RESEARCH STATEMENT YUHAO HU

Transcription

1 RESEARCH STATEMENT YUHAO HU I am interested in researching in differential geometry and the geometry of partial differential equations. Earlier in my PhD studies, I learned the theory of exterior differential systems (EDS) and Cartan s method of equivalence from my advisor, Professor Robert Bryant. Later I applied these tools to study the Bäcklund problem, which I will now describe. One of the earliest and simplest findings that motivated the posing of the Bäcklund problem in the late 19th century is the first-order PDE system { zx z (1) x = λ sin(z + z), z y + z y = 1 λ sin(z z), where λ is a nonzero constant. This system is interesting partly because it enables an infinitude of solutions of the sine-gordon equation (2) u xy = 1 2 sin(2u) to be generated from a given one by solving ODE systems alone. In fact, in order for the system (1) to be compatible, both z and z must satisfy the sine-gordon equation. In addition, if z is a solution of the sine-gordon equation, then the system (1) reduces to a compatible first-order system for z; hence it can be solved using ODE techniques. Geometrically, the sine-gordon equation arises in the study of surfaces of negative constant Gauss curvature in the Euclidean space E 3. Indeed, the system (1) is only a variant of the classical Bäcklund theorem concerning such surfaces, also known for more than a century. The Bäcklund theorem says that, given a surface of negative constant Gauss curvature in E 3, one could deduce an infinitude of surfaces of the same Gauss curvature, which relate to the given one by a pseudo-spherical line congruence (see [1] or [4]), and that such line congruences can be found by solving ODE systems alone. What is important about these findings is that they suggest a new way of looking at PDEs. That is, instead of studying equivalence of PDEs under contact transformations, which may be too strict a notion, we could consider how their solutions relate. In particular, the examples described above represent a particular kind of such relations. In this case, a relation is a PDE system E of two unknown functions, say u and v. The compatibility of E gives rise to a PDE system E 1 (resp. E 2 ) in u (resp. v). Specifying a solution of E 1, the system E reduces to a system solvable using ODE techniques, producing solutions of E 2, and vice versa. A relation E satisfying these properties is traditionally known as

2 2 YUHAO HU a Bäcklund transformation, bearing the name of one of its earliest discoverers, though a Bäcklund relation would seem a more appropriate term. Roughly speaking, the Bäcklund problem asks for a classification of Bäcklund transformations. For more than a century since its discovery, and with Bianchi, Bäcklund, Darboux, Goursat and Cartan contributing to its early studies, the concept of a Bäcklund transformation has drawn interest from mathematicians and physicists alike. During this time, numerous examples of Bäcklund transformations have been discovered (see [8]), including those relating the KdV equation and those relating the Tzitzeica equation (equation arising from the study of affine spheres). Connections of Bäcklund transformations with integrable systems and soliton theory have also been found. However, examples being isolated, a classification of Bäcklund transformations remains largely unknown. For instance, we do not have an effective test that tells whether two PDEs in the plane are Bäcklund-related. The generality of Bäcklund transformations, even in the most classical case, also remains unknown. In my research, I focus on Bäcklund transformations of a particular kind, namely, those relating hyperbolic Monge-Ampère systems in the sense of [1]. Instead of treating Bäcklund transformations as PDE systems and studying them algebraically, I set them up as exterior differential systems on differentiable manifolds and study their structures through geometric invariants, to which the theory of EDS and the method of equivalence are particularly suitable. In this setting, we could associate notions of rank and homogeneity to a Bäcklund transformation. When the rank is one, I have found an upper bound for the number of initial conditions needed to determine a certain generic type of Bäcklund transformations (Theorem 1). Moreover, when the rank is one and when the two Monge-Ampère systems are both Euler-Lagrange, I have found simple obstructions to the existence of a Bäcklund transformation as well as certain new views on classical results (e.g., Proposition 1). When the rank is two, I have partially classified the homogeneous Bäcklund transformations (Theorems 2,3). Extension and refinement of these results remain my short-term goal. My long-term goal is to develop a method that effectively uncovers the structure of Bäcklund transformations, thus making progress towards solving the general Bäcklund problem. In the following paragraphs, I will explain my work and my future plan in further detail. 1. The Geometry of Bäcklund Transformations An accepted geometric formulation of a Bäcklund transformation uses the concept of an exterior differential system (EDS). An EDS is a pair (M, I), where M is a differentiable manifold, and I Ω M an ideal closed under exterior differentiation. A submanifold f : S M is called an integral manifold of (M, I) if any differential form in I pulls back

