Related papers: Density and spectrum of minimal submanifolds in sp…

We show that intrinsic and extrinsic area density bounds are equivalent, with matching asymptotic values, for complete, connected, smooth minimal immersions $i:\Sigma^d\to\mathbb{R}^N$ of any dimension and codimension. Combining our results…

We study a new bi-Lipschitz invariant \lambda(M) of a metric space M; its finiteness means that Lipschitz functions on an arbitrary subset of M can be linearly extended to functions on M whose Lipschitz constants are enlarged by a factor…

We investigate specific intrinsic curvatures $\rho_k$ (where $1\leq k\leq n$) that interpolate between the minimum Ricci curvature $\rho_1$ and the normalized scalar curvature $\rho_n=\rho$ of $n$-dimensional Riemannian manifolds. For…

We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show…

We classify minimal hypersurfaces in $R^n \times S^m$, $n,m \geq 2$, which are invariant by the canonical action of $O(n) \times O(m)$. We also construct compact and noncompact examples of invariant hypersurfaces of constant mean curvature.…

The aim of this paper is twofold. On the one hand, the study of gradient Schr\"{o}dinger operators on manifolds with density $\phi$. We classify the space of solutions when the underlying manifold is $\phi-$parabolic. As an application, we…

We consider triholomorphic maps from an almost hyper-Hermitian manifold $\mathcal{M}^{4m}$ into a hyperK\"ahler manifold $\mathcal{N}^{4n}$. This means that $u \in W^{1,2}$ satisfies a quaternionic del-bar equation. We work under the…

We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic $3$-manifold $\mathcal{N}$. We also obtain a least area, incompressible, properly embedded, finite topology, $2$-sided surface. We prove a…

We consider a compact $C^\infty$ stratified 2D variety $M$ in $\mathbb{R}^3$ and its $\epsilon$ neighborhood $M_\epsilon$, which we call a "fattened open book structure". Assuming absence of zero-dimensional strata, i.e. "corners", we show…

We study the density of functions which are holomorphic in a neighbourhood of the closure $\overline{\Omega}$ of a bounded non-smooth pseudoconvex domain $\Omega$, in the Bergman space $ H^2(\Omega ,\varphi)$ with a plurisubharmonic weight…

The paper deals with a nontrivial density result for $C^m(\overline{\Omega})$ functions, with $m\in{\mathbb N}\cup\{\infty\}$, in the space $$W^{k,\ell,p}(\Omega;\Gamma)= \left\{u\in W^{k,p}(\Omega): u_{|\Gamma}\in…

We establish the following Hadamard--Stoker type theorem: Let $f:M^n\rightarrow\mathscr{H}^n\times\mathbb R$ be a complete connected hypersurface with positive definite second fundamental form, where $\mathscr H^n$ is a Hadamard manifold.…

In this paper, we define natural capacities using a relative volume of graphs over manifolds, which can be characterized by solutions of bounded variation to Dirichlet problems of minimal hypersurface equation. Using the capacities, we…

We investigate complete minimal submanifolds $f\colon M^3\to\Hy^n$ in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already…

We develop a min-max theory for the construction of capillary surfaces in 3-manifolds with smooth boundary. In particular, for a generic set of ambient metrics, we prove the existence of nontrivial, smooth, almost properly embedded surfaces…

Let $f:N\rightarrow (M,g)$ be an oriented (or spin), complete, stable, minimal, immersed hypersurface. In this paper we establish various vanishing theorems for the space of $L^2$-harmonic forms and spinors (in the spin case) under suitable…

We introduce the $R$ cut-off covering spectrum and the cut-off covering spectrum of a complete length space or Riemannian manifold. The spectra measure the sizes of localized holes in the space and are defined using covering spaces called…

We prove that the answer to the "zero-in-the-spectrum" conjecture, in its form, suggested by J. Lott, is negative. Namely, we show that for any n > 5 there exists a closed n-dimensional manifold M, so that zero does not belong to the…

Let $M\subset S^{n+1}$ be the hypersurface generated by rotating a hypersurface $M_0$ contained in the interior of the unit ball of $\mathbb{R}^{n-k+1}$. More precisely, $M=\{(\sqrt{1-|m|^2}\, y, m):y\in S^k, m\in M_0\}$. We derive the…

In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below, positive injectivity radius and spectral gap b. We introduce a sequence X^1(M), X^2(M), ... of new Hardy spaces on M,…