Related papers: Smooth and PL-Rigidity Problems on Locally Symmetr…

We show that for some negatively curved solvable Lie groups, all self quasiisometries are almost isometries. We prove this by showing that all self quasisymmetric maps of the ideal boundary (of the solvable Lie groups) are bilipschitz with…

Harmonic mappings into Teichmuller spaces appear in the study of manifolds which are fibrations whose fibers are Riemann surfaces. In this article we will study the existence and uniquenesses questions of harmonic mappings into Teichmuller…

We consider geodesics in both Riemannian and Lorentzian manifolds with metrics of low regularity. We discuss existence of extremal curves for continuous metrics and present several old and new examples that highlight their subtle…

We describe all the self quasisymmetric maps on the ideal boundary of a particular negatively curved solvable Lie group. As applications, we prove a Liouville type theorem, and derive some rigidity properties for quasiisometries of the…

This is a short survey of Riemannian geometric applications of Lp-cohomology of thick spaces, p not equal to 2.

Spacelike surfaces in the Lorentz-Minkowski space L^3 can be endowed with two different Riemannian metrics, the metric inherited from L^3 and the one induced by the Euclidean metric of R^3. It is well known that the only surfaces with zero…

We introduce a smooth mapping of some discrete space-time symmetries into quasi-continuous ones. Such transformations are related with q-deformations of the dilations of the Euclidean space and with the non-commutative space. We work out…

In this note we discuss the geometry of Riemannian surfaces having a discrete set of singular points. We assume the conformal structure extends through the singularities and the curvature is integrable. Such points are called \emph{simple…

We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the…

Local minimizers for the anisotropic isoperimetric problem in the small-volume regime on closed Riemannian manifolds are shown to be geodesically convex and small smooth perturbations of tangent Wulff shapes, quantitatively in terms of the…

This paper studies the reduction by symmetry of variational problems on Lie groups and Riemannian homogeneous spaces. We derive the reduced equations of motion in the case of Lie groups endowed with a left-invariant metric, and on Lie…

These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach-Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach…

We discuss negatively curved homogeneous spaces admitting a simply transitive group of isometries, or equivalently, negatively curved left-invariant metrics on Lie groups. Negatively curved spaces have a remarkably rich and diverse…

We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally…

In this work, radially symmetric kink-like solutions in the presence of impurities are investigated for both flat and curved $D+1$ spacetimes, with geometry generated by a rotationally invariant background metric. We have examined the…

In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call {\it quasi-$k$-curved metrics}. Quasi-$k$-curved metrics generalize the metrics of space forms. We construct explicit…

We study rigidity problems for Riemannian and semi-Riemannian manifolds with metrics of low regularity. Specifically, we prove a version of the Cheeger-Gromoll splitting theorem \cite{CheegerGromoll72splitting} for Riemannian metrics and…

The lens data of a Riemannian manifold with boundary is the collection of lengths of geodesics with endpoints on the boundary together with their incoming and outgoing vectors. We show that negatively-curved Riemannian manifolds with…

This is a final step in a local classification of pseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric.

This paper starts by introducing results from geometric measure theory to prove symmetric decreasing rearrangement inequalities on $\mathbb{R}^n$, which give multiple proofs of the isoperimetric and P\'{o}lya-Szeg\H{o} inequalities. Then we…