相关论文: Minimal tori in five-dimensional sphere in $C^3$
This is an expanded version of my plenary lecture at the 8th European Congress of Mathematics in Portoro\v{z} on 23 June 2021. The main part of the paper is a survey of recent applications of complex-analytic techniques to the theory of…
Combining the intrinsic and extrinsic geometry, we generalize Einstein manifolds to Integral-Einstein (IE) submanifolds. A Takahashi-type theorem is established to characterize minimal hypersurfaces with constant scalar curvature (CSC) in…
We give necessary conditions on complete embedded \cmc surfaces with three or four ends subject to reflection symmetries. The respective submoduli spaces are two-dimensional varieties in the moduli spaces of general \cmc surfaces. We…
We deal with minimal surfaces in the unit sphere $S^3$, which are one-parameter families of circles. Minimal surfaces in $\R^3$ foliated by circles were first investigated by Riemann, and a hundred years later Lawson constructed examples of…
For $m\in \IN, m\geq 1,$ we determine the irreducible components of the $m-th$ jet scheme of a toric surface $S.$ For $m$ big enough, we connect the number of a class of these irreducible components to the number of exceptional divisors on…
The minimal surface equation $Q$ in the second order contact bundle of $R^3$, modulo translations, is provided with a complex structure and a canonical vector-valued holomorphic differential form $Omega$ on $Q\0$. The minimal surfaces $M$…
This paper is devoted to a study of the connection between the immersion functions of two-dimensional surfaces in Euclidean or hyperbolic spaces and classical orthogonal polynomials. After a brief description of the soliton surfaces…
We introduce the moduli space of spectral curves of constant mean curvature (\cmc\hspace{-5pt}) cylinders of finite type in the round unit 3-sphere. The subset of spectral curves of mean-convex Alexandrov embedded cylinders is explicitly…
The equivariant CR minimal immersions from the round $3$-sphere $S^3$ into the complex projective space $\mathbb CP^n$ have been classified by the third author explicitly (J London Math Soc 68: 223-240, 2003). In this paper, by employing…
In this paper we find examples of slant surfaces in the nearly Kahler six sphere. First, we characterize two-dimensional small and great spheres which are slant. Their description is given in terms of the associative 3-form in $\Im \OO .$…
We analyze spectral minimal $k$-partitions for the torus. In continuation with what we have obtained for thin annuli or thin strips on a cylinder (Neumann case), we get similar results for anisotropic tori.
We study Gromov's problem concerning minimal normal curvature immersions in the unit ball. In particular, we determine the minimal possible value of the normal curvature of an $S^n\times S^1$. We also prove a differentiable sphere theorem…
The edge-to-edge tilings of the sphere by congruent quadrilaterals of Type $a^2bc$ are classified as $3$ classes: a sequence of two-parameter families of $2$-layer earth map tilings with $2n$ $(n\ge3)$ tiles, a one-parameter family of…
The fundamental group of $M = \sharp_n (S^2\times S^1)$ is $F_n$, the free group with $n$ generators. There is a 1-1 correspondence between the equivalence classes of $\mathbb{Z}$-- splittings of $F_n$ and homotopy classes of embedded…
We construct eternal mean curvature flows of tori in perturbations of the standard unit sphere $\Bbb{S}^3$. This has applications to the study of the Morse homologies of area functionals over the space of embedded tori in $\Bbb{S}^3$.
In this paper we have proved several approximation theorems for the family of minimal surfaces in R^3 that imply, among other things, that complete minimal surfaces are dense in the space of all minimal surfaces endowed with the topology of…
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…
We show that the theory of isothermic surfaces in $\E^3$ -- one of the oldest branches of differential geometry -- can be reformulated within the modern theory of completely integrable (soliton) systems. This enables one to study the…
We present a general construction of embedded minimal and constant mean curvature surfaces in $\mathbb{S}^n$ and one-phase free boundaries joined by a smooth interpolation by capillary hypersurfaces. This framework recovers all known…
We study finite order invariants of null-homotopic immersions of a closed orientable surface into an aspherical orientable 3-manifold. We give the foundational constructions, and classify all order one invariants.