We can talk about two kinds of stability of the Ricci flow at Ricci-flat metrics. One of them is a linear stability, defined with respect to Perelman's functional $\mathcal{F}$ (see [1, page 5]). The other one is a dynamical stability, and it refers to a convergence of a Ricci flow starting at any metric in a neighborhood of a considered Ricci-flat metric. We show that dynamical stability implies linear stability. We also show that a linear stability together with the integrability assumption implies dynamical stability. As a corollary, we get a stability result for $K3$-surfaces, part of which has been done in [11, Corollary 4.15, Theorem 4.16]. Our stability result applies to Calabi-Yau manifolds as well

We construct cohomology groups with compact support $H_c^i(X_{\mathrm{ar}},\mathbb{Z}(n))$ for separated schemes of finite type over a finite field which generalize Lichtenbaum's Weil-étale cohomology groups for smooth and projective schemes (see [22]). In particular, if Tate's conjecture holds, and rational and numerical equivalence agree up to torsion, then the groups $H_c^i(X_{\mathrm{ar}},\mathbb{Z}(n))$ are finitely generated, form an integral model of $l$-adic cohomology with compact support, and admit a formula for the special values of the $\zeta$-function of $X$

Let $M$ be a smooth Riemannian manifold that is the union of a compact part and a finite number of Euclidean ends, ${\mathbb{R}}^n\setminus B(0,R)$ for some $R>0$, each of which carries the standard metric. Our main result is that the Riesz transform on $M$ is bounded from $L^p(M)\to L^p(M;T^{*}M)$ for $1<p<n$ and unbounded for $p\ge n$ if there is more than one end. It follows from known results that in such a case, the Riesz transform on $M$ is bounded for $1<p\le 2$ and unbounded for $p>n$; the result is new for $2<p\le n$. We also give some heat kernel estimates on such manifolds.

We then consider the implications of boundedness of the Riesz transform in $L^p$ for some $p>2$ for a more general class of manifolds. Assume that $M$ is an $n$-dimensional complete manifold satisfying the Nash inequality and with an $O(r^n)$ upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on $L^p$ for some $p>2$ implies a Hodge–de Rham interpretation of the $L^p$-cohomology in degree $1$ and that the map from $L^2$- to $L^p$-cohomology in this degree is injective

This article is concerned with uniform convergence in the multiplicative ergodic theorem on aperiodic subshifts. If such a subshift satisfies a certain condition, originally introduced by Boshernitzan [6], [7], every locally constant $\mathrm{SL}(2,\mathbb{R})$-valued cocycle is uniform. As a consequence, the corresponding Schrödinger operators exhibit Cantor spectrum of Lebesgue measure zero

John Franks and Michael Handel [FH2] have recently proved that any nontrivial Hamiltonian diffeomorphism of a closed surface of genus at least one has periodic orbits of arbitrarily large period. They proved a similar result for a nontrivial area-preserving diffeomorphism of a sphere with at least three fixed points. We extend these results to the case of the homeomorphisms. When the genus is at least one, we prove, moreover, that the periodic orbits may be chosen contractible if the set of contractible fixed points is contained in a disk. When the surface is a sphere, we extend the result to the case of a nontrivial homeomorphism with no wandering points. The proofs make use of an equivariant foliated version of Brouwer's plane translation theorem (see [B, Proposition 2.1]) and some properties of the linking number of fixed points

We study self-adjoint operators of the form $H_{\omega }=H_0+\sum \omega (n)({\delta }_n|·){\delta }_n$, where the ${\delta }_n$'s are a family of orthonormal vectors and the $\omega (n)$'s are independent random variables with absolutely continuous probability distributions. We prove a general structural theorem that provides in this setting a natural decomposition of the Hilbert space as a direct sum of mutually orthogonal closed subspaces, which are a.s. invariant under $H_{\omega }$, and that is helpful for the spectral analysis of such operators. We then use this decomposition to prove that the singular spectrum of $H_{\omega }$ is a.s. simple