Related papers: Consequences of ergodic frame flow for rank rigidi…

This paper presents hyperbolic rank rigidity results for rank 1, nonpositively curved spaces. Let $M$ be a compact, rank 1 manifold with nonpositive sectional curvature and suppose that along every geodesic in $M$ there is a parallel vector…

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature -1 with the geodesic. If in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac14]$,…

Let $M$ be a closed, negatively curved Riemannian manifold of dimension $n \neq 4, 8$ with strictly $1/4$-pinched sectional curvature. We prove, that if the frame flow is ergodic and the sum of its unstable and stable bundles together with…

In this paper, we prove that if the geodesic flow of a complete manifold without conjugate points with sectional curvatures bounded below by $-c^2$ is of Anosov type, then the constant of contraction of the flow is $\geq e^{-c}$. Moreover,…

Let $(M,g,J)$ be a closed K\"ahler manifold with negative sectional curvature and complex dimension $m := \dim_{\mathbb{C}} M \geq 2$. In this article, we study the unitary frame flow, that is, the restriction of the frame flow to the…

In this paper, we establish new geometric rigidity results through the study of Lyapunov exponent level sets via invariant measures. First, we prove that for a manifold $M$ without focal points, if the zero Lyapunov exponent level set has…

In this paper, we study rigidity problems between Lyapunov exponents along periodic orbits and geometric structures. More specifically, we prove that for a surface M without focal points, if the value of the Lyapunov exponents is constant…

We consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a Riemannian metric $g$ of negative curvature with fixed total area. The second author has shown that the topological entropy of geodesic flow for $g$ is greater than…

G. Pipoli and C. Sinestrari considered the mean curvature flow starting from a closed submanifold in the complex projective space. They proved that if the submanifold is of small codimension and satisfies a suitable pinching condition for…

It is known that the frame flow on a closed $n$-dimensional Riemannian manifold with negative sectional curvature is ergodic if $n$ is odd and $n \neq 7$. In this paper we study its ergodicity in the remaining cases. For $n$ even and $n…

We show that groups with a mild form of non-positive curvature (a navigable path system) satisfy the weak rank rigidity conjecture: they either have linear divergence or a Morse element. This class includes discrete groups of projective…

We prove the following entropy-rigidity result in finite volume: if $X$ is a negatively curved manifold with curvature $-b^2\leq K_X \leq -1$, then $Ent_{top}(X) = n-1$ if and only if $X$ is hyperbolic. In particular, if $X$ has the same…

In this paper, we prove that if the initial submanifold $M_0$ of dimension $n(\ge6)$ satisfies an optimal pinching condition, then the mean curvature flow of arbitrary codimension in hyperbolic spaces converges to a round point in finite…

We consider the inverse mean curvature flow in smooth Riemannian manifolds of the form $([R_{0},\infty)\times S^n,\bar{g})$ with metric $\bar{g}=dr^2+{\vartheta}^2(r){\sigma}$ and non-positive radial sectional curvature. We prove, that for…

We prove the existence of compact surfaces with prescribed constant mean curvature in asymptotically flat and asymptotically hyperbolic manifolds. More precisely, let $(M^3,g)$ be an asymptotically flat manifold with scalar curvature $R\ge…

We prove the mean curvature flow of a spacelike graph in $(\Sigma_1\times \Sigma_2, g_1-g_2)$ of a map $f:\Sigma_1\to \Sigma_2$ from a closed Riemannian manifold $(\Sigma_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold…

Let $X$ be a Hadamard manifold, and $\Gamma$ a non-elementary discrete group of isometries of $X$ which contains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold $M=X/\Gamma$ to the behavior of…

Let $M$ be a smooth compact surface of nonpositive curvature, with genus $\geq 2$. We prove the ergodicity of the geodesic flow on the unit tangent bundle of $M$ with respect to the Liouville measure under the condition that the set of…

We show that a compact Riemannian $3$-manifold $M$ with strictly convex simply connected boundary and sectional curvature $K\leq a\leq 0$ is isometric to a convex domain in a complete simply connected space of constant curvature $a$,…

We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such example as the product metric…