Related papers: On Isolated Umbilic Points

Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat K\"ahler metric) as one approaches a large complex…

We investigate rigidity and stability properties of critical points of quadratic curvature functionals on the space of Riemannian metrics. We show it is possible to "gauge" the Euler-Lagrange equations, in a self-adjoint fashion, to become…

We propose two conjectures about Ricci-flat metrics: Conjecture 1: A Ricci-flat projectively induced metric is flat. Conjecture 2: A Ricci-flat metric on an $n$-dimensional complex manifold such that the $a_{n+1}$ coefficient of the TYZ…

We consider the problem of preserving weighted Riemannian metrics of positive Bakry-\'Emery Ricci curvature along surgery. We establish two theorems of this type: One for connected sums, and one for surgeries along higher-dimensional…

We obtain an exhaustive classification of totally umbilical surfaces in unimodular and non-unimodular simply-connected 3-dimensional Lie groups endowed with arbitrary left-invariant Riemannian metrics. This completes the classification of…

It was recently shown by R. Souam and E. Toubiana that the (non constantly curved) Berger spheres do not contain totally umbilic surfaces. Nevertheless in this article we show, by perturbative arguments, that all analytic metrics…

For compact Riemannian manifolds with convex boundary, B.White proved the following alternative: Either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small…

We classify all smooth flat Riemannian metrics on the two-dimensional plane. In the complete case, it is well-known that these metrics are isometric to the Euclidean metric. In the incomplete case, there is an abundance of…

A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold that is locally modeled on $R^n$ modulo the action of a finite group. Orbifolds have proven interesting in a variety of settings. Spectral geometers have…

We prove that knowing the length of geodesics joining points on the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.

The ratio of convexity radius over injectivity radius may be made arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. This is proved using Gulliver's method of constructing manifolds with…

We investigate a class of metrics for 2-manifolds in which, except for a discrete set of singular points, the metric is locally isometric to an L_1 (or equivalently L_infinity) metric, and show that with certain additional conditions such…

For any smooth Riemannian metric on an $(n+1)$-dimensional compact manifold with boundary $(M,\partial M)$ where $3\leq (n+1)\leq 7$, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by…

The aim of this paper is to study ultralimits of pointed metric measure spaces (possibly unbounded and having infinite mass). We prove that ultralimits exist under mild assumptions and are consistent with the pointed measured…

Let $M$ be a compact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary $\partial M$. Assume that the mean curvature $H$ of the boundary $\partial M$ satisfies $H \geq (n-1) k >0$ for some positive…

This paper investigates the failure of certain metric measure spaces to be infinitesimally Hilbertian or quasi-Riemannian manifolds, by constructing examples arising from a manifold $M$ endowed with a Riemannian metric $g$ that is possibly…

The conjecture of D.Blair says that there are no nonflat Riemannian metrics of nonpositive curvature compatible with a contact structure. We prove this conjecture for a certain class of contact structures on closed 3-dimensional manifolds…

In these notes we generalize the notion of a (pseudo) metric measuring the distance of two points, to a (pseudo) n-metric which assigns a value to a tuple of n points. We present two principles of constructing pseudo n-metrics. The first…

We introduce a framework for Riemannian diffeology. To this end, we use the tangent functor in the sense of Blohmann and one of the options of a metric on a diffeological space in the sense of Iglesias-Zemmour. As a consequence, the…

We show that the Morse index of a closed minimal hypersurface in a four-dimensional Riemannian manifold cannot be bound in terms of the volume and the topological invariants of the hypersurface itself by presenting a method for constructing…