Rank-two stable sheaves with odd determinant on Fano threefolds of genus nine

According to Mukai and Iliev, a smooth prime Fano threefold $X$ of genus $9$ is associated with a surface $\mathbb{P }(\mathcal{V })$ , ruled over a smooth plane quartic $\varGamma $ , and the derived category of $\varGamma $ embeds into that of $X$ by a theorem of Kuznetsov. We use this setup to study the moduli spaces of rank- $2$ stable sheaves on $X$ with odd determinant. For each $c_2 \ge 7$ , we prove that a component of their moduli space $\mathsf{M}_X(2,1,c_2)$ is birational to a Brill–Noether locus of vector bundles with fixed rank and degree on $\varGamma $ , having enough sections when twisted by $\mathcal{V }$ . For $c_2=7$ , we prove that $\mathsf{M}_X(2,1,7)$ is isomorphic to the blow-up of the Picard variety $\text{ Pic}^{2}({\varGamma })$ along the curve parametrizing lines contained in $X$ .     

Epimorphic images of the [5,3,5] Coxeter group

We classify the normal subgroups of the Coxeter group $\varGamma =[5,3,5]$ , and of its even subgroup $\varGamma ^+$ , with quotient isomorphic to a finite simple group $L_2(q)$ . There are infinitely many such normal subgroups of $\varGamma ^+$ , each uniformising a compact orientable hyperbolic $3$ -manifold tessellated by dodecahedra; we determine the isometry groups of these manifolds and the symmetry groups of their tessellations. By contrast there is a single such normal subgroup of $\varGamma $ , uniformising a compact non-orientable $3$ -orbifold with isometry group $PGL_2(19)$ .     

On the compactness of the set of invariant Einstein metrics

Let $M = G/H$ be a connected simply connected homogeneous manifold of a compact, not necessarily connected Lie group $G$ . We will assume that the isotropy $H$ -module $\mathfrak{g/h }$ has a simple spectrum, i.e. irreducible submodules are mutually non-equivalent. There exists a convex Newton polytope $N=N(G,H)$ , which was used for the estimation of the number of isolated complex solutions of the algebraic Einstein equation for invariant metrics on $G/H$ (up to scaling). Using the moment map, we identify the space $\mathcal{M }_1$ of invariant Riemannian metrics of volume 1 on $G/H$ with the interior of this polytope $N$ . We associate with a point ${x \in \partial N}$ of the boundary a homogeneous Riemannian space (in general, only local) and we extend the Einstein equation to $\partial N$ . As an application of the Alekseevsksky–Kimel’fel’d theorem, we prove that all solutions of the Einstein equation associated with points of the boundary are locally Euclidean. We describe explicitly the set $T\subset \partial N$ of solutions at the boundary together with its natural triangulation. Investigating the compactification ${\overline{\mathcal{M }}}_{1} = N$ of $\mathcal{M }_1$ , we get an algebraic proof of the deep result by Böhm, Wang and Ziller about the compactness of the set $\mathcal{E }_1 \subset \mathcal{M }_1$ of Einstein metrics. The original proof by Böhm, Wang and Ziller was based on a different approach and did not use the simplicity of the spectrum. In Appendix, we consider the non-symmetric flag manifolds $G/H$ with the second Betti number $b_2=1$ . We calculate the normalized volumes $2,6,20,82,344$ of the corresponding Newton polytopes and discuss the number of complex solutions of the algebraic Einstein equation and the finiteness problem.     

A Characterization of Compact Operators Via the Non-Connectedness of the Attractors of a Family of IFSs

In this paper we present a result which establishes a connection between the theory of compact operators and the theory of iterated function systems. For a Banach space $X$ , $S$ and $T$ bounded linear operators from $X$ to $X$ such that $\Vert S\Vert , \Vert T\Vert <1$ and $w\in X$ , let us consider the IFS $\mathcal S _{w}=(X,f_{1},f_{2})$ , where $f_{1},f_{2}:X\rightarrow X$ are given by $f_{1}(x)=S(x)$ and $f_{2}(x)=T(x)+w$ , for all $x\in X$ . On one hand we prove that if the operator $S$ is compact, then there exists a family $(K_{n})_{n\in \mathbb N }$ of compact subsets of $X$ such that $A_{\mathcal S _{w}}$ is not connected, for all $w\in X-\bigcup _{n\in \mathbb N } K_{n}$ . On the other hand we prove that if $H$ is an infinite dimensional Hilbert space, then a bounded linear operator $S:H\rightarrow H$ having the property that $\Vert S\Vert <1$ is compact provided that for every bounded linear operator $T:H\rightarrow H$ such that $\Vert T\Vert <1$ there exists a sequence $(K_{T,n})_{n}$ of compact subsets of $H$ such that $A_{\mathcal S _{w}}$ is not connected for all $w\in H-\bigcup _{n}K_{T,n}$ . Consequently, given an infinite dimensional Hilbert space $H$ , there exists a complete characterization of the compactness of an operator $S:H\rightarrow H$ by means of the non-connectedness of the attractors of a family of IFSs related to the given operator. Finally we present three examples illustrating our results.     

