arXiv 1207.43672012-07math.DGhep-thmath-phmath.APmath.MP

The adiabatic limit of wave map flow on a two torus

J. M. Speight

备注：: 33 pages. Version accepted for publication. Includes updated discussion of the literature on blow-up

英文摘要：

The two-sphere valued wave map flow on a Lorentzian domain R x Sigma, where Sigma is any flat two-torus, is studied. The Cauchy problem with initial data tangent to the moduli space of holomorphic maps Sigma -> S^2 is considered, in the limit of small initial velocity. It is proved that wave maps, in this limit, converge in a precise sense to geodesics in the moduli space of holomorphic maps, with respect to the L^2 metric. This establishes, in a rigorous setting, a long-standing informal conjecture of Ward.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

The spectrum of the Laplace-Beltrami operator on the two-dimensional torus in adiabatic limit

We obtain an asymptotic formula for the eigenvalue distribution function of the Laplace-Beltrami operator on the two-dimensional torus in the adiabatic limit given by a Kronecker foliation. Related problems in number theory are discussed.

2006-12-22math.DGmath.SP

Integer points in domains and adiabatic limits

We prove an asymptotic formula for the number of integer points in a family of bounded domains in the Euclidean space with smooth boundary, which remain unchanged along some linear subspace and stretch out in the directions, orthogonal to this subspace. A more precise estimate for the remainder is obtained in the case when the domains are strictly convex. Using these results, we improved the remainder estimate in the adiabatic limit formula (due to the first author) for the eigenvalue distribution function of the Laplace operator associated with a bundle-like metric on a compact manifold equip…

2011-04-14math.DGmath.NTmath.SP

调和带理论：从2-环面到复射影空间的非零度调和映射的刚性

Harmonic band theory: rigidity of non-zero degree harmonic maps from 2-torus to complex projective space

Osaka Metropolitan University（大阪公立大学）；Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等技术学院）；Advanced Institute for Materials Research (WPI-AIMR), Tohoku University（东北大学物质材料研究高层研究所（WPI-AIMR））

研究非零度调和映射的刚性，通过全纯嵌入与完备线性系统关联的holomorphic嵌入构造，证明其刚性，并扩展到无特殊超振荡点的全纯嵌入，附加对Fubini-Study辛形式拉回的假设。

2026-04-27cond-mat.mes-hallmath-phmath.AG

零亏格超稳定映射的模空间上的环面作用

Torus Actions on Moduli Spaces of Super Stable Maps of Genus Zero

本研究在零亏格固定树型的超J-全纯曲线等三类模空间上构造了光滑$\u2102^*$-作用并保证约化空间环面不变，还给出固定点法丛的显式描述，明确了构造的核心步骤。

2023-06-16math-phmath.AGmath.DG

Conformal maps from a 2-torus to the 4-sphere

We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type operators parameterized over the complexified dual torus. The kernel line bundle of this family over the spectral curve describes the generalized Darboux transforms of the conformally immersed torus. If the spectral curve has finite genus the kernel bundle can be extended to the compa…

2007-12-14math.DG

Biharmonic maps from tori into a 2-sphere

Biharmonic maps are generalizations of harmonic maps. A well-known result of Eells and Wood on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere (whatever the metrics chosen) in the homotopy class of maps of Brower degree $\pm 1$. It would be interesting to know if there exists any biharmonic map in that homotopy class of maps. In this paper, we obtain some classifications on biharmonic maps from a torus into a sphere, where the torus is provided with a flat or a class of non-flat metrics whilst the sphere is provided with the standard metric. Ou…

2014-06-18math.DG

更多相关论文收起更多论文

Torus invariant transverse Kähler foliations

In this paper, we show the convexity of the image of a moment map on a transverse symplectic manifold equipped with a torus action under a certain condition. We also study properties of moment maps in the case of transverse Kähler manifolds. As an application, we give a positive answer to the conjecture posed by Cupit-Foutou and Zaffran.

2015-09-14math.CVmath.DGmath.SG

A convexity theorem for torus actions on contact manifolds

We show that the cone associated with a moment map for an action of a torus on a contact compact connected manifold is a convex polyhedral cone and that the moment map has connected fibers provided the dimension of the torus is bigger than 2 and that no orbit is tangent to the contact distribution. This may be considered as a version of the Atiyah - Guillemin - Sternberg convexity theorem for torus actions on symplectic cones and as a direct generalization of the convexity theorem of Banyaga and Molino for completely integrable torus actions on contact manifolds.

2001-03-06math.DGmath.SG

Convexity for Hamiltonian torus actions on $b$-symplectic manifolds

In [GMPS] we proved that the moment map image of a $b$-symplectic toric manifold is a convex $b$-polytope. In this paper we obtain convexity results for the more general case of non-toric hamiltonian torus actions on $b$-symplectic manifolds. The modular weights of the action on the connected components of the exceptional hypersurface play a fundamental role: either they are all zero and the moment map behaves as in classic symplectic one, or they are all nonzero and the moment map behaves as in the toric $b$-symplectic case studied in [GMPS].

2016-05-11math.DGmath.SG

Stability of the positive mass theorem and torus rigidity theorems under integral curvature bounds

Work of D. Stern and Bray-Kazaras-Khuri-Stern provide differential-geometric identities which relate the scalar curvature of Riemannian 3-manifolds to global invariants in terms of harmonic functions. These quantitative formulas are useful for stability results and show promise for more applications of this type. In this paper, we analyze harmonic maps to flat model spaces in order to address conjectures concerning the geometric stability of the positive mass theorem and the Geroch conjecture. By imposing integral Ricci curvature and isoperimetric bounds, we leverage the previously mentioned f…

2022-10-09math.DG

A connecting theorem for geodesic flows on the 2-torus

We use a result of J. Mather on the existence of connecting orbits for compositions of monotone twist maps of the cylinder to prove the existence of connecting geodesics on the unit tangent bundle $ST^2$ of the 2-torus in regions without invariant tori.

2021-02-05math.DGmath.DS

超凯勒四维平坦环面上辛梯度流的局部存在性

Local Existence Of The Symplectic Gradient Flow On The Hyperkähler Four-dimensional Flat Torus

通过引入矩映射并利用DeTurck技巧将辛梯度流转化为严格抛物型流，证明了实四维环面辛同胚群的局部可缩性，方法基于Yann Rollin的思想。

2025-04-23math.DGmath.SG

The Albanese Map of sGG Manifolds and Self-Duality of the Iwasawa Manifold

We prove that the three-dimensional Iwasawa manifold $X$, viewed as a locally holomorphically trivial fibration by elliptic curves over its two-dimensional Albanese torus, is self-dual in the sense that the base torus identifies canonically with its dual torus under a sesquilinear duality, the Jacobian torus of $X$, while the fibre identifies with itself. To this end, we derive elements of Hodge theory for arbitrary sGG manifolds, introduced in earlier joint work of the author with L. Ugarte, to construct in an explicit way the Albanese torus and map of any sGG manifold. These definitions coin…

2017-06-29math.AGmath.CVmath.DG

Centered complexity one Hamiltonian torus actions

We consider symplectic manifolds with Hamiltonian torus actions which are "almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are "centered" and the moment map is proper. In particular, this classifies the moment map preimages of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmanians Gr^+(2,R^5) and Gr^+(2,R^6) by two equal symplectic balls.

2000-02-23math.AGmath.DGmath.SG