By adapting the test functions introduced by Choi-Daskaspoulos \cite{c-d} and Brendle-Choi-Daskaspoulos \cite{b-c-d} and exploring properties of the $k$-th elementary symmetric functions $\sigma_{k}$ intensively, we show that for any fixed $k$ with $1\leq k\leq n-1$, any strictly convex closed hypersurface in $\mathbb{R}^{n+1}$ satisfying $\sigma_{k}^{\alpha}=\langle X,\nu \rangle$, with $\alpha\geq \frac{1}{k}$, must be a round sphere.

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A Riemannian or Finsler metric on a compact manifoldM gives rise to a length function on the free loop space M, whose critical points are the closed geodesics in the given metric. If X is a homology class on M, the “minimax” critical level cr(X) is a critical value. Let M be a sphere of dimension > 2, and fix a metric g and a coefficient field G. We prove that the limit as deg(X) goes to infinity of cr(X)/ deg(X) exists. We call this limit = (M, g,G) the global mean frequency of M. As a consequence we derive resonance statements for closed geodesics on spheres; in particular either all homology on  of sufficiently high degreee lies hanging on closed geodesics of mean frequency (length/average index) , or there is a sequence of infinitely many closed geodesics whose mean frequencies converge to . The proof uses the ChasSullivan product and results of Goresky-Hingston [7].

In this paper, we prove a scalar curvature rigidity result for geodesic balls in Sn. This result contrasts sharply with the counterexamples to Min-Oo’s conjecture constructed in [7].

By using variational method and under generic condition, we show that Arnold diffusion exists in a priori hyperbolic and timeperiodic Hamiltonian systems with multiple degrees of freedom.