Related papers: Gaussian Harmonic Forms and Two-Dimensional Self-S…

3-dimensional Harvey Lawson submanifolds were introduced in an earlier paper by Akbulut-Salur, as examples of Lagrangian-type manifolds inside G2 manifold. In this paper, we first show that the space of deformations of a smooth, compact,…

In this paper, we provide tight lower bounds for the oracle complexity of minimizing high-order H\"older smooth and uniformly convex functions. Specifically, for a function whose $p^{th}$-order derivatives are H\"older continuous with…

Let G be a semisimple Lie group with no compact factors, K a maximal compact subgroup of G, and $\Gamma$ a lattice in G. We study automorphic forms for $\Gamma$ if G is of real rank one with some additional assumptions, using dynamical…

This study aims at contributing to lower bounds for empirical compatibility constants or empirical restricted eigenvalues. This is of importance in compressed sensing and theory for $\ell_1$-regularized estimators. Let $X$ be an $n \times…

In this paper, we prove a pinching theorem for $n-$dimensional closed self-shrinkers of the mean curvature flow. If the squared norm of the second fundamental form of a closed self-shrinker of arbitrary codimension satisfies: $ |…

The 2D Discrete Gaussian model gives each height function $\eta : \mathbb{Z}^2\to\mathbb{Z}$ a probability proportional to $\exp(-\beta \mathcal{H}(\eta))$, where $\beta$ is the inverse-temperature and $\mathcal{H}(\eta) = \sum_{x\sim…

On a closed Riemannian surface $(M,\bar g)$ with negative Euler characteristic, we study the problem of finding conformal metrics with prescribed volume $A>0$ and the property that their Gauss curvatures $f_\lambda= f + \lambda$ are given…

We classify properly immersed self-shrinkers of the mean curvature flow in arbitrary codimension under a quadratic pinching condition of Andrews-Baker type on the second fundamental form that is preserved along the flow. Under this…

Kohonen's Self-Organizing Maps (SOMs) have proven to be a successful data-reduction method to identify the intrinsic lower-dimensional sub-manifold of a data set that is scattered in the higher-dimensional feature space. Motivated by the…

We present a lower bound for a fragmentation norm and construct a bi-Lipschitz embedding $I\colon \mathbb{R}^n\to\mathrm{Ham}(M)$ with respect to the fragmentation norm on the group $\mathrm{Ham}(M)$ of Hamiltonian diffeomorphisms of a…

Given a surface $\Sigma$ in $\mathbb{R}^3$ diffeomorphic to $S^2$, Struwe (Acta Math., 1988) proved that for almost every $H$ below the mean curvature of the smallest sphere enclosing $\Sigma$, there exists a branched immersed disk which…

Let $x:M^m\to \bar M$, with $m\geq 3$, be an isometric immersion of a complete noncompact manifold $M$ in a complete simply-connected manifold $\bar M$ with sectional curvature satisfying $-c^2\leq K_{\bar M}\leq 0$, for some constant $c$.…

Let $F:\Sigma^n \times [0,T)\to \R^{n+m}$ be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps $\gamma:(\Sigma^n, g_t)\to G(n, m)$ form a harmonic heat flow with respect to the…

It is shown that the reduced particle dynamics of 2+1 dimensional gravity in the maximally slicing gauge is of hamiltonian nature. We give the exact diffeomorphism which transforms the expression of the spinning cone geometry in the Deser,…

In this short paper we extend the classical Hoffman-Meeks Halfspace Theorem to self-shrinkers, that is: "Let $P $ be a hyperplane passing through the origin. The only properly immersed self-shrinker $\Sigma$ contained in one of the closed…

The purpose of this paper is to study complete self-shrinkers of mean curvature flow in Euclidean spaces. In the paper, we give a complete classification for 2-dimensional complete Lagrangian self-shrinkers in Euclidean space $\mathbb R^4$…

This is a part of our joint program. The purpose of this paper is to study smooth toroidal compactifications of Siegel varieties and their applications, we also try to understand the K\"ahler-Einstein metrics on Siegel varieties through the…

We establish geometric lower bounds for the smallest positive eigenvalue of the Hodge Laplacian in the class of non-convex domains given by Euclidean annular regions with a convex outer boundary and a spherical inner boundary. These bounds…

The Gauss map $g$ of a surface $\Sigma$ in $\mathbb{R}^4$ takes its values in the Grassmannian of oriented 2-planes of $\mathbb{R}^4$: $G^+(2,4)$. We give geometric criteria of stability for minimal surfaces in $\mathbb{R}^4$ in terms of…

In part II we constructed the lower bound, in the spirit of $\Gamma$- $\liminf$ for some general classes of singular perturbation problems, with or without the prescribed differential constraint, taking the form E_\e(v):=\int_\Omega…