Super currents and tropical geometry

We introduce the formalism of positive super currents on ${\mathbb{R}^{n}}$ , in strong analogy with the theory of positive currents in ${\mathbb{C}^{n}}$ . We consider intersection of currents and Lelong numbers, and as an application we show that the formalism can be used to describe tropical varieties. This is similar in spirit to the fact that in complex analysis the current of integration of an analytic variety can be identified with a closed, positive current.     

Boundary Value Problem of the Nonstationary Stokes System in the Half Space

In this paper, we study the boundary value problem of the nonstationary Stokes system in \({\mathbb R}^{n}_+\times (0,T)\) . We generalize the result of (Hofmann and Nystrom Methods Appl. Anal. 9(1), 12–98, 2002; Koch and Solonnikov J. Math. Sci. 106, 3042–3072, 2001; Shen Am. J. Math. 113(2), 293–373, 1991) to a general anisotropic Besov space.     

Topological Aspects of Differential Chains

In this paper we investigate the topological properties of the space of differential chains $\,^{\prime}\mathcal{B}(U)$ defined on an open subset U of a Riemannian manifold M. We show that $\,^{\prime}\mathcal {B}(U)$ is not generally reflexive, identifying a fundamental difference between currents and differential chains. We also give several new brief (though non-constructive) definitions of the space $\,^{\prime}\mathcal{B}(U) $ , and prove that it is a separable ultrabornological (DF)-space. Differential chains are closed under dual versions of the fundamental operators of the Cartan calculus on differential forms (Harrison in Geometric Poincare lemma, Jan 2011, submitted; Operator calculus??the exterior differential complex, Jan 2011, submitted). The space has good properties, some of which are not exhibited by currents $\mathcal{B}'(U)$ or? $\mathcal{D}'(U)$ . For example, chains supported in finitely many points are dense in $\,^{\prime}\mathcal{B}(U)$ for all open U?M, but not generally in the strong dual topology of? $\mathcal{B}'(U)$ .     

Complete bounded null curves immersed in {\mathbb {C}^3} and {\rm {SL}(2,\mathbb {C})}

We construct a simply connected complete bounded mean curvature one surface in the hyperbolic 3-space ${\mathcal {H}^3}$ . Such a surface in ${\mathcal {H}^3}$ can be lifted as a complete bounded null curve in ${\rm {SL}(2,\mathbb {C})}$ . Using a transformation between null curves in ${\mathbb {C}^3}$ and null curves in ${\rm {SL}(2,\mathbb {C})}$ , we are able to produce the first examples of complete bounded null curves in ${\mathbb {C}^3}$ . As an application, we can show the existence of a complete bounded minimal surface in ${\mathbb {R}^3}$ whose conjugate minimal surface is also bounded. Moreover, we can show the existence of a complete bounded immersed complex submanifold in ${\mathbb {C}^2}$ .     

Tunneling for a Class of Difference Operators

We analyze a general class of difference operators ${H_\varepsilon = T_\varepsilon + V_\varepsilon}$ on ${\ell^2((\varepsilon \mathbb {Z})^d)}$ where ${V_\varepsilon}$ is a multi-well potential and ${\varepsilon}$ is a small parameter. We decouple the wells by introducing certain Dirichlet operators on regions containing only one potential well, and we shall treat the eigenvalue problem for ${H_\varepsilon}$ as a small perturbation of these comparison problems. We describe tunneling by a certain interaction matrix, similar to the analysis for the Schr?dinger operator [see Helffer and Sj?strand in Commun Partial Differ Equ 9:337–408, 1984], and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix.     

Ergodicity of laminations with singularities in Kähler surfaces

Let $\mathcal F $ be a holomorphic foliation on $\mathcal M $ , a homogeneous compact Kähler surface, with only hyperbolic singularities. Let $\mathcal L $ be a closed set saturated by leaves of the foliation, containing singularities and with every leaf dense on it. If there are no positive closed currents directed by $\mathcal L $ , then there is a unique positive harmonic current directed by $\mathcal L $ of mass one. This result was previously obtained for $\mathbb CP ^2$ by Fornæss and Sibony and we obtain the result for the rest of homogeneous compact Kähler surfaces.     

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.     

Locally homogeneous rigid geometric structures on surfaces

We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection ${\nabla}$ on a two dimensional torus is complete and, up to a finite cover, homogeneous. Let ${\nabla}$ be a unimodular real analytic affine connection on a real analytic compact connected surface M. If ${\nabla}$ is locally homogeneous on a nontrivial open set in M, we prove that ${\nabla}$ is locally homogeneous on all of M.     

Generalized Rotation Surfaces in {\mathbb E^4}

In the present study we consider generalized rotation surfaces imbedded in an Euclidean space of four dimensions. We also give some special examples of these surfaces in ${\mathbb E^4}$ . Further, the curvature properties of these surfaces are investigated. We give necessary and sufficient conditions for generalized rotation surfaces to become pseudo-umbilical. We also show that every general rotation surface is Chen surface in ${\mathbb E^4}$ . Finally we give some examples of generalized rotation surfaces in ${\mathbb E^4}$ .     