The Calabi homomorphism, Lagrangian paths and special Lagrangians

Let $\mathcal{O }$ be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold $(X,\omega ).$ We define a functional $\mathcal{C }:\mathcal{O } \rightarrow \mathbb{R }$ for each differential form $\beta $ of middle degree satisfying $\beta \wedge \omega = 0$ and an exactness condition. If the exactness condition does not hold, $\mathcal{C }$ is defined on the universal cover of $\mathcal{O }.$ A particular instance of $\mathcal{C }$ recovers the Calabi homomorphism. If $\beta $ is the imaginary part of a holomorphic volume form, the critical points of $\mathcal{C }$ are special Lagrangian submanifolds. We present evidence that $\mathcal{C }$ is related by mirror symmetry to a functional introduced by Donaldson to study Einstein–Hermitian metrics on holomorphic vector bundles. In particular, we show that $\mathcal{C }$ is convex on an open subspace $\mathcal{O }^+ \subset \mathcal{O }.$ As a prerequisite, we define a Riemannian metric on $\mathcal{O }^+$ and analyze its geodesics. Finally, we discuss a generalization of the flux homomorphism to the space of Lagrangian submanifolds, and a Lagrangian analog of the flux conjecture.     

Asymptotics for Some Combinatorial Characteristics of the Convex Hull of a Poisson Point Process in the Clifford Torus

Let $\mathcal P _\lambda $ be a homogeneous Poisson point process of rate $\lambda $ in the Clifford torus $T^2\subset \mathbb E ^4$ . Let $(f_0, f_1, f_2, f_3)$ be the $f$ -vector of conv $\,\mathcal P _\lambda $ and let $\bar{v}$ be the mean valence of a vertex of the convex hull. Asymptotic expressions for $\mathsf E \, f_1$ , $\mathsf E \, f_2$ , $\mathsf E \, f_3$ and $\mathsf E \, \bar{v}$ as $\lambda \rightarrow \infty $ are proved in this paper.     

Boundedness of Generalized Riesz Transforms on Orlicz–Hardy Spaces Associated to Operators

Let ${\Phi}$ be a continuous, strictly increasing and concave function on (0, ∞) of critical lower type index ${p_\Phi^- \in(0,\,1]}$ . Let L be an injective operator of type ω having a bounded H _∞ functional calculus and satisfying the k-Davies–Gaffney estimates with ${k \in {\mathbb Z}_+}$ . In this paper, the authors first introduce an Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ in terms of the non-tangential L-adapted square function and then establish its molecular characterization. As applications, the authors prove that the generalized Riesz transform ${D_{\gamma}L^{-\delta/(2k)}}$ is bounded from the Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz space ${L^{\widetilde{\Phi}}(\mathbb{R}^n)}$ when ${p_\Phi^- \in (0, \frac{n}{n+ \delta - \gamma}]}$ , ${0 < \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz–Hardy space ${H^{\widetilde \Phi}(\mathbb{R}^n)}$ when ${p_\Phi^-\in (\frac{n}{n + \delta+ \lfloor \gamma \rfloor- \gamma},\,\frac{n}{n+ \delta- \gamma}]}$ , ${1\le \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the weak Orlicz–Hardy space ${WH^\Phi(\mathbb{R}^n)}$ when ${\gamma = \delta}$ and ${p_\Phi=n/(n + \lfloor \gamma \rfloor)}$ or ${p_\Phi^-=n/(n + \lfloor \gamma \rfloor)}$ with ${p_\Phi^-}$ attainable, where ${\widetilde{\Phi}}$ is an Orlicz function whose inverse function ${\widetilde{\Phi}^{-1}}$ is defined by ${\widetilde{\Phi}^{-1}(t):=\Phi^{-1}(t)t^{\frac{1}{n}(\gamma- \delta)}}$ for all ${t \in (0,\,\infty)}$ , ${p_\Phi}$ denotes the strictly critical lower type index of ${\Phi}$ , ${\lfloor \gamma \rfloor}$ the maximal integer not more than ${\gamma}$ and ${(p_-(L),\,p_+(L))}$ the range of exponents ${p \in[1,\, \infty]}$ for which the semigroup ${\{e^{-tL}\}_{t >0 }}$ is bounded on ${L^p(\mathbb{R}^n)}$ .     

