arXiv subjects
Publications and source records attributed to Joe S. Wang.
We propose a formally completely integrable extension of heat hierarchy based on the space of symmetries isomorphic to the Weyl algebra $\mathcal{A}_1$. The extended heat hierarchy will be the basic model for the analysis of the extension of KP hierarchy, and other integrable equations.
We propose an extension of the structure equation for constant mean curvature (CMC) surfaces in a three dimensional Riemannian space form to the associated CMC hierarchy of evolution equations by the higher-order commuting symmetries. Via the canonical formal Killing field, considered as an infinitely prolonged and loop algebra valued Gauß$\,$ map, the CMC hierarchy is obtained by the assembly of a pair of Adler-Kostant-Symes bi-Hamiltonian hierarchies to the original CMC system. The infinite sequence of higher-order conservation laws of the CMC system admits the corresponding extension, and we find a formula for the generating series of the representative 1-forms. We also introduce a class of generalized (complexified) CMC surfaces as the phase space of the CMC hierarchy.
Continuing the previous work, we propose a further extension of the structure equation for a truncated CMC hierarchy by the non-commuting, truncated Virasoro algebra of non-local symmetries. Via a canonical dressing transformation, we first define a wave function for the CMC hierarchy. This leads to a pair of additional formal Killing fields, and the corresponding spectral Killing field is defined by a purely algebraic formula up to an integrable extension. The extended CMC hierarchy is obtained by packaging these data into the associated affine Kac-Moody algebra valued Killing fields. The log of tau function of the extended CMC hierarchy is defined as the central component of the affine extension of the spectral Killing field. We give a closed formula for the tau function in terms of the determinant of the spectral Killing field.
The differential system for minimal Lagrangian surfaces in a $2_{\mathbb{C}}$-dimensional, non-flat, complex space form is an elliptic system defined on the bundle of oriented Lagrangian planes. This is a 6-symmetric space associated with the Lie group SL(3,$\mathbb{C}$), and the minimal Lagrangian surfaces arise as the primitive maps. Utilizing this property, we derive the differential algebraic inductive formulas for a pair of loop algebra $\mathfrak{sl}(3,\mathbb{C})[[λ]]$-valued canonical formal Killing fields. As a result, we give a complete classification of the (infinite sequence of) Jacobi fields for the minimal Lagrangian system. We also obtain an infinite sequence of higher-order conservation laws from the components of the formal Killing fields.
We propose an extension of the differential system for constant mean curvature (CMC) surfaces in a three dimensional space form to an associated hierarchy of evolution equations by the higher-order commuting symmetries. The infinite sequence of higher-order conservation laws of CMC surfaces admit the corresponding extension to the conservation laws of the entire hierarchy. A class of generalized CMC surfaces are introduced as the phase space of the hierarchy. Via a canonical dressing transformation, we define a wave function for CMC hierarchy. This leads to a pair of additional non-local formal Killing fields and the associated spectral (Virasoro) Killing field.
We propose the Legendrian web in a contact three manifold as a second order generalization of the planar web. An Abelian relation for a Legendrian web is analogously defined as an additive equation among the first integrals of its foliations. For a class of Legendrian $\, d$-webs defined by simple second order ODE's, we give an algebraic construction of $$ ρ_d = \frac{(d-1)(d-2)(2d+3)}{6}$$ linearly independent Abelian relations. We then employ the method of local differential analysis and the theory of linear differential systems to show that $\, ρ_d$ is the maximum rank of a Legendrian $\, d$-web. In the complex analytic category, we give a possible projective geometric interpretation of $ρ_d$ as an analogue of Castelnuovo bound for degree 2d surfaces in the 3-quadric $\mathbb{Q}^3\subset\mathbb{P}^4$ via the duality between $\mathbb{P}^3$ and $\mathbb{Q}^3$ associated with the rank two complex simple Lie group Sp$(2,\mathbb{C})$. The Legendrian 3-webs of maximum rank three are analytically characterized, and their explicit local normal forms are found. For an application, we give an alternative characterization of a Darboux super-integrable metric as a two dimensional Riemannian metric $\,g_+$ which admits a mate metric $\, g_-$ such that a Legendrian 3-web naturally associated with the geodesic foliations of the pair $\, g_{\pm}$ has maximum rank.