Large collections of curves pairwise intersecting exactly once

Let \(\Omega =(\omega _{j})_{j\in I}\) be a maximum size collection of pairwise non-isotopic simple closed curves on the closed, orientable, genus \(g\) surface \(S_{g}\) , such that \(\omega _{i}\) and \(\omega _{j}\) intersect exactly once for \(i\ne j\) . We show that for \(g\ge 3\) , there exists atleast two such collections up to the action of the mapping class group, answering a question posed by Malestein, Rivin and Theran. As a consequence, we show that the automorphism group of the systole graph for \(S_{g}, g\ge 3\) (whose vertices are isotopy classes of simple closed curves, and whose edges correspond to pairs of curve intersecting once) does not act transitively on maximal complete subgraphs.     

A new family of surfaces of general type with K^2=7 and p_g=0

We construct a new family of smooth minimal surfaces of general type with $K^2=7$ and $p_g=0$ . We show that a surface in this family has ample canonical divisor and birational bicanonical morphism. We also prove that these surfaces satisfy Bloch’s conjecture.     

Evolution of spacelike surfaces in \mathrm{AdS}_3 by their Lagrangian angle

We study spacelike hypersurfaces $M$ in an anti-De Sitter spacetime $N$ of constant sectional curvature $-\kappa , \kappa >0$ that evolve by the Lagrangian angle of their Gauß maps. In the two dimensional case we prove a convergence result to a maximal spacelike surface, if the Gauß curvature $K$ of the initial surface $M\subset N$ and the sectional curvature of $N$ satisfy $|K|<\kappa $ .     

A groupoid generalisation of Leavitt path algebras

Let \(G\) be a locally compact, Hausdorff, étale groupoid whose unit space is totally disconnected. We show that the collection \(A(G)\) of locally-constant, compactly supported complex-valued functions on \(G\) is a dense \(*\) -subalgebra of \(C_c(G)\) and that it is universal for algebraic representations of the collection of compact open bisections of \(G\) . We also show that if \(G\) is the groupoid associated to a row-finite graph or \(k\) -graph with no sources, then \(A(G)\) is isomorphic to the associated Leavitt path algebra or Kumjian–Pask algebra. We prove versions of the Cuntz–Krieger and graded uniqueness theorems for \(A(G)\) .     

Non-Veech surfaces in {\mathcal{H}^{\rm hyp}(4)} are generic

We show that every surface in the component \({\mathcal{H}^{\rm hyp}(4)}\) , that is the moduli space of pairs \({(M,\omega)}\) where M is a genus three hyperelliptic Riemann surface and \({\omega}\) is an Abelian differential having a single zero on M, is either a Veech surface or a generic surface, i.e. its \({{\rm GL}^{+}(2,\mathbb{R})}\) -orbit is either a closed or a dense subset of \({\mathcal{H}^{\rm hyp}(4)}\) . The proof develops new techniques applicable in general to the problem of classifying orbit closures, especially in low genus. Combined with work of Matheus and the second author, a corollary is that there are at most finitely many non-arithmetic Teichmüller curves (closed orbits of surfaces not covering the torus) in \({\mathcal{H}^{\rm hyp}(4)}\) .     

Counting and Sampling Minimum Cuts in Genus g Graphs

Let \(G\) be a directed graph with \(n\) vertices embedded on an orientable surface of genus \(g\) with two designated vertices \(s\) and \(t\) . We show that computing the number of minimum \((s,t)\) -cuts in \(G\) is fixed-parameter tractable in \(g\) . Specifically, we give a \(2^{O(g)} n^2\) time algorithm for this problem. Our algorithm requires counting sets of cycles in a particular integer homology class. That we can count these cycles is an interesting result in itself as there are no prior results that are fixed-parameter tractable and deal directly with integer homology. We also describe an algorithm which, after running our algorithm to count minimum cuts once, can sample an \((s,t)\) -minimum cut uniformly at random in \(O(n \log n)\) time per sample.     

Formal conserved quantities for isothermic surfaces

Isothermic surfaces in \(S^n\) are characterised by the existence of a pencil \(\nabla ^t\) of flat connections. Such a surface is special of type \(d\) if there is a family \(p(t)\) of \(\nabla ^t\) -parallel sections whose dependence on the spectral parameter \(t\) is polynomial of degree \(d\) . We prove that any isothermic surface admits a family of \(\nabla ^t\) -parallel sections which is a formal Laurent series in \(t\) . As an application, we give conformally invariant conditions for an isothermic surface in \(S^3\) to be special.     

h-analogue of Fibonacci numbers

We introduce the \(h\) -analogue of Fibonacci numbers for non-commutative \(h\) -plane. For \(h h'= 1\) and \(h = 0\) , these are just the usual Fibonacci numbers as it should be. We show that the Laplace integral transforms for both the Fibonacci and Chebyshev polynomials are the \(h\) -Fibonacci numbers. We also derive a collection of identities for these numbers.     

The Number of Singular Vector Tuples and Uniqueness of Best Rank-One Approximation of Tensors

In this paper we discuss the notion of singular vector tuples of a complex-valued \(d\) -mode tensor of dimension \(m_1\times \cdots \times m_d\) . We show that a generic tensor has a finite number of singular vector tuples, viewed as points in the corresponding Segre product. We give the formula for the number of singular vector tuples. We show similar results for tensors with partial symmetry. We give analogous results for the homogeneous pencil eigenvalue problem for cubic tensors, i.e., \(m_1=\cdots =m_d\) . We show the uniqueness of best approximations for almost all real tensors in the following cases: rank-one approximation; rank-one approximation for partially symmetric tensors (this approximation is also partially symmetric); rank- \((r_1,\ldots ,r_d)\) approximation for \(d\) -mode tensors.