Compact almost Ricci solitons with constant scalar curvature are gradient

The aim of this note is to prove that any compact non-trivial almost Ricci soliton $\big (M^n,\,g,\,X,\,\lambda \big )$ with constant scalar curvature is isometric to a Euclidean sphere $\mathbb {S}^{n}$ . As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreover, the vector field $X$ decomposes as the sum of a Killing vector field $Y$ and the gradient of a suitable function.     

A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold

It is a classical fact that the cotangent bundle \(T^* {\mathcal {M}}\) of a differentiable manifold \({\mathcal {M}}\) enjoys a canonical symplectic form \(\Omega ^*\) . If \(({\mathcal {M}},\mathrm{J} ,g,\omega )\) is a pseudo-Kähler or para-Kähler \(2n\) -dimensional manifold, we prove that the tangent bundle \(T{\mathcal {M}}\) also enjoys a natural pseudo-Kähler or para-Kähler structure \(({\tilde{\hbox {J}}},\tilde{g},\Omega )\) , where \(\Omega \) is the pull-back by \(g\) of \(\Omega ^*\) and \(\tilde{g}\) is a pseudo-Riemannian metric with neutral signature \((2n,2n)\) . We investigate the curvature properties of the pair \(({\tilde{\hbox {J}}},\tilde{g})\) and prove that: \(\tilde{g}\) is scalar-flat, is not Einstein unless \(g\) is flat, has nonpositive (resp. nonnegative) Ricci curvature if and only if \(g\) has nonpositive (resp. nonnegative) Ricci curvature as well, and is locally conformally flat if and only if \(n=1\) and \(g\) has constant curvature, or \(n>2\) and \(g\) is flat. We also check that (i) the holomorphic sectional curvature of \(({\tilde{\hbox {J}}},\tilde{g})\) is not constant unless \(g\) is flat, and (ii) in \(n=1\) case, that \(\tilde{g}\) is never anti-self-dual, unless conformally flat.     

Independence Under the G-Expectation Framework

We show that, for two non-trivial random variables \(X\) and \(Y\) under a sublinear expectation space, if \(X\) is independent from \(Y\) and \(Y\) is independent from \(X\) , then \(X\) and \(Y\) must be maximally distributed.     

Salem numbers in dynamics on Kähler threefolds and complex tori

Let \(X\) be a compact Kähler manifold of dimension \(k\!\le \! 4\) and \(f{:}X\!\rightarrow \! X\) a pseudo-automorphism. If the first dynamical degree \(\lambda _1(f)\) is a Salem number, we show that either \(\lambda _1(f)=\lambda _{k-1}(f)\) or \(\lambda _1(f)^2=\lambda _{k-2}(f)\) . In particular, if \({\dim }(X)=3\) then \(\lambda _1(f)=\lambda _2(f)\) . We use this to show that if \(X\) is a complex 3-torus and \(f\) is an automorphism of \(X\) with \(\lambda _1(f)>1\) , then \(f\) has a non-trivial equivariant holomorphic fibration if and only if \(\lambda _1(f)\) is a Salem number. If \(X\) is a complex 3-torus having an automorphism \(f\) with \(\lambda _1(f)=\lambda _2(f)>1\) but is not a Salem number, then the Picard number of \(X\) must be 0, 3 or 9, and all these cases can be realized.     

Hyperreflexivity of bounded N-cocycle spaces of banach algebras

Let \(X\) and \(Y\) be Banach spaces, \(n\in \mathbb {N}\) , and \(B^n(X,Y)\) the space of bounded \(n\) -linear maps from \(X\times \ldots \times X\) ( \(n\) -times) into \(Y\) . The concept of hyperreflexivity has already been defined for subspaces of \(B(X,Y)\) , where \(X\) and \(Y\) are Banach spaces. We extend this concept to the subspaces of \(B^n(X,Y)\) , taking into account its \(n\) -linear structure. We then investigate when \(\mathcal {Z}^n(A,X)\) , the space of all bounded \(n\) -cocycles from a Banach algebra \(A\) into a Banach \(A\) -bimodule \(X\) , is hyperreflexive. Our approach is based on defining two notions related to a Banach algebra, namely the strong property \((\mathbb {B})\) and bounded local units, and then applying them to find uniform criterions under which \(\mathcal {Z}^n(A,X)\) is hyperreflexive. We also demonstrate that these criterions are satisfied in variety of examples including large classes of C \(^*\) -algebras and group algebras and thereby providing various examples of hyperreflexive \(n\) -cocyle spaces. One advantage of our approach is that not only we obtain the hyperreflexivity for bounded \(n\) -cocycle spaces in different cases but also our results generalize the earlier ones on the hyperreflexivity of bounded derivation spaces, i.e. when \(n=1\) , in the literature. Finally, we investigate the hereditary properties of the strong property \((\mathbb {B})\) and b.l.u. This allows us to come with more examples of bounded \(n\) -cocycle spaces which are hyperreflexive.     

