Mostrando recursos 1 - 20 de 115

  1. From spatially periodic instantons to singular monopoles

    Charbonneau, Benoit
    The main result is a computation of the Nahm transform of a $\SUt$-instanton over $\R\times T^3$, called spatially-periodic instanton. It is a singular monopole over $T^3$, a solution to the Bogomolny equation, whose rank is computed and behavior at the singular points is described.

  2. Mass endomorphism and spinorial Yamabe type problems on conformally flat manifolds

    Ammann, Bernd; Humbert, Emmanuel; Morel, Bertrand
    1Let $M$ be a compact manifold equipped with a Riemannian metric $g$ and a spin structure $\si$. We let $\lamin (M,[g],\si)= \inf_{\tilde{g} \in [g] } \lambda_1^+(\tilde{g}) Vol(M,\tilde{g})^{1/n} $ where $\lambda_1^+(\tilde{g})$ is the smallest positive eigenvalue of the Dirac operator $D$ in the metric $\tilde{g}$. A previous result stated that $\lamin(M,[g],\si) \leq \lamin(\mS^n) =\frac{n}{2} \om_n^{{1/n}}$ where $\om_n$ stands for the volume of the standard $n$-sphere. %The inequality is strict when $n \geq 7$ and $(M,g)$ is not conformally flat. In this paper, we study this problem for conformally flat manifolds of dimension $n \geq 2$ such that $D$ is invertible. E.g., we show that strict inequality holds in dimension $n\equiv 0,1,2\mod 4$ if...

  3. Stability conditions and the braid group

    Thomas, R. P.
    We find stability conditions [6,3] on some derived categories of differential graded modules over a graded algebra studied in [12,10]. This category arises in both derived Fukaya categories and derived categories of coherent sheaves. This gives the first examples of stability conditions on the A-model side of mirror symmetry, where the triangulated category is not naturally the derived category of an abelian category. The existence of stability conditions, however, gives many such abelian categories, as predicted by mirror symmetry. ¶ In our examples in 2 dimensions, we completely describe a connected component of the space of stability conditions. It is the universal cover of the configuration space $C_{k+1}^0$ of $k+1$ points in...

  4. Counting Curves in Elliptic Surfaces by Symplectic Methods

    Lee, Junho
    We explicitly compute family GW invariants of elliptic surfaces for primitive classes. That involves establishing a TRR formula and a symplectic sum formula for elliptic surfaces and then determining the GW invariants using an argument from [9]. In particular, as in [2], these calculations also confirm the well-known Yau--Zaslow Conjecture [22] for primitive classes in $K3$ surfaces.

  5. Rigidity and Non-rigidity Results on the Sphere

    Hang, Fengbo; Wang, Xiaodong

  6. Semistable Degeneration of Toric Varieties and Their Hypersurfaces

    Hu, Shengda
    We provide a construction of examples of semistable degeneration via toric geometry. The applications include a higher dimensional generalization of classical degeneration of $K3$ surface into 4 rational components, an algebraic geometric version of decomposing $K3$ as the fiber sum of two $E(1)$'s as well as it's higher dimensional generalizations and many other new examples.

  7. From Sparks to Grundles --- Differential Characters

    Harvey, Reese; Lawson, Blaine
    We introduce a new homological machine for the study of secondary geometric invariants. The objects, called spark complexes, occur in many areas of mathematics. The theory is applied here to establish the equivalence of a large family of spark complexes which appear naturally in geometry, topology and physics. These complexes are quite different. Some of them are purely analytic, some are simplicial, some are of ?ech-type, and many are mixtures. However, the associated theories of secondary invariants are all shown to be canonically isomorphic. Numerous applications and examples are explored.

  8. Degeneration of Kähler-Einstein Manifolds II: the Toroidal case

    Ruan, Wei--Dong
    In this paper, we prove that the Kähler-Einstein metrics for a toroidal canonical degeneration family of Kähler manifolds with ample canonical bundles Gromov--Hausdorff converge to the complete Kähler-Einstein metric on the smooth part of the central fiber when the base locus of the degeneration family is empty. We also prove the incompleteness of the Weil--Peterson metric in this case.

  9. Growth of solutions to the minimal surface equation over domains in a half plane

    Weitsman, Allen
    We consider minimal graphs $\ u=u(x,y)>0$ over unbounded domains D with $\ u=0$ on $\partial D$. We shall study the rates at which $u$ can grow when $D$ is contained in a half plane.

  10. Rigidity of differentiable structure for new class of line arrangements

    Wang, Shaobo; Yau, Stephen S.-T.
    An arrangement of hyperplanes is a finite collection of {\bf C}-linear subspaces of codimension one in a complex vector space ${\bf C}^l$. For such an arrangement ${\cal A}$, there is a natural projective arrangement ${\cal A}^*$ of hyperplanes in ${\bf CP}^{l-1}$ associated to it. Let $M({\cal A})={\bf C}^l- \bigcup_{H \in {\cal A}} H$ and $M({\cal A}^*)= {\bf CP}^{l-1}- \bigcup_{H^* \in {\cal A}^*}H^*$. One of central topics in the theory of arrangements is to find connections between the topology or differentiable structure of $M({\cal A})$ (or $M({\cal A}^*)$) and the combinatorial geometry of ${\cal A}$. A partial solution to this problem was...

