Related papers: Conformal Structures and Necksizes of Embedded Con…

Constant mean curvature (CMC) surfaces in space forms can be described by their associated $\mathbb C^*$-family of flat $SL(2,\mathbb C)$-connections $\nabla^\lambda$. In this paper we consider the asymptotic behavior (for $\lambda\to0$) of…

The Gauss map of a conformal minimal immersion of an open Riemann surface $M$ into $\mathbb{R}^n$, $n\ge 3$, is a holomorphic map $M\to{\bf Q}^{n-2}\subset \mathbb{CP}^{n-1}$. Denote by ${\rm CMI}_{\rm full}(M,\mathbb{R}^n)$ and…

We prove that on a compact Riemannian manifold of dimension $3$ or higher, with positive Ricci curvature, the Allen--Cahn min-max scheme (implemented by the first author and N. Wickramasekera in 2020), with prescribing function taken to be…

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…

This article is concerned with the geometry of germs of real analytic surfaces in $(\mathbb{C}^2,0)$ having an isolated Cauchy-Riemann (CR) singularity at the origin. These are perturbations of {\it Bishop quadrics}. There are two kinds of…

We investigate surfaces with constant harmonic-mean curvature one (HMC-1 surfaces) in hyperbolic three-space. We allow them to have certain kinds of singularities, and discuss some global properties. As well as flat surfaces and surfaces…

An embedded cubic graph consisting of segments of geodesics such that the angles at any vertex are equal to $2\pi/3$ is a closed local minimal net. This net is regular if all segments of geodesics are equal. The problem of classification of…

We prove a converse to well-known results by E. Cartan and J. D. Moore. Let $f\colon M^n_c\to\Q^{n+p}_{\tilde c}$ be an isometric immersion of a Riemannian manifold with constant sectional curvature $c$ into a space form of curvature…

Let $ X $ be an oriented, closed manifold with $ \dim X \geqslant 2 $. Let $ (Z, \partial Z) $ be an oriented, compact manifold with (possibly empty) smooth boundary and $ \dim Z \geqslant 2 $. In this article, we show that if the…

It is still an open question whether a compact embedded hypersurface in the Euclidean space R^{n+1} with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of…

Constant mean curvature surfaces in $S^3$ can be studied via their associated family of flat connections. In the case of tori this approach has led to a deep understanding of the moduli space of all CMC tori. For compact CMC surfaces of…

Consider a strictly convex bounded regular domain $C$ of $\R^3$. For any arbitrary finite topological type we find a compact Riemann surface $\mathcal{M}$, an open domain $M\subset \mathcal{M}$ with the fixed topological type, and a…

In [20], Ros and Vergasta proved that an immersed orientable compact stable constant mean curvature surface $\Sigma$ with free boundary in a closed ball $B\subset\mathbb{R}^3$ must be a planar equator, a spherical cap or a surface of genus…

Given $I,B\in\mathbb{N}\cup \{0\}$, we investigate the existence and geometry of complete finitely branched minimal surfaces $M$ in $\mathbb{R}^3$ with Morse index at most $I$ and total branching order at most $B$. Previous works of…

For each $k\geq 3$, we construct a 1-parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space $\mathbb{H}^2\times\mathbb{R}$ with genus $1$ and $k$ embedded ends asymptotic to vertical…

Anatomical structures such as the hippocampus, liver, and bones can be analyzed as orientable, closed surfaces. This permits the computation of volume, surface area, mean curvature, Gaussian curvature, and the Euler-Poincar\'e…

Using connections to random matrix theory and orthogonal polynomials, we develop a framework for obtaining explicit closed-form formulae for the number, $\mathscr{N}_{g}(2\nu,j)$, of connected $2\nu$-valent labeled graphs with $j$ vertices…

We investigate the structure of conformally rigid graphs. Graphs are conformally rigid if introducing edge weights cannot increase (decrease) the second (last) eigenvalue of the Graph Laplacian. Edge-transitive graphs and distance-regular…

Riemannian manifolds of quasi-constant sectional curvatures (QC-manifolds) are divided into two basic classes: with positive or negative horizontal sectional curvatures. We prove that the Riemannian QC-manifolds with positive horizontal…

We prove that on any compact manifold $M^n$ with boundary, there exist a conformal class $C$ such that for any riemannian metric $g\in C$, $\lambda_1(M^n,g)Vol(M^n,g)^{2/n}< n.Vol(S^n,g_{\textrm{can}})^{2/n}$ and $\sigma_1(M,g,\rho)\mathcal…