The second pinching theorem for hypersurfaces with constant mean curvature in a sphere

We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng–Terng, Wei–Xu, Zhang, and Ding–Xin to the case of hypersurfaces with small constant mean curvature. Let $M^n$ be a compact hypersurface with constant mean curvature $H$ in $S^{n+1}$ . Denote by $S$ the squared norm of the second fundamental form of $M$ . We prove that there exist two positive constants $\gamma (n)$ and $\delta (n)$ depending only on $n$ such that if $|H|\le \gamma (n)$ and $\beta (n,H)\le S\le \beta (n,H)+\delta (n)$ , then $S\equiv \beta (n,H)$ and $M$ is one of the following cases: (i) $S^{k}\Big (\sqrt{\frac{k}{n}}\Big )\times S^{n-k}\Big (\sqrt{\frac{n-k}{n}}\Big )$ , $\,1\le k\le n-1$ ; (ii) $S^{1}\Big (\frac{1}{\sqrt{1+\mu ^2}}\Big )\times S^{n-1}\Big (\frac{\mu }{\sqrt{1+\mu ^2}}\Big )$ . Here $\beta (n,H)=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)} \sqrt{n^2H^4+4(n-1)H^2}$ and $\mu =\frac{n|H|+\sqrt{n^2H^2+ 4(n-1)}}{2}$ .     

Ranks of backward shift invariant subspaces of the Hardy space over the bidisk

Let $\{\varphi _n(z)\}_{n\ge 0}$ be a sequence of inner functions satisfying that $\zeta _n(z):=\varphi _n(z)/\varphi _{n+1}(z)\in H^\infty (z)$ for every $n\ge 0$ and $\{\varphi _n(z)\}_{n\ge 0}$ has no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace $\mathcal{M }$ of $H^2(\mathbb{D }^2)$ . The ranks of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }_z$ and $\mathcal{F }^*_z$ respectively are determined, where $\mathcal{F }_z$ is the fringe operator on $\mathcal{M }\ominus w\mathcal{M }$ . Let $\mathcal{N }= H^2(\mathbb{D }^2)\ominus \mathcal{M }$ . It is also proved that the rank of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }^*_z$ equals to the rank of $\mathcal{N }$ for $T^*_z$ and $T^*_w$ .     

Global symplectic coordinates on gradient Kähler–Ricci solitons

A classical result of McDuff [14] asserts that a simply connected complete Kähler manifold $(M,g,\omega )$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $\Psi \ : M \rightarrow \mathbb{R }^{2n}$ (where $n$ is the complex dimension of $M$ ), satisfying the following property (proved by E. Ciriza in [4]): the image $\Psi (T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $\Psi (p)=0$ , is a complex linear subspace of $\mathbb C ^n\simeq \mathbb{R }^{2n}$ . The aim of this paper is to exhibit, for all positive integers $n$ , examples of $n$ -dimensional complete Kähler manifolds with non-negative sectional curvature globally symplectomorphic to $\mathbb{R }^{2n}$ through a symplectomorphism satisfying Ciriza’s property.     

Dimension Reduction for Finite Trees in \varvec{\ell _1}

We show that every $n$ -point tree metric admits a $(1+\varepsilon )$ -embedding into $\ell _1^{C(\varepsilon ) \log n}$ , for every $\varepsilon > 0$ , where $C(\varepsilon ) \le O\big ((\frac{1}{\varepsilon })^4 \log \frac{1}{\varepsilon })\big )$ . This matches the natural volume lower bound up to a factor depending only on $\varepsilon $ . Previously, it was unknown whether even complete binary trees on $n$ nodes could be embedded in $\ell _1^{O(\log n)}$ with $O(1)$ distortion. For complete $d$ -ary trees, our construction achieves $C(\varepsilon ) \le O\big (\frac{1}{\varepsilon ^2}\big )$ .     

On characterizations of real hypersurface in complex space form with Codazzi type structure Lie operator

In this paper, we prove that if $(\nabla _{X} L_{\xi })Y= (\nabla _{Y} L_{\xi })X$ holds on $M$ , then $M$ is a Hopf hypersurface, where $L_\xi $ denote the induced operator from the Lie derivative with respect to the structure vector field $\xi $ . We characterize such Hopf hypersurfaces of $M_n(c)$ .     

Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold $M$ . Under the assumption that the sectional curvature $K^M$ is strictly positive, we prove the existence of a smooth immersion $f:{\mathbb {S}}^2 \rightarrow M$ minimizing the $L^2$ integral of the second fundamental form. Assuming instead that $K^M \le 2$ and that there is some point $\overline{x} \in M$ with scalar curvature $R^M(\overline{x}) > 6$ , we obtain a smooth minimizer $f:{\mathbb {S}}^2 \rightarrow M$ for the functional $\int \frac{1}{4}|H|^2+1$ , where $H$ is the mean curvature.     

Yangians and quantum loop algebras

Let $\mathfrak{g }$ be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra $U_\hbar (L\mathfrak g )$ of $\mathfrak{g }$ degenerates to the Yangian ${Y_\hbar (\mathfrak g )}$ . We strengthen this result by constructing an explicit algebra homomorphism $\Phi $ from $U_\hbar (L\mathfrak g )$ to the completion of ${Y_\hbar (\mathfrak g )}$ with respect to its grading. We show moreover that $\Phi $ becomes an isomorphism when ${U_\hbar (L\mathfrak g )}$ is completed with respect to its evaluation ideal. We construct a similar homomorphism for $\mathfrak{g }=\mathfrak{gl }_n$ and show that it intertwines the actions of $U_\hbar (L\mathfrak gl _{n})$ and $Y_\hbar (\mathfrak gl _{n})$ on the equivariant $K$ -theory and cohomology of the variety of $n$ -step flags in ${\mathbb{C }}^d$ constructed by Ginzburg–Vasserot.     

Fundamental groups of positively curved manifolds with symmetry

We obtain a classification for the fundamental groups of closed $n$ -manifolds of positive sectional curvature which admit an isometric $T^k$ -action with $k \ge \frac{n}{6}+1 (n \ne 11, 15)$ .     

A decomposition of the universal embedding space for the near polygon {{\mathbb H}_n}

Let ${{\mathbb H}_n, n \geq 1}$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n?+?2 vertices, and let e denote the absolutely universal embedding of ${{\mathbb H}_n}$ into PG(W), where W is a ${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$ -dimensional vector space over the field ${{\mathbb F}_2}$ with two elements. For every point z of ${{\mathbb H}_n}$ and every ${i \in {\mathbb N}}$ , let Δ _i (z) denote the set of points of ${{\mathbb H}_n}$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of ${{\mathbb H}_n, W}$ can be written as a direct sum ${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$ such that the following four properties hold for every ${i \in \{0,\ldots,n \}}$ : (1) ${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$ ; (2) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$ ; (3) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$ ; (4) ${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$ .     

\mathfrak D -parallelism of normal and structure Jacobi operators for hypersurfaces in complex two-plane Grassmannians

In this paper, we give non-existence theorems for Hopf hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C }^{m+2})$ with $\mathfrak D $ -parallel normal Jacobi operator ${\bar{R}}_N$ and $\mathfrak D $ -parallel structure Jacobi operator $R_{\xi }$ if the distribution $\mathfrak D $ or $\mathfrak D ^{\bot }$ component of the Reeb vector field is invariant by the shape operator, respectively.     

Geometric rigidity for sequences of W^{2,2} conformal immersions

We analyse sequences of discs conformally immersed in $ \mathbb{R }^ n$ with energy $ \int _{ D} |A_k |^ 2 \le \gamma _n$ , where $ \gamma _n = 8\pi $ if $ n=3$ and $ \gamma _n = 4 \pi $ when $n\ge 4$ . We show that if such sequences do not weakly converge to a conformal immersion, then by a sequence of dilations we obtain a complete minimal surface with bounded total curvature, either Enneper’s minimal surface if $ n=3$ or Chen’s minimal graph if $ n \ge 4$ . In the papers, (Kuwert and Li, Comm Anal Geom 20(2), 313–340, 2012; Rivière, Adv Calculus Variations 6(1), 1–31, 2013) it was shown that if a sequence of immersed tori diverges in moduli space then $\liminf _ {k\rightarrow \infty } \mathcal W ( f_k )\ge 8\pi $ . We apply the above analysis to show that in $ \mathbb{R }^3$ if the sequence diverges so that $ \lim _{ k \rightarrow \infty } \mathcal W (f_k) =8\pi $ then there exists a sequence of Möbius transforms $ \sigma _{k}$ such that $ \sigma _k\circ f _k$ converges weakly to a catenoid.     

Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions

We consider the distribution of the orbits of the number 1 under the $\beta $ -transformations $T_\beta $ as $\beta $ varies. Mainly, the size of the set of $\beta >1$ for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. The dimension of the following set $E\big (\{\ell _n\}_{n\ge 1}, x_0\big )=\Big \{\,\beta >1: |T^n_{\beta }1-x_0|<\beta ^{-\ell _n}, \hbox { for infinitely many}, \, n\in \mathbb{N }\,\Big \}$ is determined, where $x_0$ is a given point in $[0,1]$ and $\{\ell _n\}_{n\ge 1}$ is a sequence of integers tending to infinity as $n\rightarrow \infty $ . For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterise the lengths and the distribution of cylinders (the set of $\beta $ with a common prefix in the expansion of 1) in the parameter space $\{\,\beta \in \mathbb{R }: \beta >1\,\}$ .     

A construction of integer-valued polynomials with prescribed sets of lengths of factorizations

For an arbitrary finite non-empty set $S$ of natural numbers greater $1$ , we construct $f\in \text{ Int }(\mathbb{Z })=\{g\in \mathbb{Q }[x]\mid g(\mathbb{Z })\subseteq \mathbb{Z }\}$ such that $S$ is the set of lengths of $f$ , i.e., the set of all $n$ such that $f$ has a factorization as a product of $n$ irreducibles in $\text{ Int }(\mathbb{Z })$ . More generally, we can realize any finite non-empty multi-set of natural numbers greater 1 as the multi-set of lengths of the essentially different factorizations of $f$ .     

On the Bloch–Kato conjecture for elliptic modular forms of square-free level

Let $\kappa \ge 6$ be an even integer, $M$ an odd square-free integer, and $f \in S_{2\kappa -2}(\Gamma _0(M))$ a newform. We prove that under some reasonable assumptions that half of the $\lambda $ -part of the Bloch–Kato conjecture for the near central critical value $L(\kappa ,f)$ is true. We do this by bounding the $\ell $ -valuation of the order of the appropriate Bloch–Kato Selmer group below by the $\ell $ -valuation of algebraic part of $L(\kappa ,f)$ . We prove this by constructing a congruence between the Saito–Kurokawa lift of $f$ and a cuspidal Siegel modular form.     

Multiple P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in {\varvec{R}}^{2n}

In this paper, let $n$ be a positive integer and $P=diag(-I_{n-\kappa },I_\kappa ,-I_{n-\kappa },I_\kappa )$ for some integer $\kappa \in [0, n]$ , we prove that for any compact convex hypersurface $\Sigma $ in $\mathbf{R}^{2n}$ with $n\ge 2$ there exist at least two geometrically distinct P-invariant closed characteristics on $\Sigma $ , provided that $\Sigma $ is P-symmetric, i.e., $x\in \Sigma $ implies $Px\in \Sigma $ . This work is shown to extend and unify several earlier works on this subject.     

On adding a variable to a Frobenius manifold and generalizations

Let $\pi :V\rightarrow M$ be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure $(\circ _{M},e_{M},g_{M})$ and typical fiber has the structure of a Frobenius algebra $(\circ _{V},e_{V},g_{V})$ . Using a connection $D$ on the bundle $\pi : V{\,\rightarrow \,}M$ and a morphism $\alpha :V\rightarrow TM$ , we construct an almost Frobenius structure $(\circ , e_{V},g)$ on the manifold $V$ and we study when it is Frobenius. In particular, we describe all (real) positive definite Frobenius structures on $V$ obtained in this way, when $M$ is a semisimple Frobenius manifold with non-vanishing rotation coefficients. In the holomorphic setting, we add a real structure $k_{M}$ on $M$ and a real structure $k_{V}$ on the bundle $\pi : V \rightarrow M$ . Using $k_{M}$ , $k_{V}$ and $D$ we define a real structure $k$ on the manifold $V$ . We study when $k$ , together with an almost Frobenius structure $(\circ , e_{V}, g) $ , satisfies the tt*- equations. Along the way, we prove various properties of adding variables to a Frobenius manifold, in connection with Legendre transformations and $tt^{*}$ -geometry.     

