Related papers: $K$-surfaces with free boundaries

We study expansions near the boundary of solutions to the Dirichlet problem for the constant mean curvature equation in the hyperbolic space. With a characterization of remainders of the expansion by multiple integrals, we establish optimal…

For a bounded closed convex set $K$, in this note, we study the FPP for $\alpha$-H\"older nonexpansive maps, i.e. mappings $T\colon K\to K$ for which $\|T x -Ty\| \leq\| x - y\|^\alpha$ for all $x, y\in K$, $\alpha\in (0,1)$. First, we note…

We remark on two different free-boundary problems for holomorphic curves in nearly-K\"{a}hler 6-manifolds. First, we observe that a holomorphic curve in a geodesic ball $B$ of the round 6-sphere that meets $\partial B$ orthogonally must be…

The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in $\mathbb R^n$. By classical results of Caffarelli, the free boundary is $C^\infty$ outside a set of singular points. Explicit examples…

In [17], Labourie initiated the study of the dynamical properties of the space of $k$-surfaces, that is, suitably complete immersed surfaces of constant extrinsic curvature in $3$-dimensional manifolds, which he presented as a…

We investigate the boundary behavior of variational solutions of Dirichlet problems for prescribed mean curvature equations at smooth boundary points where certain boundary curvature conditions are satisfied (which preclude the existence of…

A long-standing conjecture of De Giorgi asserts that every monotone solution of the Allen--Cahn equation in \(\mathbb{R}^{n+1}\) is one-dimensional if \(n \leq 7\). A stronger version of the conjecture, also widely studied and often called…

Given a compact Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$ and $\partial M\neq\emptyset$, the free boundary min-max theory built by Martin Man-Chun Li and Xin Zhou shows the existence of a smooth almost properly embedded…

We establish rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we show that any free boundary minimal hypersurface of Morse index I admits an I-parameter family…

We study the free boundary Euler equations with surface tension in three spatial dimensions, showing that the equations are well-posed if the coefficient of surface tension is positive. Then we prove that under natural assumptions, the…

We prove that any complete non-compact K\"ahler surface with positive sectional curvature is biholomorphic to $\mathbb{C}^2$, establishing the two dimensional case of the weaker form of Yau's uniformisation conjecture. In contrast to all…

We consider a complete biharmonic hypersurface with nowhere zero mean curvature vector field $\phi:(M^m,g)\rightarrow (S^{m+1},h)$ in a sphere. If the squared norm of the second fundamental form $B$ is bounded from above by m, and $\int_M…

These lecture notes are concerned with the solvability of the second boundary value problem of the prescribed affine mean curvature equation and related regularity theory of the Monge-Amp\`ere and linearized Monge-Amp\`ere equations. The…

We study a two-phase free boundary problem in which the two-phases satisfy an impenetrability condition. Precisely, we have two ordered positive functions, which are harmonic in their supports, satisfy a Bernoulli condition on the one-phase…

On a finite-volume hyperbolic $3$-manifold, we establish an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and the genus bounds. This area bound implies…

We show that for a very general and natural class of curvature functions (for example the curvature quotients $(\sigma_n/\sigma_l)^{\frac{1}{n-l}}$) the problem of finding a complete spacelike strictly convex hypersurface in de Sitter space…

We study conformally flat surfaces with prescribed Gaussian curvature, described by solutions $u$ of the PDE: $\Delta u(x)+K(x)\exp(2u(x))=0$, with $K(x)$ the Gauss curvature function at $x\in\RR^2$. We assume that the integral curvature is…

In this article, we study a free boundary isometric embedding problem for abstract Riemannian two-manifolds with the topology of the disc. Under the assumption of positive Gauss curvature and geodesic curvature of the boundary being equal…

Surfaces of constant negative curvature in Euclidean space can be described by either the sine-Gordon equation for the angle between asymptotic directions, or a Monge-Ampere equation for the graph of the surface. We present the explicit…

Let $M$ be a compact connected surface with boundary. We prove that the signal condition given by the Gauss-Bonnet theorem is necessary and sufficient for a given smooth function $f$ on $\partial M$ (resp. on $M$) to be geodesic curvature…