Related papers: Min-max for sweepouts by curves

We generalize and strengthen the theorem of Gromov that every compact Riemannian manifold of diameter at most D has a set of generators g_1,...,g_k of length at most 2D and relators of the form g_ig_m = g_j . In particular, we obtain an…

We establish existence of steepest descent curves emanating from almost every point of a regular locally Lipschitz quasiconvex functions, where regularity means that the sweeping process flow induced by the sublevel sets is reversible. We…

A finite simple graph is called a 2-graph if all of its unit spheres S(x) are cyclic graphs of length 4 or larger. A 2-graph G is Eulerian if all vertex degrees of G are even. An edge refinement of a graph splits an edge (a,b) to two edges…

We introduce a direct generalization of the Weinstein conjecture to closed, Lichnerowicz exact, locally conformally symplectic manifolds, (for short $\lcs$ manifolds). This conjectures existence of certain 2-curves in the manifold, which we…

A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and…

Given a compact manifold with boundary with unknown Riemannian metric. The problem is to reconstruct the metric in a class of conformal metrics from knowledge of lengths of all closed geodesics (kinematic data). An integral inequality is…

Let $(M,g_0)$ be a closed Riemannian manifold of dimension $n \geq 25$ with positive Yamabe invariant $Y(M,g_0)>0$ and positive fourth-order invariant $Y_4(M,g_0)>0$. We show that, arbitrarily $C^1$-close to $g_0$, there exists a Riemannian…

Recall that the radius of a compact metric space $(X, dist)$ is given by $rad\ X = \min_{x\in X} \max_{y\in X} dist(x,y)$. In this paper we generalize Berger's $\frac{1}{4}$-pinched rigidity theorem and show that a closed, simply connected,…

We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in $\R^n$ for every $n\geq 9$. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular,…

I prove a scalar curvature rigidity theorem for spheres. In particular, I prove that geodesic balls of radii strictly less than $\frac{\pi}{2}$ in $n+1~(n\geq 2)$ dimensional unit sphere can be rigid under smooth deformations that increase…

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…

On the domain of a Riemannian submersion, we consider variations (i.e., smooth one-parameter families) of Riemannian metrics, for which the submersion is Riemannian and which all keep the metric induced on its fibers fixed. We obtain a…

We investigate isoperimetric inequalities for Lipschitz 2-spheres in CAT(0) spaces, proving bounds on the volume of efficient null-homotopies. In one dimension lower, it is known that a quadratic inequality with a constant smaller than…

Consider a quadratic rational self-map of the Riemann sphere such that one critical point is periodic of period 2, and the other critical point lies on the boundary of its immediate basin of attraction. We will give explicit topological…

We study min-max theory for area functional among hypersurfaces constrained in a smooth manifold with boundary. A Schoen-Simon-type regularity result is proved for integral varifolds which satisfy a variational inequality and restrict to a…

In Riemannian computing applications, it is crucial to map manifold data to a Euclidean domain, where vector space arithmetic is available, and back. Classical manifold theory guarantees the existence of such mappings, called charts and…

The length of the geodesic between two data points along a Riemannian manifold, induced by a deep generative model, yields a principled measure of similarity. Current approaches are limited to low-dimensional latent spaces, due to the…

This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being…

We prove the result stated in the title; it is equivalent to the existence of a regular point of the sub-Riemannian exponential mapping. We also prove that the metric is analytic on an open everywhere dense subset in the case of a complete…

Introduced by Seifert and Threlfall, cylindrical neighborhoods of isolated critical points of smooth functions is an essential tool in the Lusternik- Schnirelmann theory. We conjecture that every isolated critical point of a smooth function…