Classification of conformal minimal immersions of constant curvature from S^2 to { HP}^2

In this paper, we study geometry of conformal minimal two-spheres immersed in quaternionic projective spaces. We firstly use Bahy-El-Dien and Wood’s results to obtain some characterizations of the harmonic sequences generated by conformal minimal immersions from \(S^2\) to the quaternionic projective space \({ HP}^2\) . Then we give a classification theorem of linearly full totally unramified conformal minimal immersions of constant curvature from \(S^2\) to the quaternionic projective space \({ HP}^2\) .     

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.     

Finiteness of non-parabolic ends on submanifolds in spheres

We study a complete noncompact submanifold \(M^n\) in a sphere \(\mathbb {S}^{n+p}\) . We prove that the dimension of the space of \(L^2\) harmonic \(1\) -forms on \(M\) is finite and there are finitely many non-parabolic ends on \(M\) if the total curvature of \(M\) is finite and \(n\ge 3\) . This result is an improvement of Fu–Xu theorem on submanifolds in spheres and a generalized version of Cavalcante, Mirandola and Vitorio’s result on submanifolds in Hadamard manifolds.     

Intrinsically p-biharmonic maps

For a compact Riemannian manifold \(N\) , a domain \(\Omega \subset \mathbb {R}^m\) and for \(p\in (1, \infty )\) , we introduce an intrinsic version \(E_p\) of the \(p\) -biharmonic energy functional for maps \(u : \Omega \rightarrow N\) . This requires finding a definition for the intrinsic Hessian of maps \(u : \Omega \rightarrow N\) whose first derivatives are merely \(p\) -integrable. We prove, by means of the direct method, existence of minimizers of \(E_p\) within the corresponding intrinsic Sobolev space, and we derive a monotonicity formula. Finally, we also consider more general functionals defined in terms of polyconvex functions.     

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.     

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 \(g\) .     

Regularity of minimal hypersurfaces with a common free boundary

Let \(N\) be a Riemannian manifold and consider a stationary union of three or more \(C^{1,\mu }\) hypersurfaces-with-boundary \(M_k \subset N\) with a common boundary \(\Gamma \) . We show that if \(N\) is smooth, then \(\Gamma \) is smooth and each \(M_k\) is smooth up to \(\Gamma \) (real analytic in the case \(N\) is real analytic). Consequently we strengthen a result of Wickramasekera for stable codimension 1 integral varifolds regularity to conclude that under the stronger hypothesis that \(V\) is a stationary, stable, integral \(n\) -varifold in an \((n+1)\) -dimensional, smooth (real analytic) Riemannian manifold such that the support of \(\Vert V\Vert \) is nowhere locally the union of three or more smooth (real analytic) hypersurfaces-with-boundary meeting along a common boundary, the singular set of \(V\) is empty if \(n \le 6\) , is discrete if \(n = 7\) , and has Hausdorff dimension at most \(n-7\) if \(n \ge 8\) .     

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.     

On weakly closed subgroups of finite groups

Suppose that \(G\) is a finite group and \(H\) , \(K\) are subgroups of \(G\) . We say that \(H\) is weakly closed in \(K\) with respect to \(G\) if, for any \(g \in G\) such that \(H^{g}\le K\) , we have \(H^{g}=H\) . In particular, when \(H\) is a subgroup of prime-power order and \(K\) is a Sylow subgroup containing it, \(H\) is simply said to be a weakly closed subgroup of \(G\) or weakly closed in \(G\) . In the paper, we investigate the structure of finite groups by means of weakly closed subgroups.     

Additive mappings satisfying algebraic conditions in rings

Let \(R\) be any \((n+1)!\) -torsion free ring and \(F,D: R\rightarrow R\) be additive mappings satisfying \(F(x^{n+1})=(\alpha (x))^nF(x)+\sum \nolimits _{i=1}^n (\alpha (x))^{n-i}(\beta (x))^iD(x)\) for all \(x\in R\) , where \(n\) is a fixed integer and \(\alpha \) , \(\beta \) are automorphisms of \(R\) . Then, \(D\) is Jordan left \((\alpha , \beta )\) -derivation and \(F\) is generalized Jordan left \((\alpha , \beta )\) -derivation on \(R\) and if additive mappings \(F\) and \(D\) satisfying \(F(x^{n+1})=F(x)(\alpha (x))^n+\sum \nolimits _{i=1}^n (\beta (x))^iD(x)(\alpha (x))^{n-i}\) for all \(x\in R\) . Then, \(D\) is Jordan \((\alpha , \beta )\) -derivation and \(F\) is generalized Jordan \((\alpha , \beta )\) -derivation on \(R\) . At last some immediate consequences of the above theorems have been given.     

Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies

Let \(K\subset \mathbb R ^N\) be a convex body containing the origin. A measurable set \(G\subset \mathbb R ^N\) with positive Lebesgue measure is said to be uniformly \(K\) -dense if, for any fixed \(r>0\) , the measure of \(G\cap (x+r K)\) is constant when \(x\) varies on the boundary of \(G\) (here, \(x+r K\) denotes a translation of a dilation of \(K\) ). We first prove that \(G\) must always be strictly convex and at least \(C^{1,1}\) -regular; also, if \(K\) is centrally symmetric, \(K\) must be strictly convex, \(C^{1,1}\) -regular and such that \(K=G-G\) up to homotheties; this implies in turn that \(G\) must be \(C^{2,1}\) -regular. Then for \(N=2\) , we prove that \(G\) is uniformly \(K\) -dense if and only if \(K\) and \(G\) are homothetic to the same ellipse. This result was already proven by Amar et al. in 2008 . However, our proof removes their regularity assumptions on \(K\) and \(G\) , and more importantly, it is susceptible to be generalized to higher dimension since, by the use of Minkowski’s inequality and an affine inequality, avoids the delicate computations of the higher-order terms in the Taylor expansion near \(r=0\) for the measure of \(G\cap (x+r\,K)\) (needed in 2008).     