3 RESEARCH STATEMENT 3 to vanish on S. In familiar terms, an EDS corresponds to a system of partial differential equations, and an integral manifold corresponds to a solution of that system. For example, the sine- Gordon equation (2) can be established as an EDS on the 1-jet space J 1 (R 2, R) = R 5 with coordinates (x, y, u, p, q), where p, q are variables representing the x and y partials of u, respectively. The associated differential ideal I is generated by the differential forms θ = du pdx qdy, Θ 1 = (dp 1 2 sin(2u)dy) dx, Θ 2 = (dq 1 2 sin(2u)dx) dy. It is easy to see that, on any integral surface where dx, dy are independent forms, the vanishing of θ, Θ 1 and Θ 2 implies that u = u(x, y) satisfies the equation (2). I would also like to inform the reader that this exterior differential system generated by θ, Θ 1, Θ 2 is an example of a hyperbolic Monge-Ampère system, which, generally speaking, is a differential system on a 5-manifold, its differential ideal being generated by a contact 1-form, its exterior differential and another two form, such that each integral surface is foliated by two distinct families of characteristic curves. In terms of these, and up to certain transversality conditions, a Bäcklund transformation relating (M, I) and ( M, Ī) is formulated as a double fibration π (N, B) π (M, I) ( M, Ī) such that the preimage of any integral manifold of either (M, I) or ( M, Ī) is foliated by integral manifolds of (N, B), and such that any integral manifold of (N, B) projects via π (resp. π) to be an integral manifold of (M, I) (resp. ( M, Ī)). Often, one would, for simplicity, refer to (N, B) as the Bäcklund transformation. When M and M have the same dimension, I define r = dim N dim M to be the rank of the Bäcklund transformation. A reason for making this definition and studying Bäcklund transformations of various ranks is that while being Bäcklund-related at a particular rank may not, a priori, be transitive in the sense of a relation, being Bäcklund-related is. Another reason is that interesting Bäcklund transformations also arise in higher ranks, such as the Bäcklund transformation relating the Tzizeica equation to itself (see [8]), which is of rank-2. The classical transformation that links minimal surfaces in the standard 3-sphere S 3 with surfaces in E 3 with mean curvature 1 can be interpreted as a rank-5 Bäcklund

4 4 YUHAO HU transformation. Composition of Bäcklund transformations also yields a Bäcklund transformation of higher rank. In general, the subgroup of Diff(N) that leaves B invariant is called the symmetry group of the Bäcklund transformation. If the symmetry group acts transitively, we say that (N, B) is homogeneous. In [6], Clelland, using the method of equivalence, classified all the rank-1 homogeneous Bäcklund transformations relating hyperbolic Monge-Ampère systems, finding fifteen cases. In [5], Clelland and Ivey, classified all the rank-1 Bäcklund transformations that relate a certain hyperbolic Monge-Ampère system to the system representing the wave equation z xy = 0. Their classification coincides with the well-known Goursat-Vessiot list of Darboux integrable systems. These are some results that inspired my research. My research is mainly in two parts. The first part is the study of general rank-1 Bäcklund transformations relating hyperbolic Monge-Ampère systems; the second, the study of homogeneous rank-2 Bäcklund transformations relating hyperbolic Monge-Ampère systems. These will be briefly refered to as rank-1 general and rank-2 homogeneous Rank-1 General. In this part, my research objective is to answer the following questions. I. Thinking abstractly of a 6-manifold with a Bäcklund structure, only assuming that it is a rank-1 Bäcklund transformation relating two hyperbolic Monge-Ampère systems, how many such structures are there up to diffeomorphism? II. Which pairs of hyperbolic Monge-Ampère systems are related by a rank-1 Bäcklund transformation? In the context of the first question, a Bäcklund transformation (N, B) is a 6-manifold with a differential ideal B. In particular, B is algebraically generated as B = {θ, η, ω 1 ω 2, ω 3 ω 4 }, where θ, η and the ω s comprise a local coframe on N. Not every manifold admitting such a coframe can be regarded as a Bäcklund transformation. In fact, as long as {dθ, dη} has rank two modulo θ and η, it is shown in [6] that we could always adjust these 1-forms such that B is still generated in the form as above and such that the following structure equations hold: dθ = A 1 ω 1 ω 2 + ω 3 ω 4, mod θ, dη = ω 1 ω 2 + A 2 ω 3 ω 4, mod η, where A 1 A 2 1. Once this is the case, in order for B to represent a Bäcklund transformation, we need the additional condition that each characteristic system (such as {θ, ω 1, ω 2 }) is well-defined on a corresponding 5-dimensional quotient of N. This puts further restrictions on the structure equations.

