Related papers: Quantitative rigidity results for conformal immers…

Given a sequence of properly embedded minimal surfaces in a $3$-manifold with local bounds on area and genus, we prove subsequential convergence, smooth away from a discrete set, to a smooth embedded limit surface, possibly with…

We establish a Mittag-Leffler-type theorem with approximation and interpolation for meromorphic curves $M\to \mathbb{C}^n$ ($n\geq 3$) directed by Oka cones in $\mathbb{C}^n$ on any open Riemann surface $M$. We derive a result of the same…

Let $M$ be an open Riemann surface and $n\ge 3$ be an integer. In this paper we establish some generic properties (in Baire category sense) in the space of all conformal minimal immersions $M\to\mathbb{R}^n$ endowed with the compact-open…

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere…

In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that…

We study ends of an oriented, immersed, non-compact, complete Willmore surfaces, which are critical points of the integral of the square of the mean curvature, in asymptotically flat spaces of any dimension; assuming the surface has…

In this paper, we firstly extend Theorem 5.1.1 in \cite {Helein} due to H\'elein to a rescaled branched conformal immersed sequence(c.f. Theorem 1.5). By virtue of this local convergence theorem, we study the blowup behavior of a sequence…

For a smooth immersion $f$ from the punctured disk $D\backslash\{0\}$ into $\mathbb{R}^n$ extendable continuously at the puncture, if its mean curvature is square integrable and the measure of $f(D)\cap B_{r_k}=o(r_k)$ for a sequence…

In this paper, we study complete minimal surfaces in $\mathbb{R}^4$ with three embedded planar ends parallel to those of the union of the Lagrangian catenoid and the plane passing through its waist circle. We show that any complete,…

We study minimal immersions of closed surfaces (of genus $g \ge 2$) in hyperbolic 3-manifolds, with prescribed data $(\sigma, t\alpha)$, where $\sigma$ is a conformal structure on a topological surface $S$, and $\alpha dz^2$ is a…

This paper is devoted to the study of the global properties of harmonically immersed Riemann surfaces in $\mathbb{R}^3.$ We focus on the geometry of complete harmonic immersions with quasiconformal Gauss map, and in particular, of those…

A compactness theorem is proved for a family of K\"{a}hler surfaces with constant scalar curvature and volume bounded from below, diameter bounded from above, Ricci curvature bounded and the signature bounded from below. Furthermore, a…

We study Willmore surfaces of constant Moebius curvature $K$ in $S^4$. It is proved that such a surface in $S^3$ must be part of a minimal surface in $R^3$ or the Clifford torus. Another result in this paper is that an isotropic surface…

We derive an infinitesimal rigidity lemma for the strain tensor of surfaces with their curvatures changing sign. As an application, we obtain the optimal constant in the first Korn inequality scales like $h^{4/3}$ for such shells of mixed…

We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding…

The goal of this article is to study minimal surfaces in $\mathbb{M}^2 \times \mathbb{R}$ having finite total curvature, where $\mathbb{M}^2$ is a Hadamard manifold. The main result gives a formula to compute the total curvature in terms of…

In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a 3-sphere which has scalar curvature greater than or equal to 6 and is not round must have an…

Using a new estimate for the Peng-Terng invariant and the multiple-parameter method, we verify a rigidity theorem on the stronger version of Chern Conjecture for minimal hypersurfaces in spheres. More precisely, we prove that if $M$ is a…

We revise the steady vortex surface theory following the recent finding of asymmetric vortex sheets (AM,2021). These surfaces avoid the Kelvin-Helmholtz instability by adjusting their discontinuity and shape. The vorticity collapses to the…

We consider surfaces which minimize a nonlocal perimeter functional and we discuss their interior regularity and rigidity properties, in a quantitative and qualitative way, and their (perhaps rather surprising) boundary behavior. We present…