New Tools for Classifying Hamiltonian Circle Actions with Isolated Fixed Points

For every compact almost complex manifold \((\mathsf {M},\mathsf {J})\) equipped with a \(\mathsf {J}\) -preserving circle action with isolated fixed points, a simple algebraic identity involving the first Chern class is derived. This enables us to construct an algorithm to obtain linear relations among the isotropy weights at the fixed points. Suppose that \(\mathsf {M}\) is symplectic and the action is Hamiltonian. If the manifold satisfies an extra so-called positivity condition, then this algorithm determines a family of vector spaces that contain the admissible lattices of weights. When the number of fixed points is minimal, this positivity condition is necessarily satisfied whenever \(\dim (\mathsf {M})\le 6\) and, when \(\dim (\mathsf {M})=8\) , whenever the \(S^1\) -action extends to an effective Hamiltonian \(T^2\) -action, or none of the isotropy weights is \(1\) . Moreover, there are no known examples with a minimal number of fixed points contradicting this condition, and their existence is related to interesting questions regarding fake projective spaces. We run the algorithm for \(\dim (\mathsf {M})\le 8\) , quickly obtaining all the possible families of isotropy weights. In particular, we simplify the proofs of Ahara and Tolman for \(\dim (\mathsf {M})=6\) and, when \(\dim (\mathsf {M})=8\) , we prove that the equivariant cohomology ring, Chern classes, and isotropy weights agree with those of \({\mathbb {C}}P^4\) with the standard \(S^1\) -action (thereby proving the symplectic Petrie conjecture in this setting).     

Bounding the exponent of a verbal subgroup

We deal with the following conjecture. If \(w\) is a group word and \(G\) is a finite group in which any nilpotent subgroup generated by \(w\) -values has exponent dividing \(e\) , then the exponent of the verbal subgroup \(w(G)\) is bounded in terms of \(e\) and \(w\) only. We show that this is true in the case where \(w\) is either the \(n\text{ th }\) Engel word or the word \([x^n,y_1,y_2,\ldots ,y_k]\) (Theorem A). Further, we show that for any positive integer \(e\) there exists a number \(k=k(e)\) such that if \(w\) is a word and \(G\) is a finite group in which any nilpotent subgroup generated by products of \(k\) values of the word \(w\) has exponent dividing \(e\) , then the exponent of the verbal subgroup \(w(G)\) is bounded in terms of \(e\) and \(w\) only (Theorem B).     

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 \(X\) .     

New M-ary sequences with low autocorrelation from interleaved technique

Let \(p\) and \(q\) be two odd primes with \(p=Mf+1\) and \(M\) is even. A new construction of \(M\) -ary sequences of period \(pq\) with low periodic autocorrelation is presented in this paper based on interleaving the \(M\) -ary power residue sequence of period \(p\) according to the quadratic residue with respect to \(q\) . This construction can generate the well-known twin-prime sequence and generalized cyclotomy sequence of order two if \(M=2\) . For \(M=4\) , a new class of quaternary sequences of period \(pq\) with maximal nontrivial autocorrelation value being either \(\sqrt{5}\) or \(3\) is obtained. This achieves the best known results for such kind of quaternary sequences.     

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.     

On extensions of completely simple semigroups by groups

An example of an extension of a completely simple semigroup \(U\) by a group \(H\) is given which cannot be embedded into the wreath product of \(U\) by \(H\) . On the other hand, every central extension of \(U\) by \(H\) is shown to be embeddable in the wreath product of \(U\) by \(H\) , and any extension of \(U\) by \(H\) is proved to be embeddable in a semidirect product of a completely simple semigroup \(V\) by \(H\) where the maximal subgroups of \(V\) are direct powers of those of \(U\) .     

Note on prehomogeneous vector spaces

Let \(V\) be a complex prehomogeneous vector space under the action of a linear algebraic group \(G\) . Assume the poset of orbit closures in the Zariski topology \(\{\overline{Gx}:x\in V\}\) coincides with a (partial) flag \(V_0=0<V_1<\dots <V_k=V\) in \(V\) . Then for any Borel subgroup \(B\) of \(G\) , the poset \(\{\overline{B x}:x\in V\}\) coincides with a full flag in \(V\) .     

Endomorphism rings of bimodules

Let \(M\) be an \(R\) - \(R\) -bimodule over a semi-prime right and left Goldie ring \(R\) . We investigate how non-singularity conditions on \(M_R\) are related to such conditions on \(_RM\) . In particular, we say an \(R\) - \(R\) -bimodule \(M\) such that \(_RM\) and \(M_R\) are non-singular has the right essentiality property if \(IM_R\) is essential in \(M_R\) for all essential right ideals \(I\) of \(R\) , and investigate several questions related to this property.