Conformal deformations of immersed discs in $R^3$ and elliptic boundary value problems

An example of a minimal surface of genus one with two catenoid ends and one Enneper end

In this paper we construct an example of a complete immersed minimal surface in $\mathbb{R}^3$ of genus one with two embedded catenoid-type ends, one Enneper-type end and total Gauss curvature $-16π.$ The proof of the existence of this example, was obtained using the Weierstrass representation, the theory of elliptic functions and explicitly solving the period problem.

带有浸入特殊拉格朗日链的切锥的唯一性

Uniqueness of tangent cones with immersed special Legendrian links

该研究在特定假设下证明了$\boldsymbol{\text{C}}^3$中带有横截二重极点的紧浸入特殊拉格朗日链的特殊拉格朗日切锥的唯一性，核心方法基于离散Lojasiewicz型不等式，源于胶合障碍机制。

Immersed solutions of Plateau's problem for piecewise smooth boundary curves with small total curvature

We provide a new proof of the classical result that any closed rectifiable Jordan curve Gamma in space being piecewise of class C^2 bounds at least one immersed minimal surface of disc-type, under the additional assumption that the total curvature of Gamma is smaller than 6*Pi. In contrast to the methods due to Osserman, Gulliver and Alt, our proof relies on a polygonal approximation technique, using the existence of immersed solutions of Plateau's problem for polygonal boundary curves, provided by the first author's accomplishment of Garnier's ideas.

Contracting Convex Immersed Closed Plane Curves with Slow Speed of Curvature

We study the contraction of a convex immersed plane curve with speed (1/α)k^{α}, where αin(0,1] is a constant and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution. We also discuss a special symmetric case of type two blow-up and show that it converges to a translational self-similar solution. In the case of curve shortening flow (i.e., when α=1), this translational self-similar solution is the familiar "Grim Reaper" (a terminology due to M. Grayson [GR]).

Fundamental tone estimates for elliptic operators in divergence form and geometric applications

We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the $L_{r}$ operator associated to immersed hypersurfaces with locally bounded $(r+1)$-th mean curvature $H_{r+1}$ of the space forms $\mathbb{N}^{n+1}(c)$ of curvature $c$. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of $\mathbb{N}^{n+1}(c)$ with $H_{r+1}>0$ in terms of the $r$-th and $(r+1)$-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues…

Curvature estimates for immersed hypersurfaces in Riemannian manifolds

We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general Riemannian manifold. The estimate has a direct consequence for the Weyl isometric embedding problem of $(\mathbb S^2, g)$ in $3$-dimensional warped product space $(N^3, \bar g)$. We also discuss isometric embedding problem in spaces with horizon in general relativity, like the Anti-de Sitter-Schwarzschild manifolds and the Reissner-Nordström manifolds.