We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves we prove that such a solution is locally Hölder continuous in the interior and continuous up to the boundary. Our results generalize corresponding results of Douglas and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces.
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. Below this value, in contrast, we construct examples.
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.
We study existence and structure of $P-$area minimizing surfaces in the Heisenberg group under Dirichlet and Neumann boundary conditions. We show that there exists an underlying vector field $N$ that characterized existence and structure of $P$-area minimizing surfaces. This vector field exists even if there is no $P$-area minimizing surface satisfying the prescribed boundary conditions. We prove that if $\partial Ω$ satisfies a so called Barrier condition, it is sufficient to guarantee existence of such surfaces. Our approach is completely different from previous methods in the literature and…
In this paper we continue to study the connection among the area minimizing problem, certain area functional and the Dirichlet problem of minimal surface equations in a class of conformal cones with a similar motivation from \cite{GZ20}. These cones are certain generalizations of hyperbolic spaces. We describe the structure of area minimizing $n$-nteger multiplicity currents in bounded $C^2$ conformal cones with prescribed $C^1$ graphical boundary via a minimizing problem of these area functionals. As an application we solve the corresponding Dirichlet problem of minimal surface equations unde…
The Plateau-Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. Th…
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.
In this paper we study the area minimizing problem in some kinds of conformal cones. This concept is a generalization of the cones in Eulcidean spaces and the cylinders in product manifolds. We define a non-closed-minimal (NCM) condition for bounded domains. Under this assumption and other necessary conditions we establish the existence of bounded minimal graphs in mean convex conformal cones. Moreover those minimal graphs are the solutions to corresponding area minizing problems. We can solve the area minimizing problem in non-mean convex translating conformal cones if these cones are contain…
We study the stability of partitions in convex domains involving simultaneous coexistence of three phases, viz. triple junctions. We present a careful derivation of the formula for the second variation of area, written in a suitable form with particular attention to boundary and spine terms, and prove, in contrast to the two phase case, the existence of stable partitions involving a disconnected phase.
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 fact, we show that every Lawson-Osserman cone of type (n, p, 2) constructed in [XYZ] is area-minimizing.
We introduce and study a notion of relative 1-homotopy type for Sobolev maps from a surface to a metric space spanning a given collection of Jordan curves. We use this to establish the existence and local Hölder regularity of area minimizing surfaces in a given relative 1-homotopy class in proper geodesic metric spaces admitting a local quadratic isoperimetric inequality. If the underlying space has trivial second homotopy group then relatively 1-homotopic maps are relatively homotopic. We also obtain an analog for closed surfaces in a given 1-homotopy class. Our theorems generalize and streng…
We show that for an area minimizing $m$-dimensional integral current $T$ of codimension at least 2 inside a sufficiently regular Riemannian manifold, the upper Minkowski dimension of the interior singular set is at most $m-2$. This provides a strengthening of the existing $(m-2)$-dimensional Hausdorff dimension bound due to Almgren and De Lellis & Spadaro. As a by-product of the proof, we establish an improvement on the persistence of singularities along the sequence of center manifolds taken to approximate $T$ along blow-up scales.
We construct a Riemannian metric $g$ on $\mathbb{R}^4$ (arbitrarily close to the euclidean one) and a smooth simple closed curve $Γ\subset \mathbb R^4$ such that the unique area minimizing surface spanned by $Γ$ has infinite topology. Furthermore the metric is almost Kähler and the area minimizing surface is calibrated.
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 scalar curvature geometry saying that any (n-1)-dim. homology class of a compact manifold M^n with positive scalar curvature can be represented by a smooth (!) compact hypersurface that admits a positive scalar curvature metric.
We prove some existence and non-existence results for complete area minimizing surfaces in the homogeneous space $\mathbb{E}(-1,τ)$. As one of our main results, we present sufficient conditions for a curve $Γ$ in $\partial_{\infty} \mathbb{E}(-1,τ)$ to admit a solution to the asymptotic Plateau problem, in the sense that there exists a complete area minimizing surface in $\mathbb{E}(-1,τ)$ having $Γ$ as its asymptotic boundary.
A correspondence is established between a class of minimal immersed surfaces of $\mathbb{S}^3(2)$ and area-minimizing unit vector fields defined on the antipodally punctured unit sphere $\mathbb{S}^2\backslash\{N,S\}$. As a consequence, we establish a stability relation for Lawson cylinders in $\mathbb{S}^3(2)$.
We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing in the radial direction any given trace along appropriately chosen directions. In contrast to previous epiperimetric inequalities for minimal surfaces (e.g. those of Reifenberg, Taylor and White), we need no a priori assumptions on the structure of the cone (e.g. integrability). Moreover, if the cone is integrable (not only through rotations), we recover the classical epiperimetric inequality. As a consequence we deduce a new $\varepsilon$-regularity result for almost …
This paper is the continuation of the previous one \cite{Cui2021}, where we re-proved the area-minimization of cones over Grassmannians of $n$-planes $G(n,m;\mathbb{F})(\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H})$, Cayley plane $\mathbb{O}P^2$ from the point view of Hermitian orthogonal projectors, and gave area-minimizing cones associated to oriented real Grassmannians $\widetilde{G}(n,m;\mathbb{R})$. In this paper, we make a further step on showing that the cones, of dimension no less than $\mathbf{8}$, over minimal products of $G(n,m;\mathbb{F})$ are area-minimizing. Moreover, those cones …
