Related papers: Isometric embedding with nonnegative Gauss curvatu…

Given a metric space X, one defines its Wasserstein space W2(X) as a set of sufficiently decaying probability measures on X endowed with a metric defined from optimal transportation. In this article, we continue the geometric study of W2(X)…

We present a formal supersymmetric solution of type IIB supergravity generalizing previously known solutions corresponding to D3 branes to geometries without an orthogonal split between parallel and transverse directions. The metric is…

This note pertains to isometric embeddings endowed with certain geometric properties. We study two embedding problems for a Riemannian manifold $M$ which is diffeomorphic to $\RR^n$ and admits a Bieberbach group $\Gamma$ acting by…

This paper concerns the elastic structures which exhibit non-zero strain at free equilibria. Many growing tissues (leaves, flowers or marine invertebrates) attain complicated configurations during their free growth. Our study departs from…

Let $(M; g)$ be a smooth compact Riemiannian manifold without boundary and $g_{k}$ be a metric conformal to $g$. Suppose $vol(M; g_{k})+||R_{k}||_{L^{p}(M;g_{k})} < C$, where $R_{k}$ is the scalar curvature and $p > \frac{n}{2}$. We will…

We consider the problem of exact and inexact matching of weighted undirected graphs, in which a bijective correspondence is sought to minimize a quadratic weight disagreement. This computationally challenging problem is often relaxed as a…

If a graph $G$ can be embedded on the torus, and be embedded linklessly in $\mathbb{R}^3$, it's not known whether or not we can always find a linkless embedding of $G$ contained in the standard (unknotted) torus; We show that, for orders 9…

We consider the Grassmann graphs and dual polar graphs over the same finite field and show that, up to graph automorphism, for every dual polar graph there is the unique isometric embedding in the corresponding Grassmann graph.

For any n-dimensional compact Riemannian manifold (M,g), we construct a canonical t-family of isometric embeddings I_{t}: M->R^{q(t)}, with t>0 sufficiently small and q(t)>>t^{-n/2}. This is done by intrinsically perturbing the heat kernel…

We announce results about flat (linkless) embeddings of graphs in 3-space. A piecewise-linear embedding of a graph in 3-space is called {\it flat} if every circuit of the graph bounds a disk disjoint from the rest of the graph. We have…

Geometry of conformal minimal two-spheres immersed in $G(2,6;\mathbb{R})$ is studied in this paper by harmonic maps. We construct a non-homogeneous constant curved minimal two-sphere in $G(2,6;\mathbb{R})$, and give a classification theorem…

Given a closed subset $\La$ of the open unit ball $B_1\subset \real^n$, $n \geq 3$, we will consider a complete Riemannian metric $g$ on $\bar{B_1} \setminus \La$ of constant scalar curvature equal to $n(n-1)$ and conformally related to the…

We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in $\R^n$ for every $n\geq 9$. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular,…

The geometry of conformal minimal two-spheres immersed in G(2,6;R) is studied in this paper by harmonic maps. Then in most cases, we determine the linearly full reducible conformal minimal immersions from S^2 to G(2,8;R) identified with the…

Let (M,g) be a compact oriented Einstein 4-manifold. If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M,g) is CP2, equipped with its standard Fubini-Study metric.

A fundamental result in global analysis and nonlinear elasticity asserts that given a solution $\mathfrak{S}$ to the Gauss--Codazzi--Ricci equations over a simply-connected closed manifold $(\mathcal{M}^n,g)$, one may find an isometric…

We continue the study of the properties of graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to the ball of radius $r$ in some fixed vertex-transitive graph $F$, for various choices of $F$ and $r$. This is…

We $\delta$-approximate strictly short (e.g. constant) maps between Riemannin manifolds $f_0:X^m\to Y^N$ for $N>>m^2/2$ by $C^\infty$-smooth isometric immersions $f_\delta:X^m\to Y^N$ with curvatures $curv(f_\delta) < \frac {\sqrt…

We consider the edge-isoperimetric problem on the graph of the infinite grid $\mathbb{N}^{2}$ in the $\ell_{\infty}$ metric. We first show that the solutions are not nested, so that techniques other than compressions have to be used. We…

Let $\varphi:F_1\to F_2$ be an injective morphism of free groups. If $\varphi$ is geometric (i.e. induced by an inclusion of oriented compact connected surfaces with nonempty boundary), then we show that $\varphi$ is an isometric embedding…