Embeddedness of least area minimal hypersurfaces

@article{Song2018EmbeddednessOL,
  title={Embeddedness of least area minimal hypersurfaces},
  author={Antoine Song},
  journal={Journal of Differential Geometry},
  year={2018}
}
  • A. Song
  • Published 9 November 2015
  • Mathematics
  • Journal of Differential Geometry
E. Calabi and J. Cao showed that a closed geodesic of least length in a two-sphere with nonnegative curvature is always simple. Using min-max theory, we prove that for some higher dimensions, this result holds without assumptions on the curvature. More precisely, in a closed $(n+1)$-manifold with $2 \leq n \leq 6$, a least area closed minimal hypersurface exists and any such hypersurface is embedded. As an application, we give a short proof of the fact that if a closed three-manifold $M$ has… 
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References

SHOWING 1-10 OF 23 REFERENCES
Minimal hypersurfaces of least area
In this paper, we study closed embedded minimal hypersurfaces in a Riemannian $(n+1)$-manifold ($2\le n\le 6$) that minimize area among such hypersurfaces. We show they exist and arise either by…
MIN-MAX MINIMAL HYPERSURFACE
In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in [AF2, P] corresponding to the fundamental class of a Riemannian manifold (Mn+1, g) of positive Ricci…
Existence of infinitely many minimal hypersurfaces in positive Ricci curvature
In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min–max theory for the area functional to…
Rigidity of min-max minimal spheres in three-manifolds
In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a 3-sphere which has scalar curvature greater than or…
Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature
Let N be a three dimensional Riemannian manifold. Let E be a closed embedded surface in N. Then it is a question of basic interest to see whether one can deform : in its isotopy class to some…
On the existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary metric
  • F. Smith
  • Mathematics
    Bulletin of the Australian Mathematical Society
  • 1983
First, by modifying the minimax techniques of Pitts [2] , i t is shown that there exists in N a stationary 2-varifold V which can be written as the (varifold) limit of embedded 2-spheres and which…
A Course in Minimal Surfaces
Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential…
Deforming three-manifolds with positive scalar curvature
In this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact 3-manifold is path-connected. The proof uses the Ricci flow with surgery, the conformal…
A regularity theorem for minimizing hypersurfaces modulo
It is proved that an (n 1)-dimensional, area-minimizing flat chain modulo v in RW, with smooth extremal boundary of at most v/2 components, has an interior singular set of Hausdorff dimension at most…
The existence of embedded minimal hypersurfaces
We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces in closed smooth (n + 1)-dimensional Riemannian manifolds, a theorem proved first by Pitts for 2 ≤ n ≤ 5 and…
...
1
2
3
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