Related papers: Minimal graphs of arbitrary codimension in Euclide…

Generalizing the Cauchy-Riemann equations, we construct the Osserman system of the first order for a pair $\left(f(x, y), g(x,y) \right)$ of two ${\mathbb{R}}$-valued functions on the domain $\Omega \subset {\mathbb{R}}^{2}$. The graph…

We consider the following extension of the classical Liouville theorem: A calibration $\omega \in \Lambda^n \mathbb{R}^m$, where $3 \le n \le m$, has the Liouville property if a Sobolev mapping $F\colon \Omega \to \mathbb{R}^m$, where…

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 admitting a "no-hole" condition on the…

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…

We show geodesic completeness of certain compact locally symmetric pseudo-Riemannian manifolds of signature $(2,n)$. Our model space $\mathbf{X}$ is a $1$-connected, indecomposable symmetric space of signature $(2,n)$, that admits a unique…

Given a bounded Lipschitz domain $\omega\subset\mathbb{R}^{d-1}$ and a lower semicontinuous function $W:\mathbb{R}^N\to\mathbb{R}_+\cup\{+\infty\}$ that vanishes on a finite set and that is bounded from below by a positive constant at…

Let $(M,g)$ be a non-compact riemannian $n$-manifold with bounded geometry at order $k\geq\frac{n}{2}$. We show that if the spectrum of the Laplacian starts with $q+1$ discrete eigenvalues isolated from the essential spectrum, and if the…

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold…

We discover some very general configuration results for constructing area-minimizing cones. In particular, given any closed minimal submanifold in some Euclidean sphere, every cone over the minimal product of sufficiently many copies of the…

Suppose that there exists a discrete subset $X$ of a complete, connected, $n$-dimensional Riemannian manifold $M$ such that the Riemannian distances between points of $X$ correspond to the Euclidean distances of a net in $\mathbb{R}^{n}$.…

Consider an $m$-dimensional area minimizing mod$(2Q)$ current $T$, with $Q\in\mathbb{N}$, inside a sufficiently regular Riemannian manifold of dimension $m + 1$. We show that the set of singular density-$Q$ points with a flat tangent cone…

Liebmann's Theorem asserts that a compact, connected, convex surface with constant mean curvature (CMC) in the Euclidean space must be a totally umbilical sphere. In this article we extend Liebmann's result to hypersurfaces with boundary.…

This article studies typical 1-Lipschitz images of $n$-rectifiable metric spaces $E$ into $\mathbb{R}^m$ for $m\geq n$. For example, if $E\subset \mathbb{R}^k$, we show that the Jacobian of such a typical 1-Lipschitz map equals 1…

We show that simply connected toric hyperK\"ahler metrics of finite topological type with maximal volume growth are generically quasi-asymptotically conical, which allows us to compute explicitly their reduced $L^2$-cohomology groups. In…

In a recent paper, Cs\"ornyei and Wilson prove that curves in Euclidean space of $\sigma$-finite length have tangents on a set of positive $\mathscr{H}^{1}$-measure. They also show that a higher dimensional analogue of this result is not…

In this paper, we show that a closed manifold $M^{n+1} (n \geq 7)$ endowed with a $C^\infty$-generic (Baire sense) metric contains infinitely many singular minimal hypersurfaces with optimal regularity. Moreover, for $2 \leq n \leq 6$, our…

We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold $M$ with only one end if $M$ has asymptotically non-negative…

In this paper we prove Schur's conjecture in $\mathbb R^d$, which states that any diameter graph $G$ in the Euclidean space $\mathbb R^d$ on $n$ vertices may have at most $n$ cliques of size $d$. We obtain an analogous statement for…

In this note, we derive a uniqueness theorem for minimal graphs of general codimension under certain restrictions closed related to the convexity (not strict convexity) of the area functional with respect to singular values, improving the…

This paper discusses the regularity of multiple-valued Dirichlet minimizing maps into the sphere. It shows that even at branched point, as long as the normalized energy is small enough, we have the energy decay estimate. Combined with the…