Related papers: An essential one sided boundary singularity for a …

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 neighbourhood of any point at which at least one…

We consider an area-minimizing integral current $T$ of codimension higher than $1$ in a smooth Riemannian manifold $\Sigma$. In a previous paper we have subdivided the set of interior singular points with at least one flat tangent cone…

F.-H. Lin studied minimal graphs of the Dirichlet problem in the hyperbolic space and proved that any such minimal graph has the same global regularity as the boundary if the dimension of the minimal graph is even and that there is an…

We prove that the combination of strict positivity of $k$-tri-Ricci curvature with non-negative $3$-intermediate Ricci curvature forces rigidity of two-sided stable free boundary minimal hypersurface in a 5-manifold with bounded geometry…

We develop a regularity and compactness theory for stable capillary minimal hypersurfaces in the half-space $\mathbb{H}^{n+1}$ with contact angle $\theta \in (0,\pi)$ and dimension $n \geq 2$. As a consequence, we obtain the generalized…

In this paper, we study minimizers of the Chon\'e--Rochet variational problem in dimension two. We first establish global $C^1$ regularity on arbitrary bounded convex domains, and then prove global $C^{1,1}$ regularity on bounded strictly…

We study domain wall solutions in d=5, N=2 supergravity coupled to a single hypermultiplet whose moduli space is described by certain inhomogeneous, toric ESD manifolds constructed recently by Calderbank and Singer. Upon gauging a generic…

We show that for a generic $8$-dimensional Riemannian manifold with positive Ricci curvature, there exists a smooth minimal hypersurface. Without the curvature condition, we show that for a dense set of 8-dimensional Riemannian metrics…

We study numerically transitional coherent structures in a boundary-layer flow with homogeneous suction at the wall (the so-called asymptotic suction boundary layer ASBL). The dynamics restricted to the laminar-turbulent separatrix is…

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}.…

For an oriented surface $S$, the singular set of a fold map $f:S\rightarrow \mathbb{R}^2$ is a collection of smooth curves, also known as fold singularities. We construct a sharp lower bound on the number of self-intersections of such fold…

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…

The formation of singularities on a free surface of a conducting ideal fluid in a strong electric field is considered. It is found that the nonlinear equations of two-dimensional fluid motion can be solved in the small-angle approximation.…

Motivated by various applications, this article develops the notion of boundary control for Maxwell's equations in the frequency domain. Surface curl is shown to be the appropriate regularization in order for the optimal control problem to…

We prove that the boundary of the trapped region in an asymptotically Euclidean Riemannian manifold of dimension at least 3 is a stable smooth minimal hypersurface except for a singular set of codimension at least 8.

We study a higher order analogue to the Alt-Caffarelli functional that arises in several shape optimization problems, among which the minimization of the critical buckling load of a clamped plate of fixed area. We obtain several regularity…

We study the Cauchy problem for the $3D$ compressible Euler equations under an arbitrary equation of state with positive speed of sound, aside from that of a Chaplygin gas. For open sets of smooth initial data with non-trivial vorticity and…

We establish a first general partial regularity theorem for area minimizing currents $\mathrm{mod}(p)$, for every $p$, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of…

A model of simplicial quantum gravity in three dimensions(3D) was investigated numerically based on the technique of dynamical triangulation (DT). We are concerned with the genus of surfaces appearing on boundaries (i.e., sections) of a 3D…

We explicitly bound T-singularities on normal projective surfaces $W$ with one singularity, and $K_W$ ample. This bound depends only on $K_W^2$, and it is optimal when $W$ is not rational. We classify and realize surfaces attaining the…