Toric surfaces,K-stability and Calabi flow

Let $X$ be a toric surface and $u$ be a normalized symplectic potential on the corresponding polygon $P$ . Suppose that the Riemannian curvature is bounded by a constant $C_1$ and $ \int _{\partial P} u d \sigma < C_2, $ then there exists a constant $C_3$ depending only on $C_1, C_2$ and $P$ such that the diameter of $X$ is bounded by $C_3$ . Moreoever, we can show that there is a constant $M > 0$ depending only on $C_1, C_2$ and $P$ such that Donaldson’s $M$ -condition holds for $u$ . As an application, we show that if $(X,P)$ is (analytic) relative $K$ -stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Calabi flow exists for all time and the Riemannian curvature is uniformly bounded along the Calabi flow.     

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 $C(\varepsilon ) \le O\big (\frac{1}{\varepsilon ^2}\big )$ .     

Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups

Let $G \subset GL(V)$ be a reductive algebraic subgroup acting on the symplectic vector space $W=(V \oplus V^*)^{\oplus m}$ , and let $\mu :\ W \rightarrow Lie(G)^*$ be the corresponding moment map. In this article, we use the theory of invariant Hilbert schemes to construct a canonical desingularization of the symplectic reduction $\mu ^{-1}(0)/\!/G$ for classes of examples where $G=GL(V)$ , $O(V)$ , or $Sp(V)$ . For these classes of examples, $\mu ^{-1}(0)/\!/G$ is isomorphic to the closure of a nilpotent orbit in a simple Lie algebra, and we compare the Hilbert–Chow morphism with the (well-known) symplectic desingularizations of $\mu ^{-1}(0)/\!/G$ .     

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.     

Generalized normal homogeneous Riemannian metrics on spheres and projective spaces

In this paper, we develop new methods to study generalized normal homogeneous Riemannian manifolds. In particular, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on spheres ${S^n}$ . We prove that for any connected (almost effective) transitive on $S^n$ compact Lie group $G$ , the family of $G$ -invariant Riemannian metrics on $S^n$ contains generalized normal homogeneous but not normal homogeneous metrics if and only if this family depends on more than one parameters and $n\ge 5$ . Any such family (that exists only for $n=2k+1$ ) contains a metric $g_\mathrm{can}$ of constant sectional curvature $1$ on $S^n$ . We also prove that $(S^{2k+1}, g_\mathrm{can})$ is Clifford–Wolf homogeneous, and therefore generalized normal homogeneous, with respect to $G$ (except the groups $G={ SU}(k+1)$ with odd $k+1$ ). The space of unit Killing vector fields on $(S^{2k+1}, g_\mathrm{can})$ from Lie algebra $\mathfrak g $ of Lie group $G$ is described as some symmetric space (except the case $G=U(k+1)$ when one obtains the union of all complex Grassmannians in $\mathbb{C }^{k+1}$ ).     

Real Clifford Algebras as Tensor Products over Centers

Denote by ${\mathcal{C}\ell_{p,q}}$ the Clifford algebra on the real vector space ${\mathbb{R}^{p,q}}$ . This paper gives a unified tensor product expression of ${\mathcal{C}\ell_{p,q}}$ by using the center of ${\mathcal{C}\ell_{p,q}}$ . The main result states that for nonnegative integers p, q, ${\mathcal{C}\ell_{p,q} \simeq \otimes^{\kappa-\delta}\mathcal{C}_{1,1} \otimes Cen(\mathcal{C}\ell_{p,q}) \otimes^{\delta} \mathcal{C}\ell_{0,2},}$ where ${p + q \equiv \varepsilon}$ mod 2, ${\kappa = ((p + q) - \varepsilon)/2, p - |q - \varepsilon| \equiv i}$ mod 8 and ${\delta = \lfloor i / 4 \rfloor}$ .     

A Few Comments on the Clifford Algebra {C\ell_2} of the Standard Euclidean Plane

This paper is a continuation of the author’s plenary lecture given at ICCA 9 which was held in Weimar at the Bauhaus University, 15–20 July, 2011. We want to study on both the mathematical and the epistemological levels the thought of the brilliant geometer W. K. Clifford by presenting a few comments on the structure of the Clifford algebra ${C\ell_2}$ associated with the standard Euclidean plane ${\mathbb{R}^2}$ . Miquel’s theorem will be given in the algebraic context of the even Clifford algebra ${C\ell^+_2}$ isomorphic to the real algebra ${\mathbb{C}}$ . The proof of this theorem will be based on the cross ratio (the anharmonic ratio) of four complex numbers. It will lead to a group of homographies of the standard projective line ${\mathbb{C}P^1 = P(\mathbb{C}^2)}$ which appeared so attractive to W. K. Clifford in his overview of a general theory of anharmonics. In conclusion it will be shown how the classical Clifford-Hopf fibration S ¹ → S ³ → S ² leads to the space of spinors ${\mathbb{C}^2}$ of the Euclidean space ${\mathbb{R}^3}$ and to the isomorphism ${{\rm {PU}(1) = \rm {SU}(2)/\{I,-I\} \simeq SO(3)}}$ .     

