Related papers: Minimal faces and Schur's Lemma for embeddings int…

Given two-dimensional Riemannian manifolds $\mathcal{M},\mathcal{N}$, we prove a lower bound on the distortion of embeddings $\mathcal{M} \to \mathcal{N}$, in terms of the areas' discrepancy $V_{\mathcal{N}}/V_{\mathcal{M}}$, for a certain…

In this paper, we develop a general existence theory for properly embedded minimal surfaces with free boundary in any compact Riemannian 3-manifold $M$ with boundary $\partial M$. The main feature of our result is that no convexity…

We prove the following comparison theorem for metrics with nonnegative scalar curvature, also known as the dihedral rigidity conjecture by Gromov: for $n\le 7$, if an $n$-dimensional prism has nonnegative scalar curvature and weakly mean…

We give a basis for the space V spanned by the lowest degree part \hat{s}_\lambda of the expansion of the Schur symmetric functions s_\lambda in terms of power sums, where we define the degree of the power sum p_i to be 1. In particular,…

We solve Chui's conjecture on the simplest fractions (i.e., sums of Cauchy kernels with unit coefficients) in weighted (Hilbert) Bergman spaces. Namely, for a wide class of weights, we prove that for every $N$, the simplest fractions with…

Given a Riemannian $\mathbb{RP}^3$ with a bumpy metric or a metric of positive Ricci curvature, we show that there either exist four distinct minimal real projective planes, or exist one minimal real projective plane together with two…

For each surface besides the sphere, projective plane, and Klein bottle, we construct a face-simple minimal quadrangulation, i.e., a simple quadrangulation on the fewest number of vertices possible, whose dual is also a simple graph. Our…

We extend Hochman's work on exponentially separated self-similar measures on $\mathbb{R}$ to the real analytic setting. More precisely, let $\Phi=\left\{ \varphi_{i}\right\} _{i\in\Lambda}$ be an iterated function system on $I:=[0,1]$…

In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper says that any short embedding in codimension one can be uniformly approximated by $C^1$ isometric embeddings. This…

Here is one of the results obtained in this paper: Let $X, Y$ be two convex sets each in a real vector space, let $J:X\times Y\to {\bf R}$ be convex and without global minima in $X$ and concave in $Y$, and let $\Phi:X\to {\bf R}$ be…

We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces in closed smooth $(n+1)$--dimensional Riemannian manifolds, a theorem proved first by Pitts for $2\leq n\leq 5$ and extended later by Schoen and Simon to…

We construct new examples of immersed minimal surfaces with catenoid ends and finite total curvature, of both genus zero and higher genus. In the genus zero case, we classify all such surfaces with at most $2n+1$ ends, and with symmetry…

We investigate the compact submanifolds in Riemannian space forms of nonnegative sectional curvature that satisfy a lower bound on the Ricci curvature, that bound depending solely on the length of the mean curvature vector of the immersion.…

In this paper we investigate surfaces in $\mathbb C P^2$ without complex points and characterize the minimal surfaces without complex points and the minimal Lagrangian surfaces by Ruh-Vilms type theorems. We also discuss the liftability of…

Let $M$ be a compact 3-dimensional Riemannian manifold with nonnegative Ricci curvature and a nonempty boundary $\partial M$. Fraser and Li \cite{Fraser&Li} established a compactness theorem for the space of compact, properly embedded…

Let $B_n$ be the $n$-dimensional unit ball given by the inequality $\|x\|\leq 1$, where $\|x\|$ is the standard Euclid norm in ${\mathbb R}^n$. For an $n$-dimensional nondegenerate simplex $S$, we denote by $E$ the ellipsoid of minimum…

We prove a weak version of the Perdomo Conjecture, namely, there is a positive constant $\delta(n)>0$ depending only on $n$ such that on any closed embedded, non-totally geodesic, minimal hypersurface $M^n$ in $\mathbb{S}^{n+1}$,…

In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface $M$ into a minimally convex domain $D\subset \mathbb{R}^3$ can be approximated, uniformly on compacts in $\mathring M=M\setminus bM$, by…

In this note, we prove a Schwarz-Pick type lemma for minimal maps between negatively curved Riemannian surfaces. More precisely, we prove that if $f:M \to N$ is a minimal map with bounded Jacobian between two complete negatively curved…

Let $M$ be a compact Riemannian manifold of nonnegative Ricci curvature and $\Sigma$ a compact embedded 2-sided minimal hypersurface in $M$. It is proved that there is a dichotomy: If $\Sigma$ does not separate $M$ then $\Sigma$ is totally…