# On the intersection of minimal hypersurfaces of $S^k$.

@article{Assimos2020OnTI, title={On the intersection of minimal hypersurfaces of \$S^k\$.}, author={Renan Assimos}, journal={arXiv: Differential Geometry}, year={2020} }

It is known since the work of Frankel that two compactly immersed minimal hypersurfaces in a manifold with positive Ricci curvature must have an intersection point. Several generalizations of this result can be found in the literature, for example in the works of Lawson, Petersen and Wilhelm, among others. In the special case of minimal hypersurfaces of $S^k$, we prove a stronger version of Frankel's theorem. Namely, we show that if two compact minimal hypersurfaces $M_1$, $M_2$ of $S^k$ and a…

## One Citation

Some Properties of the Intersection of Free Boundary Minimal Hypersurfaces in Euclidean Balls

- Mathematics
- 2021

In this work, we prove that any two free boundary minimal hypersurfaces in the unit Euclidean ball have an intersection point in any half-ball. This is a strong version of the Frankel property proved…

## References

SHOWING 1-10 OF 14 REFERENCES

Minimal surfaces in $$S^3$$: a survey of recent results

- Mathematics
- 2013

In this survey, we discuss various aspects of the minimal surface equation in the three-sphere $$S^3$$. After recalling the basic definitions, we describe a family of immersed minimal tori with…

The Geometry of Maximum Principles and a Bernstein Theorem in Codimension 2

- Mathematics
- 2018

We develop a general method to construct subsets of complete Riemannian manifolds that cannot contain images of non-constant harmonic maps from compact manifolds. We apply our method to the special…

A Bernstein type theorem for minimal hypersurfaces via Gauss maps

- Mathematics
- 2018

Let $M$ be an $n$-dimensional smooth oriented complete minimal hypersurface in $\mathbb{R}^{n+1}$ with Euclidean volume growth. We show that if the image under the Gauss map of $M$ avoids some…

A two-piece property for compact minimal surfaces in a three-sphere

- Mathematics
- 1995

We show that any 2-equator in a 3-sphere divides each embedded compact minimal surface in two connected pieces. We also shall see that closed regions in the sphere with mean convex boundary…

On Frankel’s Theorem

- MathematicsCanadian Mathematical Bulletin
- 2003

Abstract In this paper we show that two minimal hypersurfaces in a manifold with positive Ricci curvature must intersect. This is then generalized to show that in manifolds with positive Ricci…

Equivariant min-max theory

- Mathematics
- 2016

We develop an equivariant min-max theory as proposed by Pitts-Rubinstein in 1988 and then show that it can produce many of the known minimal surfaces in $\mathbb{S}^3$ up to genus and symmetry group.…

The regularity of harmonic maps into spheres and applications to Bernstein problems

- Mathematics
- 2009

We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of…

First eigenvalue of symmetric minimal surfaces in S3

- Mathematics
- 2009

Let λ 1 be the first nontrivial eigenvalue of the Laplacian on a compact surface without boundary. We show that λ 1 = 2 on compact embedded minimal surfaces in S 3 which are invariant under a finite…

Minimal Surfaces in the Round Three-Sphere by Doubling the Equatorial Two-Sphere, II

- Mathematics
- 2014

We construct closed embedded minimal surfaces in the round three-sphere, resembling two parallel copies of the equatorial two-sphere, joined by small catenoidal bridges symmetrically arranged either…