Regularization and iterative methods for monotone inverse variational inequalities

We consider the monotone inverse variational inequality: find $x\in H$ such that $$\begin{aligned} f(x)\in \Omega , \quad \left\langle \tilde{f}-f(x),x\right\rangle \ge 0, \quad \forall \tilde{f}\in \Omega , \end{aligned}$$ where $\Omega $ is a nonempty closed convex subset of a real Hilbert space $H$ and $f:H\rightarrow H$ is a monotone mapping. A general regularization method for monotone inverse variational inequalities is shown, where the regularizer is a Lipschitz continuous and strongly monotone mapping. Moreover, we also introduce an iterative method as discretization of the regularization method. We prove that both regularized solution and an iterative method converge strongly to a solution of the inverse variational inequality.     

Sufficient conditions for Willmore immersions in \mathbb{R }^3 to be minimal surfaces

We provide two sharp sufficient conditions for immersed Willmore surfaces in $\mathbb{R }^3$ to be already minimal surfaces, i.e. to have vanishing mean curvature on their entire domains. These results turn out to be particularly suitable for applications to Willmore graphs. We can therefore show that Willmore graphs on bounded $C^4$ -domains $\overline{\varOmega }$ with vanishing mean curvature on the boundary $\partial \varOmega $ must already be minimal graphs, which in particular yields some Bernstein-type result for Willmore graphs on $\mathbb{R }^2$ . Our methods also prove the non-existence of Willmore graphs on bounded $C^4$ -domains $\overline{\varOmega }$ with mean curvature $H$ satisfying $H \ge c_0>0 \,{\text{ on }}\, \partial \varOmega $ if $\varOmega $ contains some closed disc of radius $\frac{1}{c_0} \in (0,\infty )$ , and they yield that any closed Willmore surface in $\mathbb{R }^3$ which can be represented as a smooth graph over $\mathbb{S }^2$ has to be a round sphere. Finally, we demonstrate that our results are sharp by means of an examination of some certain part of the Clifford torus in $\mathbb{R }^3$ .     

Parabolicity of maximal surfaces in Lorentzian product spaces

In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form ${M^2 \times \mathbb {R}_1}$ , where M ² is a connected Riemannian surface with non-negative Gaussian curvature and ${M^2 \times \mathbb {R}_1}$ is endowed with the Lorentzian product metric ${{\langle , \rangle}={\langle , \rangle}_M-dt^2}$ . In particular, and as an application of our main result, we deduce that every maximal graph over a starlike domain ${\Omega \subseteq M}$ is parabolic. This allows us to give an alternative proof of the non-parametric version of the Calabi–Bernstein result for entire maximal graphs in ${M^2 \times \mathbb {R}_1}$ .     

An application of maximum principle to space-like hypersurfaces with constant mean curvature in anti-de Sitter space

In this paper, we study complete hypersurfaces with constant mean curvature in anti-de Sitter space ${H^{n+1}_1(-1)}$ . we prove that if a complete space-like hypersurface with constant mean curvature ${x:\mathbf M\rightarrow H^{n+1}_1(-1) }$ has two distinct principal curvatures ??, ??, and inf|?? ? ??|?>?0, then x is the standard embedding ${ H^{m} (-\frac{1}{r^2})\times H^{n-m} ( -\frac{1}{1 - r^2} )}$ in anti-de Sitter space ${ H^{n+1}_1 (-1) }$ .     

Algorithms on Minimizing the Maximum Sensor Movement for Barrier Coverage of a Linear Domain

In this paper, we study the problem of moving $n$ n sensors on a line to form a barrier coverage of a specified segment of the line such that the maximum moving distance of the sensors is minimized. Previously, it was an open question whether this problem on sensors with arbitrary sensing ranges is solvable in polynomial time. We settle this open question positively by giving an $O(n^2\log n)$ O ( n 2 log n ) time algorithm. For the special case when all sensors have the same-size sensing range, the previously best solution takes $O(n^2)$ O ( n 2 ) time. We present an $O(n\log n)$ O ( n log n ) time algorithm for this case; further, if all sensors are initially located on the coverage segment, our algorithm takes $O(n)$ O ( n ) time. Also, we extend our techniques to the cycle version of the problem where the barrier coverage is for a simple cycle and the sensors are allowed to move only along the cycle. For sensors with the same-size sensing range, we solve the cycle version in $O(n)$ O ( n ) time, improving the previously best $O(n^2)$ O ( n 2 ) time solution.