Uniqueness of \mathbb{C}^{ * } - and \mathbb{C}_{{\text{ + }}} -actions on Gizatullin Surfaces

A Gizatullin surface is a normal affine surface V over $ \mathbb{C} $ , which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with $ \mathbb{C}_{{\text{ + }}} $ -actions on V considered up to a “speed change”. Non-Gizatullin surfaces are known to admit at most one $ \mathbb{A}^{1} $ -fibration V → S up to an isomorphism of the base S. Moreover, an effective $ \mathbb{C}^{ * } $ -action on them, if it does exist, is unique up to conjugation and inversion t $ \mapsto $ t ^?1 of $ \mathbb{C}^{ * } $ . Obviously, uniqueness of $ \mathbb{C}^{ * } $ -actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations, see, e.g., [FKZ1]. In the present paper we obtain a criterion as to when $ \mathbb{A}^{{\text{1}}} $ -fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base $ S \cong \mathbb{A}^{{\text{1}}} $ . We exhibit as well large subclasses of Gizatullin $ \mathbb{C}^{ * } $ -surfaces for which a $ \mathbb{C}^{ * } $ -action is essentially unique and for which there are at most two conjugacy classes of $ \mathbb{A}^{{\text{1}}} $ -fibrations over $ \mathbb{A}^{{\text{1}}} $ .     

Hausdorff measures of subgroups of \mathbb{R }/\mathbb{Z } and \mathbb{R }

For a sequence $\underline{u}=(u_n)_{n\in \mathbb{N }}$ of integers, let $t_{\underline{u}}(\mathbb{T })$ be the group of all topologically $\underline{u}$ -torsion elements of the circle group $\mathbb{T }:=\mathbb{R }/\mathbb{Z }$ . We show that for any $s\in ]0,1[$ and $m\in \{0,+\infty \}$ there exists $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has Hausdorff dimension $s$ and $s$ -dimensional Hausdorff measure equal to $m$ (no other values for $m$ are possible). More generally, for dimension functions $f,g$ with $f(t)\prec g(t), f(t)\prec \!\!\!\prec t$ and $g(t)\prec \!\!\!\prec t$ we find $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has at the same time infinite $f$ -measure and null $g$ -measure.     

A zoo of diffeomorphism groups on \mathbb{R }^{n}

We consider the groups ${\mathrm{Diff }}_\mathcal{B }(\mathbb{R }^n)$ , ${\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)$ , and ${\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R }^n)$ of smooth diffeomorphisms on $\mathbb{R }^n$ which differ from the identity by a function which is in either $\mathcal{B }$ (bounded in all derivatives), $H^\infty = \bigcap _{k\ge 0}H^k$ , or $\mathcal{S }$ (rapidly decreasing). We show that all these groups are smooth regular Lie groups.     

Categories of bounded \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) - and \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) -modules

Let $ \mathfrak{g} $ be a reductive Lie algebra over $ \mathbb{C} $ and $ \mathfrak{k} \subset \mathfrak{g} $ be a reductive in $ \mathfrak{g} $ subalgebra. We call a $ \mathfrak{g} $ -module M a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module whenever M is a direct sum of finite-dimensional $ \mathfrak{k} $ -modules. We call a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module M bounded if there exists $ {C_M} \in {\mathbb{Z}_{{ \geqslant 0}}} $ such that for any simple finite-dimensional $ \mathfrak{k} $ -module E the dimension of the E-isotypic component is not greater than C _M dim E. Bounded $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -modules form a subcategory of the category of $ \mathfrak{g} $ -modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ - and $ \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ -modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V .     

The \mathfrak {sl}_{3}-web algebra

In this paper we use Kuperberg’s $\mathfrak {sl}_3$ -webs and Khovanov’s $\mathfrak {sl}_3$ -foams to define a new algebra $K^S$ , which we call the $\mathfrak {sl}_3$ -web algebra. It is the $\mathfrak {sl}_3$ analogue of Khovanov’s arc algebra. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of $q$ -skew Howe duality, which allows us to prove that $K^S$ is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group $K^{\oplus }_0(\mathcal {W}^S)_{\mathbb {Q}(q)}$ , to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that $K^S$ is a graded cellular algebra.     

Constant Angle Property and Canonical Principal Directions for Surfaces in {\mathbb{M}^{2}(c) \times {\mathbb{R}_{1}}}

In this paper, we study surfaces in Lorentzian product spaces ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ . We classify constant angle spacelike and timelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ . Moreover, complete classifications of spacelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ and timelike surfaces in ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ with a canonical principal direction are obtained. Finally, a new characterization of the catenoid of the 3rd kind is established, as the only minimal timelike surface with a canonical principal direction in Minkowski 3–space.     

