Related papers: On Complete Conformally flat submanifolds with nul…

In this paper we extend Efimov's Theorem by proving that any complete surface in $\mathbb{R}^3$ with Gauss curvature bounded above by a negative constant outside a compact set has finite total curvature, finite area and is properly…

Let $f\colon M^{2n}\to\mathbb{R}^{2n+4}$ be an isometric immersion of a Kaehler manifold of complex dimension $n\geq 5$ into Euclidean space with complex rank at least $5$ everywhere. Our main result is that, along each connected component…

A Lagrangian submanifold in an almost Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. An exact isotopy class of positive Lagrangian submanifolds admits a natural…

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $\partial M$, admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is…

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $\partial M$. Suppose that $(M,g)$ admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric quotient over the…

In this paper, we prove conformal positive mass theorems for asymptotically flat manifolds with charge. We apply conformal relations to show that if the conformal sum of scalar curvature is not less than the norm square of electric field…

In this work, we study various properties of embedded hypersurfaces in $1+1+2$ decomposed spacetimes with a preferred spatial direction, denoted $e^{\mu}$, which are orthogonal to the fluid flow velocity of the spacetime and admit a proper…

We prove four results towards a description, in terms of the null support function, of the set of isometric embeddings of the hyperbolic plane into Minkowski 3-space. We show that for sufficiently tame null support function, the…

We study warped products semi-Riemannian Einstein manifolds. We consider the case in that the base is conformal to an n-dimensional pseudo Euclidean space and invariant under the action of an translation group. We provide all such solutions…

We consider a complete biharmonic immersed submanifold $M$ in an Euclidean space $\mathbb{E}^N$. Assume that the immersion is proper, that is, the preimage of every compact set in $\mathbb{E}^N$ is also compact in $M$. Then, we prove that…

We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold $(M,h)$ without boundary. First, under the assumption that $(M,h)$ is the euclidean 3-space endowed with a semi-perturbed metric…

A basic question in submanifold theory is whether a given isometric immersion $f\colon M^n\to\R^{n+p}$ of a Riemannian manifold of dimension $n\geq 3$ into Euclidean space with low codimension $p$ admits, locally or globally, a genuine…

Let $R$ be a constant. Let $\mathcal{M}^R_\gamma$ be the space of smooth metrics $g$ on a given compact manifold $\Omega^n$ ($n\ge 3$) with smooth boundary $\Sigma $ such that $g$ has constant scalar curvature $R$ and $g|_{\Sigma}$ is a…

The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is…

We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n+1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary…

We prove that in Euclidean space $R^{n+1}$ any compact immersed nonnegatively curved hypersurface $M$ with free boundary on the sphere $S^n$ is an embedded convex topological disk. In particular, when the $m^{th}$ mean curvature of $M$ is…

Let $M$ be a closed aspherical manifold. Assume that the rational strong Novikov conjecture holds for $\pi_1(M)$. We show that on any spin surgery of $M$ along a region whose induced homomorphism on the fundamental group is trivial, every…

We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In…

We prove a general relative higher index theorem for complete manifolds with positive scalar curvature towards infinity. We apply this theorem to study Riemannian metrics of positive scalar curvature on manifolds. For every two metrics of…

We prove that for every $r>0$ if a non-positively curved $(p,q)$-map $M$ contains no flat submaps of radius $r$, then the area of $M$ does not exceed $Crn$ for some constant $C$. This strengthens a theorem of Ivanov and Schupp. We show that…