2024 in math.DG | (2403.08005v2)

Abstract: We prove that a stable C<sup>1,1C<sup>{1,1}-to-edge properly embedded free boundary minimal hypersurface Σ<sup>3\Sigma<sup>3 of a $4$-dimensional wedge domain Ω<sup>4θ\Omega<sup>4_{\theta} with angle θ∈(0,π]\theta\in (0,\pi] is flat.

Authors (1)
Definition Search Book Streamline Icon: https://streamlinehq.com
References (23)
  1. F. J. Almgren, Jr. Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem. Ann. of Math. (2), 84:277–292, 1966.
  2. Minimal cones and the Bernstein problem. Invent. Math., 7:243–268, 1969.
  3. Simon Brendle. The isoperimetric inequality for a minimal submanifold in Euclidean space. J. Amer. Math. Soc., 34(2):595–603, 2021.
  4. Qing Chen. Curvature estimates for stable minimal hypersurfaces in ℝ4superscriptℝ4\mathbb{R}^{4}blackboard_R start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT and ℝ5superscriptℝ5\mathbb{R}^{5}blackboard_R start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT. Ann. Global Anal. Geom., 19(2):177–184, 2001.
  5. Stable minimal hypersurfaces in ℝ4superscriptℝ4\mathbb{R}^{4}blackboard_R start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT. arXiv: Differential Geometry. To appear in Acta Mathematica, 2021.
  6. Stable anisotropic minimal hypersurfaces in 𝐑4superscript𝐑4{\bf R}^{4}bold_R start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT. Forum Math. Pi, 11:Paper No. e3, 22, 2023.
  7. Stable minimal hypersurfaces in ℝ5superscriptℝ5\mathbb{R}^{5}blackboard_R start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT. arXiv: Differential Geometry, 2024.
  8. Two Rigidity Results for Stable Minimal Hypersurfaces. Geom. Funct. Anal., 34(1):1–18, 2024.
  9. The structure of stable minimal hypersurfaces in 𝐑n+1superscript𝐑𝑛1{\bf R}^{n+1}bold_R start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT. Math. Res. Lett., 4(5):637–644, 1997.
  10. M. do Carmo and C. K. Peng. Stable complete minimal surfaces in 𝐑3superscript𝐑3{\bf R}^{3}bold_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT are planes. Bull. Amer. Math. Soc. (N.S.), 1(6):903–906, 1979.
  11. M. do Carmo and C. K. Peng. Stable complete minimal hypersurfaces. pages 1349–1358, 1982.
  12. Ennio De Giorgi. Una estensione del teorema di Bernstein. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 19:79–85, 1965.
  13. The structure of complete stable minimal surfaces in 3333-manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math., 33(2):199–211, 1980.
  14. Wendell H. Fleming. On the oriented Plateau problem. Rend. Circ. Mat. Palermo (2), 11:69–90, 1962.
  15. Robert Gulliver and H. Blaine Lawson, Jr. The structure of stable minimal hypersurfaces near a singularity. In Geometric measure theory and the calculus of variations (Arcata, Calif., 1984), volume 44 of Proc. Sympos. Pure Math., pages 213–237. Amer. Math. Soc., Providence, RI, 1986.
  16. Curvature estimates for stable free boundary minimal hypersurfaces. J. Reine Angew. Math., 759:245–264, 2020.
  17. Area minimizing hypersurfaces with isolated singularities. J. Reine Angew. Math., 362:102–129, 1985.
  18. The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space. J. Funct. Anal., 285(2):Paper No. 109945, 22, 2023.
  19. Curvature estimates for stable free boundary minimal hypersurfaces in locally wedge-shaped manifolds. 2023.
  20. Stably embedded minimal hypersurfaces. Math. Z., 255(3):493–514, 2007.
  21. A. V. Pogorelov. On the stability of minimal surfaces. Dokl. Akad. Nauk SSSR, 260(2):293–295, 1981.
  22. James Simons. Minimal varieties in riemannian manifolds. Ann. of Math. (2), 88:62–105, 1968.
  23. Curvature estimates for minimal hypersurfaces. Acta Math., 134(3-4):275–288, 1975.

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Tweets

Sign up for free to view the 2 tweets with 0 likes about this paper.

Sign Up for Free