arXiv research

In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by a simple closed curve in the boundary of M. Our main result is that for any g>=0…

We show that for a generic nullhomotopic simple closed curve C in the boundary of a compact, orientable, mean convex 3-manifold M with trivial second homology, there is a unique area minimizing disk D embedded in M where the boundary of D is C. We also show that the same is true for absolutely area minimizing surfaces.

New theorem on 3-manifolds with curvature and convex boundary.

Let M be a compact, orientable, mean convex 3-manifold with boundary. We show that the set of all simple closed curves in the boundary of M which bound unique area minimizing disks in M is dense in the space of simple closed curves in the boundary of M which are nullhomotopic in M. We also show that the set of all simp…

In this paper, we give some examples of area minimizing surfaces to clarify some well-known features of these surfaces in more general settings. The first example is about Meeks-Yau's result on embeddedness of solution to the Plateau problem. We construct an example of a simple closed curve in R^3 which lies in the bou…

It is well known that in compact local Lipschitz neighborhood retracts in Euclidean space flat convergence for integer rectifiable currents amounts just to weak convergence. In the present paper we extend this result to integral currents in complete metric spaces admitting a local cone type inequality. These include in…

We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,α}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…

We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,α}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…

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 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…

Hasse principle applied to area-minimizing submanifolds across different homology types.

Study area-minimizing subgraphs in integer lattices.

Study shows area-minimizing submanifolds are mostly rough, not smooth.

Paper proves existence of area-minimizing submanifolds on almost any manifold with fractal singular sets.

The study finds conditions for area-minimizing cones over submanifolds.

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

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

Constructs area-minimizing submanifolds with fractal singularities.

Study improves boundary smoothness for area-minimizing currents with complex boundaries.

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

Study on existence and structure of P-area surfaces in Heisenberg group.

Proves unique continuation for area minimizing currents.

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.

It has been 40 years since Lawson and Osserman introduced the three minimal cones associated with Dirichlet problems in their 1977 Acta paper [LO77]. The first cone was shown area-minimizing by Harvey and Lawson in the celebrated paper [HL82]. In this paper, we confirm that the other two are also area-minimizing. In fa…

Proves minimality of tensor varieties, generalizing previous results.

Proves interior singular set dimension for area-minimizing currents in smooth submanifolds.

We show that area minimizing polyhedral surfaces are saddle.

Solves area-minimizing surface problem for finite curves in H^2xR.

This paper proves area-minimizing cones over products of Grassmannian manifolds.

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

Study area minimizing currents in conformal cones, solving Dirichlet problems.

Examples of area-minimizing graphs with low regularity in a specific group.

Classifies area-minimizing surfaces in R^4 as algebraic.

We show that every area-minimizing hypercone and every oriented Lawlor cone in [Law91] can be realized as a tangent cone at a point of some homologically area-minimizing singular compact submanifold. In particular this generalizes the result of N. Smale [Sma99].

Entire area-minimizing surfaces of density 2 are planar or quadratic

We show the area-minimality property of all homogeneous area-minimizing hypercones in Euclidean spaces (classified by Lawlor) following Lawson's original idea in his 72' Trans. A.M.S. paper "The equivariant Plateau problem and interior regularity". Moreover, each of them enjoys (coflat) calibrations singular only at th…

Rectifies flat singular points for area-minimizing currents.

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

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

Study area-minimizing hypersurfaces in manifolds with controlled curvature.

This is the last of a series of three papers in which we give a new, shorter proof of a slightly improved version of Almgren's partial regularity of area minimizing currents in Riemannian manifolds. Here we perform a blow-up analysis deducing the regularity of area minimizing currents from that of Dir-minimizing multip…

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

We prove prove a bridge principle at infinity for area-minimizing surfaces in the hyperbolic space $\mathbb{H}^3$, and we use it to prove that any open, connected, orientable surface can be properly embedded in $\mathbb{H}^3$ as an area-minimizing surface. Moreover, the embedding can be constructed in such a way that t…

The paper proves prevalent existence and partially determines moduli space of area-minimizing surfaces with fractal singular sets.

In this paper, by constructing area-nonincreasing retractions, we prove area-minimizing properties of some cones over minimal embeddings of R-spaces.

Study proves uniqueness of tangent cones for area-minimizing currents in higher codimensions.

We give new examples of entire area-minimizing t-graphs in the subriemannian Heisenberg group H^1. Most of the examples are locally lipschitz in Euclidean sense. Some regular examples have prescribed singular set consisting of either a horizontal line or a finite number of horizontal halflines extending from a given po…

Upper bound on singular set dimension for area-minimizing currents.