The exterior differential system for constant mean curvature (CMC) surfaces in a 3-dimensional space form is an elliptic Monge-Ampere system defined on the unit tangent bundle. We determine the infinite sequence of higher-order symmetries and conservation laws via an enhanced prolongation modelled on a loop algebra valued formal Killing field. As a consequence we establish Noether's theorem for the CMC system and there is a canonical isomorphism between the symmetries and conservation laws. A geometric interpretation of the $\mathbb{S}^{1}$-family of associate surfaces leads to an integrable extension for a non-local symmetry called spectral symmetry. We show that the corresponding spectral conservation law exists as a secondary characteristic cohomology class. For a compact linear finite type CMC surface of arbitrary genus, we observe that the monodromies of the associated flat $\mathfrak{sl}(2,\mathbb{C})$-connection commute with each other. It follows that a single spectral curve is defined as the completion of the set of eigenvalues of the entire monodromies. This agrees with the recent result that a compact high genus linear finite type CMC surface necessarily factors through a branched covering of a torus. We introduce a sequence of Abel-Jacobi maps defined by the periods of conservation laws. For the class of deformations of CMC surfaces which scale the Hopf differential by a real parameter, we compute the first order truncated Picard-Fuchs equation. The resulting formulae for the first few terms exhibit a similarity with the Griffiths transversality theorem for variation of Hodge structures.
This is a preliminary note on a family of minimal surfaces in the 3-sphere defined by a compatible fourth order equation. The minimal surfaces are geometrically characterized either by having a surface of revolution like induced metric, or by having a flat structure 3-web. We observe that the structure equation un-couples for a natural choice of frame. The analysis is reduced to the associated curves in the 2-sphere defined by a rational third order ODE on the curvature.
The Gronwall conjecture states that a planar 3-web of foliations which admits more than one distinct linearizations is locally equivalent to an algebraic web. We propose an analogue of the Gronwall conjecture for the 3-web of foliations by Legendrian curves in a contact three manifold. The Legendrian Gronwall conjecture states that a Legendrian 3-web admits at most one distinct local linearization, with the only exception when it is locally equivalent to the dual linear Legendrian 3-web of the Legendrian twisted cubic in $\,\PP^3$. We give a partial answer to the conjecture in the affirmative for the class of Legendrian 3-webs of maximum rank. We also show that a linear Legendrian 3-web which is sufficiently flat at a reference point is rigid under local linear Legendrian deformation.
We give a local analytic characterization that a minimal surface in the 3-sphere $\, \ES^3 \subset \R^4$ defined by an irreducible cubic polynomial is one of the Lawson's minimal tori. This provides an alternative proof of the result by Perdomo (\emph{Characterization of order 3 algebraic immersed minimal surfaces of $S^3$},Geom. Dedicata 129 (2007), 23--34).
Gronwall conjecture states that a planar 3-web which admits more than one distinct linearization is locally equivalent to an algebraic web. We give a partial answer to the conjecture in the affirmative for the class of planar 3-webs with the web curvature that vanishes to order three at a point. The differential relation on the third order jet of web curvature provides an explicit criterion for unique linearization.
Consider an immersed Legendrian surface in the five dimensional complex projective space equipped with the standard homogeneous contact structure. We introduce a class of fourth order projective Legendrian deformation called \emph{$\,Ψ$-deformation}, and give a differential geometric characterization of surfaces admitting maximum three parameter family of such deformations. Two explicit examples of maximally $\, Ψ$-deformable surfaces are constructed; the first one is given by a Legendrian map from $\, \PP^2$ blown up at three distinct collinear points, which is an embedding away from the -2-curve and degenerates to a point along the -2-curve. The second one is a Legendrian embedding of the degree 6 del Pezzo surface, $\, \PP^2$ blown up at three non-collinear points. In both cases, the Legendrian map is given by a system of cubics through the three points, which is a subsystem of the anti-canonical system.