# Regularity of minimal surfaces near quadratic cones

@article{Edelen2019RegularityOM, title={Regularity of minimal surfaces near quadratic cones}, author={Nick Edelen and Luca Spolaor}, journal={arXiv: Differential Geometry}, year={2019} }

Hardt-Simon proved that every area-minimizing hypercone $\mathbf{C}$ having only an isolated singularity fits into a foliation of $\mathbb{R}^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $\mathbf{C}$. In this paper we prove that if a stationary $n$-varifold $M$ in the unit ball $B_1 \subset \mathbb{R}^{n+1}$ lies sufficiently close to a minimizing quadratic cone (for example, the Simons' cone $\mathbf{C}^{3,3}$), then $\mathrm{spt} M \cap B_{1/2}$ is a $C^{1,\alpha… Expand

#### 4 Citations

Uniqueness of certain cylindrical tangent cones

- Mathematics
- 2020

We show that the cylindrical tangent cone $C\times \mathbf{R}$ for an area-minimizing hypersurface is unique, where $C$ is the Simons cone $C_S= C(S^3\times S^3)$. Previously Simon proved a… Expand

Deformations of Singular Minimal Hypersurfaces I, Isolated Singularities.

- Mathematics
- 2020

Locally stable minimal hypersurface could have singularities in dimension $\geq 7$ in general, locally modeled on stable and area-minimizing cones in the Euclidean spaces. In this paper, we present… Expand

Minimal hypersurfaces with cylindrical tangent cones

- Mathematics
- 2021

First we construct minimal hypersurfaces M ⊂ R in a neighborhood of the origin, with an isolated singularity but cylindrical tangent cone C × R, for any strictly minimizing strictly stable cone C in… Expand

Regularity of free boundary minimal surfaces in locally polyhedral domains

- Mathematics
- 2020

We prove an Allard-type regularity theorem for free-boundary minimal surfaces in Lipschitz domains locally modelled on convex polyhedra. We show that if such a minimal surface is sufficiently close… Expand

#### References

SHOWING 1-10 OF 13 REFERENCES

The singular set of minimal surfaces near polyhedral cones

- Mathematics
- 2017

We adapt the method of Simon [JDG '93] to prove a $C^{1,\alpha}$-regularity theorem for minimal varifolds which resemble a cone $\bf{C}_0^2$ over an equiangular geodesic net. For varifold classes… Expand

Uniqueness of some cylindrical tangent cones

- Mathematics
- 1994

It is known ([SL1]) that if a minimal submanifold M has a tangent cone C at a singular point p, then C is the unique tangent cone of M at p, and M approaches C asymptotically in the appropriate… Expand

Transverse Singularities of Minimal Two-Valued Graphs in Arbitrary Codimension

- Mathematics
- 2014

We prove some epsilon regularity results for n-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a… Expand

On the radial behavior of minimal surfaces and the uniqueness of their tangent cones

- Mathematics
- 1981

Suppose V is an m + 1-dimensional minimal surface in R" containing 0 as an isolated singular point. What can one say about the structure of V near O? This question initiated the present investigation… Expand

Cylindrical tangent cones and the singular set of minimal submanifolds

- Mathematics
- 1993

The question of what can be said about the structure of the singular set of minimal surfaces and the extrema of other geometric variational problems has remained largely open. Indeed, for minimal… Expand

On the first variation of a varifold

- Mathematics
- 1972

Suppose M is a smooth m dimensional Riemannian manifold and k is a positive integer not exceeding m. Our purpose is to study the first variation of the k dimensional area integrand in M. Our main… Expand

Minimal surfaces with isolated singularities

- Mathematics
- 1984

For n≥3, there exists an embedded minimal hypersurface in Rn+1 which has an isolated singularity but which is not a cone. Each example constructed here is asymptotic to a given, completely arbitrary,… Expand

Asymptotics for a class of non-linear evolution equations, with applications to geometric problems

- Mathematics
- 1983

Soit Σ une variete de Riemann compacte et soit une fonction reguliere u=u(x,t), (x,t)∈ΣX(0,T) (T>0) satisfaisant une equation d'evolution soit de la forme ci-#7B-M(u)=f soit de la forme… Expand

Minimal cones and the Bernstein problem

- Mathematics, Computer Science
- 1969

DigiZeitschriften e.V. gewährt ein nicht exklusives, nicht übertragbares, persönliches and beschränktes Recht auf Nutzung dieses Dokuments, der Copyright bleibt bei den Herausgebern oder sonstigen Rechteinhaber vor. Expand