On conditional permutability and factorized groups

Two subgroups \(A\) and \(B\) of a group \(G\) are said to be totally completely conditionally permutable (tcc-permutable) if \(X\) permutes with \(Y^g\) for some \(g\in \langle X,Y\rangle \) , for all \(X \le A\) and all \(Y\le B\) . In this paper, we study finite products of tcc-permutable subgroups, focussing mainly on structural properties of such products. As an application, new achievements in the context of formation theory are obtained.     

Optimal mass transport and symmetric representations of their cost functions

We consider Monge–Kantorovich problems corresponding to general cost functions \(c(x,y)\) but with symmetry constraints on a Polish space \(X\times X\) . Such couplings naturally generate anti-symmetric Hamiltonians on \(X\times X\) that are \(c\) -convex with respect to one of the variables. In particular, if \(c\) is differentiable with respect to the first variable on an open subset \(X\) in \( \mathbb {R}^d\) , we show that for every probability measure \(\mu \) on \(X\) , there exists a symmetric probability measure \(\pi _0\) on \(X\times X\) with marginals \(\mu \) , and an anti-symmetric Hamiltonian \(H\) such that \(\nabla _2H(y, x)=\nabla _1c(x,y)\) for \( \pi _0\) -almost all \((x,y) \in X \times X.\) If \(\pi _0\) is supported on a graph \((x, Sx)\) , then \(S\) is necessarily a \(\mu \) -measure preserving involution (i.e., \(S^2=I\) ) and \(\nabla _2H(x, Sx)=\nabla _1c(Sx,x)\) for \(\mu \) -almost all \(x \in X.\) For monotone cost functions such as those given by \(c(x,y)=\langle x, u(y)\rangle \) or \(c(x,y)=-|x-u(y)|^2\) where \(u\) is a monotone operator, \(S\) is necessarily the identity yielding a classical result by Krause, namely that \(u(x)=\nabla _2H(x, x)\) where \(H\) is anti-symmetric and concave-convex.     

Sharp local estimates for the Szegö–Weinberger profile in Riemannian manifolds

We study the local Szegö–Weinberger profile in a geodesic ball \(B_g(y_0,r_0)\) centered at a point \(y_0\) in a Riemannian manifold \(({\mathcal {M}},g)\) . This profile is obtained by maximizing the first nontrivial Neumann eigenvalue \(\mu _2\) of the Laplace–Beltrami Operator \(\Delta _g\) on \({\mathcal {M}}\) among subdomains of \(B_g(y_0,r_0)\) with fixed volume. We derive a sharp asymptotic bounds of this profile in terms of the scalar curvature of \({\mathcal {M}}\) at \(y_0\) . As a corollary, we deduce a local comparison principle depending only on the scalar curvature. Our study is related to previous results on the profile corresponding to the minimization of the first Dirichlet eigenvalue of \(\Delta _g\) , but additional difficulties arise due to the fact that \(\mu _2\) is degenerate in the unit ball in \(\mathbb {R}^N\) and geodesic balls do not yield the optimal lower bound in the asymptotics we obtain.     

Approximation faible et principe de Hasse pour des espaces homogènes à stabilisateur fini résoluble

Let \(K\) be a global field and \(G\) a finite solvable \(K\) -group. Under certain hypotheses concerning the extension splitting \(G\) , we show that the homogeneous space \(V=G'/G\) with \(G'\) a semi-simple simply connected \(K\) -group has the weak approximation property. We use a more precise version of this result to prove the Hasse principle for homogeneous spaces \(X\) under a semi-simple simply connected \(K\) -group \(G'\) with finite solvable geometric stabilizer \({\bar{G}}\) , under certain hypotheses concerning the \(K\) -kernel (or \(K\) -lien) \(({\bar{G}},\kappa )\) defined by \(X\) .     

Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Fréchet manifold of all embeddings of \(M\) in \(N\) , endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\) . The Fréchet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\) , inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\) ). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\) . We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic.     

A sufficient condition for zero entropy

We prove that a diffeomorphism \(f\) defined on a compact manifold has zero topological entropy if there are \(d\in {\mathbb {N}}\) and \(K>0\) such that \(\Vert Dg^{n_x}(x)\Vert \le Kn^d_x\) for every diffeomorphism \(g\) that is \(C^1\) close to \(f\) and every periodic point \(x\) of least period \(n_x\) of \(g\) .     

A New Topological Helly Theorem and Some Transversal Results

We prove that for a topological space \(X\) with the property that \( H_{*}(U)=0\) for \(*\ge d\) and every open subset \(U\) of \(X\) , a finite family of open sets in \(X\) has nonempty intersection if for any subfamily of size \(j,\,1\le j\le d+1,\) the \((d-j)\) -dimensional homology group of its intersection is zero. We use this theorem to prove new results concerning transversal affine planes to families of convex sets.     