Geodesic Ricci solitons on unit tangent sphere bundles

In the paper, (Abbassi and Kowalski, Ann Glob Anal Geom, 38: 11–20, 2010) the authors study Einstein Riemannian $g$ natural metrics on unit tangent sphere bundles. In this study, we equip the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,g)$ with an arbitrary Riemannian $g$ natural metric $\tilde{G}$ and we show that if the geodesic flow $\tilde{\xi }$ is the potential vector field of a Ricci soliton $(\tilde{G},\tilde{\xi },\lambda )$ on $T_1M,$ then $(T_1M,\tilde{G})$ is Einstein. Moreover, we show that the Reeb vector field of a contact metric manifold is an infinitesimal harmonic transformation if and only if it is Killing. Thus, we consider a natural contact metric structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ over $T_1 M$ and we show that the geodesic flow $\tilde{\xi }$ is an infinitesimal harmonic transformation if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi },\tilde{\xi })$ is Sasaki $\eta $ -Einstein. Consequently, we get that $(\tilde{G},\tilde{\xi }, \lambda )$ is a Ricci soliton if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ is Sasaki-Einstein with $\lambda = 2(n-1) >0.$ This last result gives new examples of Sasaki–Einstein structures.     

On Wilking’s criterion for the Ricci flow

Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators $C(S)$ , which are nonnegative in a suitable sense, to every $Ad_{SO(n,\mathbb{C })}$ invariant subset $S \subset \mathbf{so}(n,\mathbb{C })$ . In this article we show that if $S$ is an $Ad_{SO(n,\mathbb{C })}$ invariant subset of $\mathbf{so}(n,\mathbb{C })$ such that $S\cup \{0\}$ is closed and $C_+(S)\subset C(S)$ denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in $C_+(S)$ also admits a metric with curvature operator in $C_+(S)$ (b) The normalized Ricci flow on any compact Riemannian manifold $M$ with curvature operator in $C_+(S)$ converges to a metric of constant positive sectional curvature. We also point out that if $S$ is an arbitrary $Ad_{SO(n,\mathbb{C })}$ subset, then $C(S)$ is contained in the cone of curvature operators with nonnegative isotropic curvature.     

Geodesics,distance, and the CAT(0) property for the manifold of Riemannian metrics

Let $\mathcal F ^a_\lambda $ be the PBW degeneration of the flag varieties of type $A_{n-1}$ . These varieties are singular and are acted upon with the degenerate Lie group $SL_n^a$ . We prove that $\mathcal F ^a_\lambda $ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\lambda $ of $\mathcal F ^a_\lambda $ . The varieties $R_\lambda $ can be viewed as towers of successive $\mathbb{P }^1$ -fibrations, thus providing an analogue of the classical Bott–Samelson–Demazure–Hansen desingularization. We prove that the varieties $R_\lambda $ are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel–Weil type theorem for $\mathcal F ^a_\lambda $ . Using the Atiyah–Bott–Lefschetz formula for $R_\lambda $ , we compute the $q$ -characters of the highest weight $\mathfrak sl _n$ -modules.     

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S _* related to the Carleson?CHunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in ${L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ and ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ on the weighted Lebesgue spaces ${L^p(\mathbb{R},w)}$ , with 1?< p <? ?? and ${w\in A_p(\mathbb{R})}$ . The Banach algebras ${L^\infty(\mathbb{R}, V(\mathbb{R}))}$ and ${PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ consist, respectively, of all bounded measurable or piecewise continuous ${V(\mathbb{R})}$ -valued functions on ${\mathbb{R}}$ where ${V(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded total variation, and the Banach algebra ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ consists of all Lipschitz ${V_d(\mathbb{R})}$ -valued functions of exponent ${\gamma \in (0,1]}$ on ${\mathbb{R}}$ where ${V_d(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded variation on dyadic shells. Finally, for the Banach algebra ${\mathfrak{A}_{p,w}}$ generated by all pseudodifferential operators a(x, D) with symbols ${a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ on the space ${L^p(\mathbb{R}, w)}$ , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ .