Related papers: A Simons' type formula for cmc surfaces in homogen…

Biconservative surfaces of Riemannian 3-space forms $N^3(\rho)$, are either constant mean curvature (CMC) surfaces or rotational linear Weingarten surfaces verifying the relation $3\kappa_1+\kappa_2=0$ between their principal curvatures…

In this paper we prove stability results for the homology of the mapping class group of a surface. We get a stability range that is near optimal, and extend the result to twisted coefficients.

In this manuscript we consider non-degenerate surfaces $\Sigma^2$ immersed in a 3-dimensional homogeneous space $\mathbb{L}^3(\kappa,\tau)$ endowed with two different metrics, the one induced by the Riemannian metric of…

A result due in its various parts to Hendrickson, Connelly, and Jackson and Jord\'an, provides a purely combinatorial characterisation of global rigidity for generic bar-joint frameworks in $\mathbb{R}^2$. The analogous conditions are known…

Let $(M,Q)$ be a compact, three dimensional manifold of strictly negative sectional curvature. Let $(\Sigma,P)$ be a compact, orientable surface of hyperbolic type (i.e. of genus at least two). Let $\theta:\pi_1(\Sigma,P)\to\pi_1(M,Q)$ be a…

We compute the homotopy type of the moduli space of flat, unitary connections over aspherical surfaces, after stabilizing with respect to the rank of the underlying bundle. Over the orientable surface M^g, we show that this space has the…

We prove some new rigidity results for proper biharmonic immersions in ${\mathbb S}^n$ of the following types: Dupin hypersurfaces; hypersurfaces, both compact and non-compact, with bounded norm of the second fundamental form; hypersurfaces…

We study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds $X$ that can be expressed as a semidirect product of $\mathbb{R}^2$ with $\mathbb{R}$ endowed with a left invariant metric. For any…

In this note we prove three rigidity results for Einstein manifolds with bounded covering geometry. (1) An almost flat manifold $(M,g)$ must be flat if it is Einstein, i.e. $\operatorname{Ric}_g=\lambda g$ for some real number $\lambda$.…

In this paper, we obtain results on rigidity of complete Riemannian manifolds with weighted Poincar\'e inequality. As an application, we prove that if $M$ is a complete $\frac{n-2}{n}$-stable minimal hypersurface in $\mathbb{R}^{n+1}$ with…

In this paper, we study Lagrangian submanifolds of the homogeneous nearly K\"ahler $6$-dimensional unit sphere $\mathbb{S}^6(1)$. As the main result, we derive a Simons' type integral inequality in terms of the second fundamental form for…

Let $M^n(n\geq3)$ be an $n$-dimensional compact Riemannian manifold with harmonic curvature and positive scalar curvature. Assume that $M^n$ satisfies some integral pinching conditions. We give some rigidity theorems on compact manifolds…

In this note we construct a first example of a closed 3-form of $\tilde G_2$-type on $S^3\times S^4$. We prove that $S^3\times S^4$ does not admit a homogeneous 3-form of $\tilde G_2$-type. Thus our example is a first example of a closed…

Let $M^{n}$ be an $n$-dimensional complete spacelike linear Weingarten submanifold immersed in a locally symmetric semi-Riemannian space $\mathbb{L}_{q}^{n+p}$ of index $q$, with parallel normalized mean curvature vector field and flat…

We define a notion of isotropic surfaces in $\mathbb{O}$, i.e. on which some canonical symplectic forms vanish. Using the cross-product in $\mathbb{O}$ we define a map $\rho\colon Gr\_2(\mathbb{O})\to S^6$ from the Grassmannian of…

Existence of Maxwell-Chern-Simons-Higgs (MCSH) vortices in a Hermitian line bundle $\L$ over a general compact Riemann surface $\Sigma$ is proved by a continuation method. The solutions are proved to be smooth both spatially and as…

We develop the deformation theory of hyperbolic cone-3-manifolds with cone-angles less than $2\pi$, i.e. contained in the interval $(0,2\pi)$. In the present paper we focus on deformations keeping the topological type of the cone-manifold…

In this paper, we show that the catenoids and the Clifford minimal hypersurfaces are the only complete minimal hypersurfaces satisfying the Simons' equation (3.9) in the space forms.

We compute properties of the interface of the 3-dimensional Ising model for a wide range of temperatures and for interface extensions up to 64 by 64. The interface tension sigma is obtained by integrating the surface energy density over the…

Under the assumption that certain Adem cohomology operation acts trivially on $H^2(M;\mathbb{Z}/2)$, we determine the homotopy types of the triple suspension $\Sigma^3M$ of a simply-connected oriented closed topological(or smooth)…