相关论文: On a conjecture of Anosov

Let $S$ be a Riemann surface of type $(p,n)$ with $3p+n>4$ and $n\geq 1$. Let $\alpha_1,\alpha_2\subset S$ be two simple closed geodesics such that $\{\alpha_1, \alpha_2\}$ fills $S$. It was shown by Thurston that most maps obtained through…

We calculate the stable pair theory of a projective surface $S$. For fixed curve class $\beta\in H^2(S)$ the results are entirely topological, depending on $\beta^2$, $\beta.c_1(S)$, $c_1(S)^2$, $c_2(S)$, $b_1(S)$ \emph{and} invariants of…

In this paper, we prove that there exist at least $n$ geometrically distinct brake orbits on every $C^2$ compact convex symmetric hypersurface $\Sg$ in $\R^{2n}$ satisfying the reversible condition $N\Sg=\Sg$ with $N=\diag (-I_n,I_n)$. As a…

A classical result of A.D. Alexandrov states that a connected compact smooth $n-$dimensional manifold without boundary, embedded in $\Bbb R^{n+1}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of…

We exhibit a continuously varying family $F_\lambda$ of homeomorphisms of the sphere $S^2$, for which each $F_\lambda$ is a measurable pseudo-Anosov map. Measurable pseudo-Anosov maps are generalizations of Thurston's pseudo-Anosov maps,…

From radial curvature geometry's standpoint, we prove a sphere theorem of the Grove-Shiohama type for a certain class of compact Finsler manifolds.

We prove that any manifold diffeomorphic to $S^3$ and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in 1983. Our approach…

In this paper we prove that the space $\cM(n,\rv,D,\Lambda):=\{(M^n,g) \text{ closed }: ~~\Ric\ge -(n-1),~\Vol(M)\ge \rv>0, \diam(M)\le D \text{ and } \int_{M}|\Rm|^{n/2}\le \Lambda\}$ has at most $C(n,\rv,D,\Lambda)$ many diffeomorphism…

We show that Caratheodory's conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f: R^2-->R whose gradient decays uniformly faster than…

We recently established a Toponogov type triangle comparison theorem for a certain class of Finsler manifolds whose radial flag curvatures are bounded below by that of a von Mangoldt surface of revolution (arXiv:1205.3913). In this article,…

Using the theory of geodesics on surfaces of revolution, we introduce the period function. We use this as our main tool in showing that any two-dimensional orbifold of revolution homeomorphic to S^2 must contain an infinite number of…

Consider a compact Riemannian manifold M of dimension n whose boundary \partial M is totally geodesic and is isometric to the standard sphere S^{n-1}. A natural conjecture of Min-Oo asserts that if the scalar curvature of M is at least…

This is a survey paper on various results relates to the following theorem first proved by A.D. Alexandrov: \textit{Let $S$ be an analytic convex sphere-homeomorphic surface in $\mathbb R^3$ and let $k_1(\boldsymbol{x})\leqslant…

The Gromov-Lawson-Rosenberg-conjecture for a group G states that a closed spin manifold M^n (n>4) with fundamental group G admits a metric with positive scalar curvature if and only if its C^*-index A(M) in KO_n(C^*_r(G)) vanishes. We prove…

An $n$-dimensional ($n\geq 2$) simply connected, compact without boundary Finsler space of positive constant sectional curvature is conformally homeomorphic to an n-sphere in the Euclidean space $\R^{n+1}$.

It is conjectured that every manifold admitting an Anosov diffeomorphism is, up to homeomorphism, finitely covered by a nilmanifold. Motivated by this conjecture, an important problem is to determine which nilmanifolds admit an Anosov…

In the previous paper, it has been proved that the generalized rotational hypersurfaces of O(n-1)-type and O (l+1) x O(m+1)-type, for which the mean curvature is any prescribed continuous function. This paper is a sequel, and a similar…

We show that if two closed hyperbolic surfaces (not necessarily orientable or even connected) have the same Laplace spectrum, then for every length they have the same number of orientation-preserving geodesics and the same number of…

Let M be a compact pseudo-umbilical submanifold of the unit sphere S. In the present note, it is shown that if the normal curvature, scalar curvature S and square of the length of second fundamental form satisfy certain conditions, then M…

In 1967, Gr\"unbaum conjectured that any $d$-dimensional polytope with $d+s\leq 2d$ vertices has at least \[\phi_k(d+s,d) = {d+1 \choose k+1 }+{d \choose k+1 }-{d+1-s \choose k+1 } \] $k$-faces. We prove this conjecture and also…