On the Equivariant Cohomology of Subvarieties of a \mathfrak{B}-Regular Variety

By a $\mathfrak{B}$ -regular variety, we mean a smooth projective variety over $\mathbb{C}$ admitting an algebraic action of the upper triangular Borel subgroup $\mathfrak{B} \subset {\text{SL}}_{2} {\left( \mathbb{C} \right)}$ such that the unipotent radical in $\mathfrak{B}$ has a unique fixed point. A result of Brion and the first author [4] describes the equivariant cohomology algebra (over $\mathbb{C}$ ) of a $\mathfrak{B}$ -regular variety X as the coordinate ring of a remarkable affine curve in $X \times \mathbb{P}^{1}$ . The main result of this paper uses this fact to classify the $\mathfrak{B}$ -invariant subvarieties Y of a $\mathfrak{B}$ -regular variety X for which the restriction map i _Y : H ^*(X) → H ^*(Y) is surjective.     

Minimal generating and normally generating sets for the braid and mapping class groups of \mathbb{D }^{2}, \mathbb S ^{2} and \mathbb{R P^2}

We consider the (pure) braid groups $B_{n}(M)$ and $P_{n}(M)$ , where $M$ is the $2$ -sphere $\mathbb S ^{2}$ or the real projective plane $\mathbb R P^2$ . We determine the minimal cardinality of (normal) generating sets $X$ of these groups, first when there is no restriction on $X$ , and secondly when $X$ consists of elements of finite order. This improves on results of Berrick and Matthey in the case of $\mathbb S ^{2}$ , and extends them in the case of $\mathbb R P^2$ . We begin by recalling the situation for the Artin braid groups ( $M=\mathbb{D }^{2}$ ). As applications of our results, we answer the corresponding questions for the associated mapping class groups, and we show that for $M=\mathbb S ^{2}$ or $\mathbb R P^2$ , the induced action of $B_n(M)$ on $H_3(\widetilde{F_n(M)};\mathbb{Z })$ is trivial, $F_{n}(M)$ being the $n^\mathrm{th}$ configuration space of $M$ .     

On a problem concerning the Banach algebra generated by the maps n\mapsto\lambda^{\binom{n}{k}}

In Jabbari and Namioka (Milan J. Math. 78:503?C522, 2010), the authors characterized the spectrum M(W) of the Weyl algebra W, i.e. the norm closure of the algebra generated by the family of functions $\{n\mapsto x^{n^{k}}; x\in\mathbb{T}, k\in\mathbb{N}\}$ , ( $\mathbb{T}$ the unit circle), with a closed subgroup of $E(\mathbb{T})^{\mathbb{N}}$ where $E(\mathbb{T})$ denotes the family of the endomorphisms of the multiplicative group $\mathbb{T}$ . But the size of M(W) in $E(\mathbb{T})^{\mathbb{N}}$ as well as the induced group operation were left as a problem. In this paper, we will give a solution to this problem.     

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 $\mathbb{R }^3$ .     

On the Hall algebra of semigroup representations over \mathbb F _1

Let $\mathrm{A }$ be a finitely generated semigroup with 0. An $\mathrm{A }$ -module over $\mathbb F _1$ (also called an $\mathrm{A }$ -set), is a pointed set $(M,*)$ together with an action of $\mathrm{A }$ . We define and study the Hall algebra $\mathbb H _{\mathrm{A }}$ of the category $\mathcal C _{\mathrm{A }}$ of finite $\mathrm{A }$ -modules. $\mathbb H _{\mathrm{A }}$ is shown to be the universal enveloping algebra of a Lie algebra $\mathfrak n _{\mathrm{A }}$ , called the Hall Lie algebra of $\mathcal C _{\mathrm{A }}$ . In the case of $\langle t \rangle $ —the free monoid on one generator $\langle t \rangle $ , the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent $\langle t \rangle $ -modules) is isomorphic to Kreimer’s Hopf algebra of rooted forests. This perspective allows us to define two new commutative operations on rooted forests. We also consider the examples when $\mathrm{A }$ is a quotient of $\langle t \rangle $ by a congruence, and the monoid $G \cup \{ 0\}$ for a finite group $G$ .     

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 ${{\rm {PU}(1) = \rm {SU}(2)/\{I,-I\} \simeq SO(3)}}$ .     

Linearity and Second Fundamental Forms for Proper Holomorphic Maps from \mathbb{B}^{n+1} to \mathbb{B}^{4n-3}

For a holomorphic proper map F from the ball $\mathbb{B}^{n+1}$ into $\mathbb{B}^{N+1}$ that is C ³ smooth up to the boundary, the image $M=F(\partial\mathbb{B}^{n})$ is an immersed CR submanifold in the sphere $\partial \mathbb{B}^{N+1}$ on which some second fundamental forms II _M and $\mathit{II}^{CR}_{M}$ can be defined. It is shown that when 4??n+1<N+1??4n?3, F is linear fractional if and only if $\mathit{II}_{M} - \mathit{II}_{M}^{CR} \equiv 0$ .     

