Related papers: On the Inoue-Bombieri construction

We prove a Poincar\'e, and a general Sobolev type inequalities for functions with compact support defined on a $k$-rectifiable varifold $V$ defined on a complete Riemannian manifold with positive injectivity radius and sectional curvature…

In this paper, we are going to show some rigidity results for complete open Riemannian manifolds with nonnegative scalar curvature. Without using the famous Cheeger-Gromoll splitting theorem we give a new proof to a rigidity result for…

We study collections of exact Lagrangian submanifolds respecting some uniform Riemannian bounds, which we equip with a metric naturally arising in symplectic topology (e.g. the Lagrangian Hofer metric or the spectral metric). We exhibit…

We show that a complete submanifold $M$ with tamed second fundamental form in a complete Riemannian manifold $N$ with sectional curvature $K_{N}\leq \kappa \leq 0$ are proper, (compact if $N$ is compact). In addition, if $N$ is Hadamard…

We show that for a Lie group $G=\R^{n}\ltimes_{\phi} \R^{m}$ with a semisimple action $\phi$ which has a cocompact discrete subgroup $\Gamma$, the solvmanifold $G/\Gamma$ admits a canonical invariant formal (i.e. all products of harmonic…

In this paper, we prove the following two results: First, we study a class of conformally invariant operators $P$ and their related conformally invariant curvatures $Q$ on even-dimensional Riemannian manifolds. When the manifold is locally…

For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We…

For any $g_1, g_2 \ge 0$, this paper shows that there is a cocompact lattice $\Gamma < \mathrm{PU}(2,1)$ such that the ball quotient $\Gamma \backslash \mathbb{B}^2$ is birational to a product $C_1 \times C_2$ of smooth projective curves…

For a polycyclic group $\Lambda$, $\text{rank} (\Lambda )$ is defined as the number of $\mathbb{Z}$ factors in a polycyclic decomposition of $\Lambda$. For a finitely generated group $G$, $\text{rank} (G)$ is defined as the infimum of $…

Gromov and Sormani conjectured that a sequence of three dimensional Riemannian manifolds with nonnegative scalar curvature and some additional uniform geometric bounds should have a subsequence which converges in some sense to a limit space…

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder $B(r)\times\R^{\ell}$ in a product Riemannian manifold $N^{n-\ell}\times\R^{\ell}$. It follows that a complete hypersurface of given constant…

Given a compact and connected four dimensional smooth Riemannian manifold $(M,g_0)$ with $k_P := \int_M Q_{g_0} dV_{g_0} <0$ and a smooth non-constant function $f_0$ with $\max_{p\in M}f_0(p)=0$, all of whose maximum points are…

We first study $f$-biharmonicity of totally umbilical hypersurfaces in a generic Riemannian manifold and then prove that any totally umbilical proper $f$-biharmonic hypersurface in a nonpositively curved manifold has to be noncompact. We…

This article studies the volume of compact quotients of reductive homogeneous spaces. Let $G/H$ be a reductive homogeneous space and $\Gamma$ a discrete subgroup of $G$ acting properly discontinuously and cocompactly on $G/H$. We prove that…

We consider complete Riemannian manifolds which satisfy a weighted Poincar\`e inequality and have the Ricci curvature bounded below in terms of the weight function. When the weight function has a non-zero limit at infinity, the structure of…

In this paper we consider $\rho$-Einstein solitons of type $M= \left(B^n, g^{*}\right) \times (F^m,g_F)$, where $\left(B^n,g^{*}\right)$ is conformal to a pseudo-Euclidean space and invariant under the action of the pseudo-orthogonal group,…

Let \Sigma be a compact Riemann surface with n distinguished points p_1,...,p_n. We prove that the set of n-tuples (\phi_1,...,\phi_n) of univalent mappings \phi_i from the open unit disc into \Sigma mapping 0 to p_i, with non-overlapping…

For a given null-cobordant Riemannian $n$-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? In [Gro99], Gromov conjectured that this dependence should be linear. We…

We prove that (apart from dimension $n=4$), each Riemannian solenoidal lamination with transitive homeomorphism group and leaves isometric to a symmetric space $X$ of noncompact type, is homeomorphic to the inverse limit of the system of…

We study a new class of rank two sub-Riemannian manifolds encompassing Riemannian manifolds, CR manifolds with vanishing Webster-Tanaka torsion, orthonormal bundles over Riemannian manifolds, and graded nilpotent Lie groups of step two.…