相关论文: On a conjecture of Anosov

We consider the problem $F=f(\nu)$ for strictly convex, closed hypersurfaces in $S^{n+1}$ and solve it for curvature functions $F$ the inverses of which are of class $(K)$.

We enumerate a necessary condition for the existence of infinitely many geometrically distinct, non-constant, prime closed geodesics on an arbitrary closed Riemannian manifold $M$. That is, we show that any Riemannian metric on $M$ admits…

A conjecture of Berger states that, for any simply connected Riemannian manifold all of whose geodesics are closed, all prime geodesics have the same length. We firstly show that the energy function on the free loop space of such a manifold…

A Finsler metric is geodesically reversible if geodesics remain geodesics after a change of orientation. Asymmetric norms on vector spaces and Funk metrics in the interior of convex bodies are examples of geodesically reversible metrics…

Let $\Sigma$ be a compact orientable surface of finite type with at least one boundary component. Let $f \in \textup{Mod}(\Sigma)$ be a pseudo Anosov mapping class. We prove a conjecture of McMullen by showing that there exists a finite…

We determine all Finsler metrics of Randers type for which the Riemannian part is a scalar multiple of the Euclidean metric, on an open subset of the Euclidean plane, whose geodesics are circles. We show that the Riemannian part must be of…

In this paper we extend Alexandrov's sphere theorems for higher-order mean curvature functions to $ W^{2,n} $-regular hypersurfaces under a general degenerate elliptic condition. The proof is based on the extension of the Montiel-Ros…

Let $M^n$ be a closed immersed hypersurface lying in a contractible ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reilly's inequality via sectional curvature upper bound of $B(p,R)$, 1st…

Let $\wedge$ be a lattice in $\mathbb{R}^n$ reduced in the sense of Korkine and Zolotareff having a basis of the form $(A_1,0,0,\ldots,0),(a_{2,1},A_2,0,\ldots,0)$, $\ldots,(a_{n,1},a_{n,2},\ldots,a_{n,n-1},A_n)$ where $A_1, A_2,\ldots,A_n$…

We show that on every compact Riemannian 2-orbifold there exist infinitely many closed geodesics of positive length.

We prove the scalar curvature rigidity for $L^\infty$ metrics on $\mathbb S^n\backslash\Sigma$, where $\mathbb S^n$ is the $n$-dimensional sphere with $n\geq 3$ and $\Sigma$ is a closed subset of $\mathbb S^n$ of codimension at least…

A Riemannian metric is of constant curvature if and only if it is locally projectively flat. There are infinitely many locally projectively flat Finsler metrics of constant curvature, that are special solutions to the Hilbert's Fourth…

We show that closed surfaces with minimal total absolute curvature in Cartan-Hadamard 3-manifolds bound flat convex bodies. This generalizes Chern-Lashof's theorem for surfaces in Euclidean space and solves a problem posed by Gromov in…

Let $M$ be a complete Sasakian sub-Riemannian $3$-manifold of constant Webster scalar curvature $\kappa$. For any point $p\in M$ and any number $\lambda\in\mathbb{R}$ with $\lambda^2+\kappa>0$, we show existence of a $C^2$ spherical surface…

Improving a singularity theorem in General Relativity by Galloway and Ling we show the following (cf.\ Theorem 1): If a globally hyperbolic spacetime $M$ satisfying the null energy condition contains a closed, spacelike Cauchy surface…

We explore the relationship between contact forms on $\mathbb S^3$ defined by Finsler metrics on $\mathbb S^2$ and the theory developed by H. Hofer, K. Wysocki and E. Zehnder in \cite{HWZ,HWZ1}. We show that a Finsler metric on $\mathbb…

We study the asymptotics of the number N(t) of geometrically distinct closed geodesics of a Riemannian or Finsler metric on a connected sum of two compact manifolds of dimension at least three with non-trivial fundamental groups and apply…

In this paper, firstly, inspired by Nat\'{a}rio's recent work \cite{Na}, we use the isoperimetric inequality to derive some Alexandrov-Fenchel type inequalities for closed convex hypersurfaces in the hyperbolic space $\H^{n+1}$ and in the…

Let $\lambda$ denote the probability Lebesgue measure on ${\mathbb T}^2$. For any $C^2$-Anosov diffeomorphism of the $2$-torus preserving $\lambda$ with measure-theoretic entropy equal to topological entropy, we show that the set of points…

We prove the existence of Alexandrov embedded closed magnetic geodesics on closed hyperbolic surfaces. Closed magnetic geodesics correspond to closed curves with prescribed geodesic curvature.