Uniqueness of integrable solutions to {\nabla \zeta=G \zeta, \zeta|_\Gamma = 0} for integrable tensor coefficients G and applications to elasticity

Let ${\Omega \subset \mathbb{R}^{N}}$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ${\partial\Omega}$ . We show that the solution to the linear first-order system $$\nabla \zeta = G\zeta, \, \, \zeta|_\Gamma = 0 \quad \quad \quad (1)$$ is unique if ${G \in \textsf{L}^{1}(\Omega; \mathbb{R}^{(N \times N) \times N})}$ and ${\zeta \in \textsf{W}^{1,1}(\Omega; \mathbb{R}^{N})}$ . As a consequence, we prove $$||| \cdot ||| : \textsf{C}_{o}^{\infty}(\Omega, \Gamma; \mathbb{R}^{3}) \rightarrow [0, \infty), \, \, u \mapsto \parallel {\rm sym}(\nabla uP^{-1})\parallel_{\textsf{L}^{2}(\Omega)}$$ to be a norm for ${P \in \textsf{L}^{\infty}(\Omega; \mathbb{R}^{3 \times 3})}$ with Curl ${P \in \textsf{L}^{p}(\Omega; \mathbb{R}^{3 \times 3})}$ , Curl ${P^{-1} \in \textsf{L}^{q}(\Omega; \mathbb{R}^{3 \times 3})}$ for some p, q > 1 with 1/p + 1/q = 1 as well as det ${P \geq c^+ > 0}$ . We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let ${\Phi \in \textsf{H}^{1}(\Omega; \mathbb{R}^{3})}$ satisfy sym ${(\nabla\Phi^\top\nabla\Psi) = 0}$ for some ${\Psi \in \textsf{W}^{1,\infty}(\Omega; \mathbb{R}^{3}) \cap \textsf{H}^{2}(\Omega; \mathbb{R}^{3})}$ with det ${\nabla\Psi \geq c^+ > 0}$ . Then, there exist a constant translation vector ${a \in \mathbb{R}^{3}}$ and a constant skew-symmetric matrix ${A \in \mathfrak{so}(3)}$ , such that ${\Phi = A\Psi + a}$ .     

Geodesics on the Moduli Space of Oriented Circles in \mathbb{S}^{3}

Let ${\mathbb{Q}^3}$ be the moduli space of oriented circles in the three dimensional unit sphere ${\mathbb{S}^3}$ . Given a natural complex structure such space becomes a three dimensional complex manifold, with a M?bius invariant Hermitian metric h of type (2, 1). Up to M?bius transformations, all geodesics with respect to the Lorentz metric g = Re(h) on ${\mathbb{Q}^3}$ are determined to form a one-parameter family of circles on a helicoid in a space form ${\mathbb{R}^3, \mathbb{H}^3}$ or ${\mathbb{S}^{3}}$ , resp. We show also that any two oriented circles in ${\mathbb{S}^3}$ are connected by countably infinitely many geodesics in ${\mathbb{Q}^3}$ .     

Finite groups with \mathbb{P }-subnormal subgroups

A subgroup $H$ of a group $G$ is called $\mathbb{P }$ -subnormal in $G$ whenever either $H=G$ or there is a chain of subgroups $H=H_0\subset H_1\subset \cdots \subset H_n=G$ such that $|H_i:H_{i-1}|$  is a prime for all $i$ . In this paper we study groups with $\mathbb{P }$ -subnormal 2-maximal subgroups, and groups with $\mathbb{P }$ -subnormal primary cyclic subgroups.     

Harmonic mappings and conformal minimal immersions of Riemann surfaces into {\mathbb {R}^{\rm N}}

We prove that for any open Riemann surface ${\mathcal{N}}$ , natural number N ≥ 3, non-constant harmonic map ${h:\mathcal{N} \to \mathbb{R}}$ ^N?2 and holomorphic 2-form ${\mathfrak{H}}$ on ${\mathcal{N}}$ , there exists a weakly complete harmonic map ${X=(X_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}$ with Hopf differential ${\mathfrak{H}}$ and ${(X_j)_{j=3,\ldots,{\sc N}}=h.}$ In particular, there exists a complete conformal minimal immersion ${Y=(Y_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}$ such that ${(Y_j)_{j=3,\ldots,{\sc N}}=h}$ . As some consequences of these results (1) there exist complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss map is non-degenerate and fails to intersect N hyperplanes of ${\mathbb{CP}^{{\sc N}-1}}$ in general position. (2) There exist complete non-proper embedded minimal surfaces in ${\mathbb{R}^{\sc N},}$ ${\forall\,{\sc N} >3 .}$      

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$