  11. Diameter control under Ricci flow

    Topping, Peter
    We estimate the diameter of a closed manifold evolving under Ricci flow in terms of a scalar curvature integral. The proof uses a new maximal function and extends some of Perelman's recent ideas.

  12. Constructing associative 3-folds by evolution equations

    Lotay, Jason

  13. Subsets of Grassmannians preserved by mean curvature flows

    Wang, Mu-Tao
    \Let $M=\Sigma_1\times \Sigma_2$ be the product of two compact Riemannian manifolds of dimension $n\geq 2 $ and two, respectively. Let $\Sigma$ be the graph of a smooth map $f:\Sigma_1\rightarrow \Sigma_2$; $\Sigma$ is an $n$-dimensional submanifold of $M$. Let ${\frak G}$ be the Grassmannian bundle over $M$ whose fiber at each point is the set of all $n$-dimensional subspaces of the tangent space of $M$. The Gauss map $\gamma:\Sigma\rightarrow \frak{G} $ assigns to each point $x\in \Sigma$ the tangent space of $\Sigma$ at $x$. This article considers the mean curvature flow of $\Sigma$ in $M$. When $\Sigma_1$ and $\Sigma_2$ are of the same non-negative curvature, we show a sub-bundle $\frak{S}$ of the Grassmannian...

  14. Constant mean curvature foliations of simplicial flat spacetimes

    Andersson, Lars
    Benedetti and Guadagnini [5] have conjectured that the constant mean curvature foliation $M_\tau$ in a $2+1$ dimensional flat spacetime $V$ with compact hyperbolic Cauchy surfaces satisfies $\lim_{\tau \to -\infty} \ell_{M_\tau} = s_{\Tree}$, where $\ell_{M_\tau}$ and $s_{\Tree}$ denote the marked length spectrum of $M_\tau$ and the marked measure spectrum of the $\Re$-tree $\Tree$, dual to the measured foliation corresponding to the translational part of the holonomy of $V$, respectively. We prove that this is the case for $n+1$ dimensional, $n \geq 2$, simplicial flat spacetimes with compact hyperbolic Cauchy surface. A simplicial spacetime is obtained from the Lorentz cone over a hyperbolic manifold by deformations corresponding to a simple measured foliation.

  15. Small knots and large handle additions

    Qiu, Ruifeng; Wang, Shicheng
    We construct a hyperbolic 3-manifold $M$ (with $\partial M$ totally geodesic) which contains no essential closed surfaces, but for any even integer $g> 0$, there are infinitely many separating slopes $r$ on $\partial M$ so that $M[r]$, the 3-manifold obtained by attaching 2-handle to $M$ along $r$, contains an essential separating closed surface of genus $g$ and is still hyperbolic. The result contrasts sharply with those known finiteness results for the cases $g=0,1$. Our 3-manifold $M$ is the complement of a simple small knot in a handlebody.

  16. On the weak limits of smooth maps for the Dirichlet energy between manifold

    Hang, Fengbo
    We identify all the weak sequential limits of smooth maps in $W^{1,2}\left( M,N\right) $. In particular, this implies a necessary and sufficient topological condition for smooth maps to be weakly sequentially dense in $W^{1,2}\left( M,N\right) $.

  17. Surface bundles versus Heegaard splittings

    Bachman, David; Schleimer, Saul
    This paper studies Heegaard splittings of surface bundles via the curve complex of the fibre. The translation distance of the monodromy is the smallest distance it moves any vertex of the curve complex. We prove that the translation distance is bounded above in terms of the genus of any strongly irreducible Heegaard splitting. As a consequence, if a splitting surface has small genus compared to the translation distance of the monodromy, then the splitting is standard.

  18. Schouten tensor and some topological properties

    Guan, Pengfei; Lin, Chang-Shou; Wang, Guofang
    In this paper, we prove a cohomology vanishing theorem on locally conformally flat manifold under certain positivity assumption on the Schouten tensor. And we show that this type of positivity of curvature is preserved under $0$-surgeries for general Riemannian manifolds, and construct a large class of such manifolds.

  19. Phase Space for the Einstein Equations

    Bartnik, Robert
    A Hilbert manifold structure is described for the phase space $\cF$ of asymptotically flat initial data for the Einstein equations. The space of solutions of the constraint equations forms a Hilbert submanifold $\cC \subset \cF$. The ADM energy-momentum defines a function which is smooth on this submanifold, but which is not defined in general on all of $\cF$. The ADM Hamiltonian defines a smooth function on $\cF$ which generates the Einstein evolution equations only if the lapse-shift satisfies rapid decay conditions. However a regularised Hamiltonian can be defined on $\cF$ which agrees with the Regge-Teitelboim Hamiltonian on $\cC$ and generates the evolution for any lapse-shift appropriately asymptotic to a (time)...

  20. General finite type {IFS} and {$M$}-matrix

    Jin, Ning; Yau, Stephen S. T.
    In [14], Ngai and Wang introduced the concept of finite type IFS to study the Hausdorff dimension of self-similar sets without open set condition. In this paper, by applying the M-matrix theory([15]), we generalize the notion of finite type IFS to the general finite type IFS. A family of IFS with $3$ parameters, but without open set condition is presented. The Hausdorff dimension of the associated attractors can be calculated by both the $M$-matrix method and the general finite type IFS method. But these IFS are not finite type except for those parameters lying in a set of measure zero.

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