Free boundary minimal hypersurfaces with spherical boundary

2016 ◽  
Vol 290 (5-6) ◽  
pp. 885-889 ◽  
Author(s):  
Sung-Ho Park ◽  
Juncheol Pyo
Keyword(s):  
Free Boundary ◽  
Minimal Hypersurfaces ◽  
Spherical Boundary

scholarly journals Curvature estimates for stable free boundary minimal hypersurfaces

2020 ◽  
Vol 2020 (759) ◽  
pp. 245-264 ◽  
Author(s):  
Qiang Guang ◽  
Martin Man-chun Li ◽  
Xin Zhou
Keyword(s):  
Free Boundary ◽  
Riemannian Manifolds ◽  
A Priori ◽  
Minimal Submanifolds ◽  
Manifolds With Boundary ◽  
Curvature Estimate ◽  
Minimal Hypersurfaces ◽  
Curvature Estimates ◽  
Minimal Laminations ◽  
Uniform Area

AbstractIn this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces satisfying a uniform area bound, which generalize the celebrated Schoen–Simon–Yau interior curvature estimates up to the free boundary. Our curvature estimates imply a smooth compactness theorem which is an essential ingredient in the min-max theory of free boundary minimal hypersurfaces developed by the last two authors. We also prove a monotonicity formula for free boundary minimal submanifolds in Riemannian manifolds for any dimension and codimension. For 3-manifolds with boundary, we prove a stronger curvature estimate for properly embedded stable free boundary minimal surfaces without a-priori area bound. This generalizes Schoen’s interior curvature estimates to the free boundary setting. Our proof uses the theory of minimal laminations developed by Colding and Minicozzi.

scholarly journals Minimal hypersurfaces in the ball with free boundary

2019 ◽  
Vol 62 ◽  
pp. 120-127 ◽  
Author(s):  
Glen Wheeler ◽  
Valentina-Mira Wheeler
Keyword(s):  
Free Boundary ◽  
Minimal Hypersurfaces

scholarly journals Compactness and generic finiteness for free boundary minimal hypersurfaces, I

2021 ◽  
Vol 310 (1) ◽  
pp. 85-114
Author(s):  
Qiang Guang ◽  
Zhichao Wang ◽  
Xin Zhou
Keyword(s):  
Free Boundary ◽  
Minimal Hypersurfaces ◽  
Generic Finiteness

scholarly journals Index estimates for free boundary minimal hypersurfaces

2017 ◽  
Vol 370 (3-4) ◽  
pp. 1063-1078 ◽  
Author(s):  
Lucas Ambrozio ◽  
Alessandro Carlotto ◽  
Ben Sharp
Keyword(s):  
Free Boundary ◽  
Minimal Hypersurfaces

scholarly journals Min–max theory for free boundary minimal hypersurfaces II: general Morse index bounds and applications

2020 ◽  
Author(s):  
Qiang Guang ◽  
Martin Man-chun Li ◽  
Zhichao Wang ◽  
Xin Zhou
Keyword(s):  
Free Boundary ◽  
Compact Manifold ◽  
Morse Index ◽  
Riemannian Metric ◽  
Upper Bounds ◽  
Minimal Hypersurfaces ◽  
Manifold With Boundary ◽  
Boundary Setting ◽  
Mean Convex ◽  
Convex Point

Abstract For any smooth Riemannian metric on an $$(n+1)$$ ( n + 1 ) -dimensional compact manifold with boundary $$(M,\partial M)$$ ( M , ∂ M ) where $$3\le (n+1)\le 7$$ 3 ≤ ( n + 1 ) ≤ 7 , we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min–max theory in the Almgren–Pitts setting. We apply our Morse index estimates to prove that for almost every (in the $$C^\infty $$ C ∞ Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M. If $$\partial M$$ ∂ M is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.

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