arXiv research

The paper proves the existence of area-minimizing hypersurfaces in AF manifolds of higher dimensions.

Singularities of area minimizing hypersurfaces can be smoothed in dimensions 9 and 10.

Study area-minimizing hypersurfaces in singular manifolds with nonnegative scalar curvature.

Study shows how certain metrics can be split into warped products.

We extend the results of Hardt and Simon on area-minimizing cones to prove that isolated singularities of stationary one-sided area-minimizing hypersurfaces can be locally perturbed away on the side that they are minimizing.

Study area-minimizing hypersurfaces in manifolds with controlled curvature.

The paper proves foliation of area-minimizing hypersurfaces in asymptotically flat manifolds.

This paper proves all Pfaffian varieties are area-minimizing except hypersurfaces.

Smooth minimizing hypersurfaces in 11D are generic, with singularities in higher dimensions.

We give an a priori bound on the (n-7)-dimensional measure of the singular set for an area-minimizing n-dimensional hypersurface, in terms of the geometry of its boundary.

In this paper we show that every area minimizing cone C^{n-1} in R^n can be approximated by entirely smooth area minimizing hypersurfaces. This extensively uses hyperbolic unfoldings of such hypersurfaces and the resulting potential theory for the Jacobi field operator. Applications include the splitting theorem in sca…

We prove that a strictly stable constant-mean-curvature hypersurface in a smooth manifold of dimension less than or equal to 7 is uniquely homologically area minimizing for fixed volume in a small L^1 neighborhood.

We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Be…

Perturbs area-minimizing hypersurfaces to reduce singular set's dimension.

The study proves that certain stable minimal hypersurfaces must be cylindrical.

This is the first in a series of papers where we develop new structural elements on singular area minimizing hypersurfaces, the skin structures. They disclose previously unapproachable and largely unexpected geometric and analytic properties of such hypersurfaces.

We show that if an asymptotically flat manifold with horizon boundary admits a global static potential, then the static potential must be zero on the boundary. We also show that if an asymptotically flat manifold with horizon boundary admits an unbounded static potential in the exterior region, then the manifold must c…

Uniqueness proven for cylindrical tangent cones in high dimensions.

Extends isoparametric foliations and area-minimizing cones in product manifolds.

Analyzes singularities of area minimizing hypersurfaces modulo p, completing the structure analysis.

The study finds generic regularity of minimal hypersurfaces in Riemannian manifolds.

Rectifies flat singular points for area-minimizing currents.

We study the constant mean curvature (CMC) hypersurfaces in hyperbolic space whose asymptotic boundaries are closed codimension-1 submanifolds in sphere at infinity. We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area m…

The paper characterizes gaps in minimal foliations on tori using energy criteria.

In this paper, we study self-expanding solutions for mean curvature flows and their relationship to minimal cones in Euclidean space. In [18], Ilmanen proved the existence of self-expanding hypersurfaces with prescribed tangent cones at infinity. If the cone is $C^{3,α}$-regular and mean convex (but not area-minimizing…

Minimal hypersurfaces with cylindrical tangent cones constructed and analyzed.

New findings show that not all area-minimizing surfaces are calibrated, even on complex manifolds.

Localized min-max method proves minimal hypersurface existence.

Low-entropy surfaces can be flowed into spheres and cylinders.

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}^…

Unique cylindrical tangent cone for Simons' hypersurface found.

Study area minimizing currents in Riemannian manifolds, proving unique structure and decay.

The doubling conjecture for positive scalar curvature is proven under certain conditions.

New results show area-minimizing surfaces have fewer singularities than expected.

This is the third in a series of papers on the geometry and analysis of singular area minimizing hypersurfaces. We show how to derive obstruction and structure theories for scalar curvature constraints without imposing dimensional or topological restrictions on the underlying manifold. To this end, we use skin structur…

Paper proves properties of minimal hypersurfaces in specific solitons.

This paper solves minimal surface equations near Hardt-Simon foliations.

We study the intrinsic geometry of area minimizing (and also of almost minimizing) hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. For any such hypersurface we define and construct a so-called S-structure which reveals some unexpected geometric and analytic properties of the …

This is the second in a series of papers where we estab- lish skin structural concepts and results for singular area minimizing hypersurfaces. Here we conformally unfold these spaces to complete Gromov hyperbolic spaces with bounded geometry and we recover their singular set as the Gromov boundary but also as the Marti…

Paper derives a formula for renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.

Hyperplanes, hyperspheres and hypercylinders in $\Bbb R^n$ with suitable densities are proved to be weighted minimizing by a calibration argument. Also calibration method is used to prove a weighted minimal hypersurface is weighted area-minimizing locally.

Characterizations of entire subsolutions for the 1-harmonic equation of a constant 1$-tension field are given with applications in geometry via transformation group theory. In particular, we prove that every level hypersurface of such a subsolution is calibrated and hence is area-minimizing over$\mathbb{R}$; and every…

We introduce and study co-dimension one area-minimizing locally rectifiable currents $T$ with $C^{1,α}$ tangentially immersed boundary: $\partial T$ is locally a finite sum of orientable co-dimension two submanifolds which only intersect tangentially with equal orientation. We show that any such $T$ is supported in a s…

We explain how to derive largeness constraints in scalar curvature geometry using some basic splitting results and the potential theory on singular area minimizing hypersurfaces. This includes a variety of results like the non-existence of positive scalar curvature metrics on enlargeable manifolds or simplified proofs …

We give a simple topological argument to show that the number of solutions of the asymptotic Plateau problem in hyperbolic space is generically unique. In particular, we show that the space of codimension-1 closed submanifolds of sphere at infinity, which bounds a unique absolutely area minimizing hypersurface in hyper…

A result of B.Solomon (On the Gauss map of an area-minimizing hypersurface. 1984. Journal of Differential Geometry, 19(1), 221-232.) says that a compact minimal hypersurface $M^k$ of the sphere $S^{k+1}$ with $H^1(M)=0$, whose Gauss map omits a neighborhood of an $S^{k-1}$ equator, is totally geodesic in $S^{k+1}$. We …

Let $M$ be a weighted manifold with boundary $\partial M$, i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of the interior weighted area for deformations by hypersurfaces with boundary in $\p…

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 l…