相关论文: The fundamental gap of simplices

We solve the classical problem of Plateau in every metric space which is $1$-complemented in an ultra-completion of itself. This includes all proper metric spaces as well as many locally non-compact metric spaces, in particular, all dual…

We work with a pre-$\lambda$-frame, which is an abstract elementary class (AEC) endowed with a collection of basic types and a non-forking relation satisfying certain natural properties with respect to models of cardinality $\lambda$. We…

We prove the following conjecture of Margulis. Let $G$ be a higher rank simple Lie group and let $\Lambda\le G$ be a discrete subgroup of infinite covolume. Then, the locally symmetric space $\Lambda\backslash G/K$ admits injected balls of…

The main result of the paper shows that the regular $n$-gon is a local minimizer for the first Dirichlet-Laplace eigenvalue among $n$-gons having fixed area for $n \in \{5,6\}$. The eigenvalue is seen as a function of the coordinates of the…

In this paper, we study the stability and minimizing properties of higher codimensional surfaces in Euclidean space associated with the $f$-weighted area-functional $$\mathcal{E}_f(M)=\int_M f(x)\; d \mathcal{H}_k$$ with the density…

We formulate several conjectures on mean convex domains in the Euclidean spaces, as well as in more general spaces with lower bonds on their scalar curvatures, and prove a few theorems motivating these conjectures.

In this paper we prove the existence of a solution to the Dirichlet problem for harmonic maps into a geodesic ball on which the squared distance function from the origin is strictly convex. This improves a celebrated theorem obtained by S.…

We apply the techniques developed by Marcus, Spielman and Srivastava, working with principal submatrices in place of rank $1$ decompositions to give an alternate proof of their results on restricted invertibility. We show that one can find…

A classical result by Marston Morse asserts that on some ellipsoids of ${\mathbb R}^3$ there exists exactly 3 closed and simple geodesics. The goal of this presentation is to prove that this rigidity result does not extend to higher…

In this paper, we first prove a compactness theorem for the space of closed embedded $f$-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry-\'{E}mery Ricci curvature. Then we give a Lichnerowicz type lower…

Let $X$ be a simplicial complex with $n$ vertices. A missing face of $X$ is a simplex $\sigma\notin X$ such that $\tau\in X$ for any $\tau\subsetneq \sigma$. For a $k$-dimensional simplex $\sigma$ in $X$, its degree in $X$ is the number of…

In this paper, we study the rigidity problem for compact minimal Legendrian submanifolds in the unit Euclidean spheres via eigenvalues of fundamental matrices, which measure the squared norms of the second fundamental form on all normal…

In this note we consider submersions from compact manifolds, homotopy equivalent to the Eschenburg or Bazaikin spaces of positive curvature. We show that if the submersion is nontrivial, the dimension of the base is greater than the…

We discuss connections between certain well-known open problems related to the uniform measure on a high-dimensional convex body. In particular, we show that the "thin shell conjecture" implies the "hyperplane conjecture". This extends a…

The width of a closed convex subset of Euclidean space is the distance between two parallel supporting planes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still…

The family of lattice simplices in $\mathbb{R}^n$ formed by the convex hull of the standard basis vectors together with a weakly decreasing vector of negative integers include simplices that play a central role in problems in enumerative…

We show that for convex domains in Euclidean space, Cheeger's isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschitz functions…

In this paper, we prove Gromov's flat corner domination conjecture in all dimensions. As a consequence, we answer positively the Stoker conjecture for convex Euclidean polyhedra in all dimensions. By applying the same techniques, we also…

In this note, we derive a uniqueness theorem for minimal graphs of general codimension under certain restrictions closed related to the convexity (not strict convexity) of the area functional with respect to singular values, improving the…

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…