Sharp estimates on the first eigenvalue of the p-Laplacian with negative Ricci lower bound

We complete the picture of sharp eigenvalue estimates for the \(p\) -Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator \(\Delta _p\) when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and put Neumann boundary conditions on it. The proof is based on a refined gradient comparison technique and a careful analysis of the underlying model spaces.     

Truncated Quillen complexes of p-groups

Let \(p\) be an odd prime and let \(P\) be a \(p\) -group. We examine the order complex of the poset of elementary abelian subgroups of \(P\) having order at least \(p^2\) . Bouc and Thévenaz showed that this complex has the homotopy type of a wedge of spheres. We show that, for each nonnegative integer \(l\) , the number of spheres of dimension \(l\) in this wedge is controlled by the number of extraspecial subgroups \(X\) of \(P\) having order \(p^{2l+3}\) and satisfying \(\Omega _1(C_P(X))=Z(X)\) . We go on to provide a negative answer to a question raised by Bouc and Thévenaz concerning restrictions on the homology groups of the given complex.     

The V-filtration for tame unit F-crystals

Let \(X\) be a smooth variety over an algebraically closed field of characteristic \(p > 0, Z\) a smooth divisor, and \(j: U=X {\setminus } Z \rightarrow X\) the natural inclusion. We introduce in an axiomatic way the notion of a \(V\) -filtration on unit \(F\) -crystals and prove such axioms determine a unique filtration. It is shown that if \(\mathcal M \) is a tame unit \(F\) -crystal on \(U\) , then such a \(V\) -filtration along \(Z\) exists on \(j_*\mathcal M \) . The degree zero component of the associated graded module is proven to be the (unipotent) nearby cycles functor of Grothendieck and Deligne under the Emerton–Kisin Riemann–Hilbert correspondence. A few applications to \(\mathbb A ^1\) and gluing are then discussed.     

On an extension of the H^{k} mean curvature flow of closed convex hypersurfaces

In this paper we prove that the \(H^{k}\) ( \(k\) is odd and larger than \(2\) ) mean curvature flow of a closed convex hypersurface can be extended over the maximal time provided that the total \(L^{p}\) integral of the mean curvature is finite for some \(p\) .     

On an extension of the Frobenius^{\prime } theorem about p-nilpotency of a finite group

Suppose that \(G\) is a finite group and \(H\) is a subgroup of \(G\) . \(H\) is said to be \(s\) -quasinormally embedded in \(G\) if for each prime \(p\) dividing the order of \(H\) , a Sylow \(p\) -subgroup of \(H\) is also a Sylow \(p\) -subgroup of some \(s\) -quasinormal subgroup of \(G\) . We fix in every non-cyclic Sylow subgroup \(P\) of \(G\) some subgroup \(D\) satisfying \(1<|D|<|P|\) and study the \(p\) -nilpotency of \(G\) under the assumption that every subgroup \(H\) of \(P\) with \(|H|=|D|\) is \(s\) -quasinormally embedded in \(G\) . Some recent results and the Frobenius \(^{\prime }\) theorem are generalized.     

Compactness and sequential completeness in some spaces of operators

Let \(X\) be a completely regular Hausdorff space and \(C_b(X)\) be the Banach lattice of all real-valued bounded continuous functions on \(X\) , endowed with the strict topologies \(\beta _\sigma ,\) \(\beta _\tau \) and \(\beta _t\) . Let \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) \((z=\sigma ,\tau ,t)\) stand for the space of all \((\beta _z,\xi )\) -continuous linear operators from \(C_b(X)\) to a locally convex Hausdorff space \((E,\xi ),\) provided with the topology \(\mathcal{T}_s\) of simple convergence. We characterize relative \(\mathcal{T}_s\) -compactness in \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) in terms of the representing Baire vector measures. It is shown that if \((E,\xi )\) is sequentially complete, then the spaces \((\mathcal{L}_{\beta _z,\xi }(C_b(X),E),\mathcal{T}_s)\) are sequentially complete whenever \(z=\sigma \) ; \(z=\tau \) and \(X\) is paracompact; \(z=t\) and \(X\) is paracompact and ?ech complete. Moreover, a Dieudonné–Grothendieck type theorem for operators on \(C_b(X)\) is given.     

A uniqueness theorem for functions in the extended Selberg class

Recently, we proved that every finite dimensional Alexandrov space is strongly locally Lipschitz contractible. In the present paper, we consider the set \(\mathcal M\) of all isometry classes of Alexandrov spaces of curvature \(\ge -1\) and of fixed dimension having upper diameter bound and lower volume bound, and prove that there exists a constant \(N\) depending on the parameters determining \(\mathcal M\) such that every space in \(\mathcal M\) can be covered by at most \(N\) strongly Lipschitz contractible balls. Also, we prove that there exists a constant \(N^\prime \) depending on \(\mathcal M\) such that every space in \(\mathcal M\) can be covered by at most \(N^\prime \) strongly Lipschitz contractible and convex regions.