Corpus ID: 237634980

# Minimal hypertori in the four-dimensional sphere

@inproceedings{Carlotto2021MinimalHI,
title={Minimal hypertori in the four-dimensional sphere},
author={Alessandro Carlotto and Mario B. Schulz},
year={2021}
}
• A. Carlotto, Mario B. Schulz
• Published 24 September 2021
• Mathematics
We prove that the four-dimensional round sphere contains a minimally embedded hypertorus, as well as infinitely many, pairwise non-isometric, immersed ones. Our analysis also yields infinitely many, pairwise non-isometric, minimally embedded hyperspheres and thus provides a self-contained solution to Chern’s spherical Bernstein conjecture in dimensions four and six.

#### References

SHOWING 1-10 OF 24 REFERENCES
Existence of infinitely many minimal hypersurfaces in closed manifolds
Using min-max theory, we show that in any closed Riemannian manifold of dimension at least 3 and at most 7, there exist infinitely many smoothly embedded closed minimal hypersurfaces. It proves aExpand
Density of minimal hypersurfaces for generic metrics.
• Mathematics
• 2017
For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that the union of all closed, smooth, embedded minimal hypersurfaces isExpand
Existence of infinitely many minimal hypersurfaces in positive Ricci curvature
• Mathematics
• 2013
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 toExpand
Compactness of minimal hypersurfaces with bounded index
We prove a compactness result for minimal hypersurfaces with bounded index and volume, which can be thought of as an extension of the compactness theorem of Choi-Schoen (Invent. Math. 1985) to higherExpand
Embedded minimal tori in S3 and the Lawson conjecture
We show that any embedded minimal torus in S3 is congruent to the Clifford torus. This answers a question posed by H. B. Lawson, Jr., in 1970.
Weyl law for the volume spectrum
• Mathematics
• 2016
Given $M$ a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum $\{\omega_p(M)\}_{p\in\mathbb{N}}$ satisfies a Weyl law that was conjectured by Gromov.
On the Multiplicity One Conjecture in min-max theory
We prove that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves,Expand
A Survey of Closed Self-Shrinkers with Symmetry
• Mathematics
• 2017
In this paper, we survey known results on closed self-shrinkers for mean curvature flow and discuss techniques used in recent constructions of closed self-shrinkers with classical rotationalExpand
Equidistribution of minimal hypersurfaces for generic metrics
• Mathematics
• Inventiones mathematicae
• 2019
AbstractFor almost all Riemannian metrics (in the $$C^\infty$$C∞ Baire sense) on a closed manifold $$M^{n+1}$$Mn+1, $$3\le (n+1)\le 7$$3≤(n+1)≤7, we prove that there is a sequence of closed, smooth,Expand