5 RESEARCH STATEMENT 5 To answer the first question, I follow Clelland [6] to establish the structure equations that represent a Bäcklund transformation. Without assuming the structure invariants to be constants, I no longer benefit from the simplifications that are found in [6]. I thus take a natural approach of applying Cartan s third theorem [2], as this theorem in general provides a criterion to determine the generality of geometric structures of certain types. Algorithmically, I start with the Bäcklund structure equations, taking step-by-step exterior differentiations, using the identity d 2 = 0 to uncover relations between the primary structure invariants, their derivatives, the derivatives of their derivatives, and so on. This criterion succeeds when a certain numerical inequality (Cartan s inequality) becomes an equality. In such a case, I will be able to answer the generality question in the form Generic rank-1 Bäcklund transformations relating hyperbolic Monge-Ampère systems depend on s k functions of k variables, where s k is the last nonzero Cartan character associated to the problem. So far, I have yet to attain such an equality, but have obtained partial information from my calculations. In particular, I have proven Theorem 1. In a generic case, a Bäcklund transformation relating hyperbolic Monge-Ampère systems can be uniquely determined by specifying initial data consisting of at most 6 functions of 3 variables. The second question has a somewhat different nature comparing to the first, so I have taken a different approach. For a pair of fixed hyperbolic Monge-Ampère systems (M, I) and ( M, Ī), a Bäcklund transformation relating them corresponds to an integral manifold of a rank-4 Pfaffian system on M M R 16. It is thus well known that, theoretically at least, the existence problem of a Bäcklund transformation relating these Monge-Ampère systems can be solved using the Cartan-Kähler theory [5]. An improvement that I have made is developing this idea so that it could also be applied to the case when the Monge- Ampère systems are arbitrary. To carry out this idea, I establish a Pfaffian system on the product of two Monge- Ampère structure bundles (c.f. [1], p.56) and a space of unknown parameters. Then I take advantage of the freedom in the Monge-Ampère structure equations to further simplify the Pfaffian system. In the language developed in [3], this Pfaffian system is integrable only if its intrinsic torsion vanishes. Based on this, I have found various obstructions, in terms of the Monge-Ampère invariants, for two hyperbolic Monge-Ampère systems to be related by a rank-1 Bäcklund transformation of a certain type. For example, when a hyperbolic Monge-Ampère system is Euler-Lagrange, its first Monge-Ampère relative invariants reduce to a 2-by-2 matrix, say S. In this case, I have obtained the following Proposition 1. If two Euler-Lagrange hyperbolic Monge-Ampère systems, (M, I) (with contact form θ) and ( M, Ī) (with contact form η), are related by a rank-1 Bäcklund transformation, on which dθ dθ and dη dη induce the same orientation on the 4-plane field θ, η, then their first Monge- Ampère relative invariants, S 1 and S 1, must be either both degenerate or both non-degenerate.

6 6 YUHAO HU A proposition like this may not represent full potential of the idea described above. In fact, one could put certain assumptions on the Monge-Ampère invariants or the form of Bäcklund transformations to simplify the torsion-vanishing conditions, thus to obtain interesting results. For example, I have found several assumptions that would necessarily lead to only a finite number of Monge-Ampère pairs that are related by Bäcklund transformations satisfying prescribed conditions. On the other hand, before one actually carries out the calculation, making these assumptions may still allow other pairs of Monge-Ampère systems to be Bäcklund-related. This tells us that the existence of Bäcklund-related pairs near a pair of Monge-Ampère systems that are known to be Bäcklund-related can be quite restricted. I believe that a more thorough analysis of this kind would lead to a rather detailed understanding of the local shape of the space of Bäcklund-related Monge-Ampère pairs. Also, it would be particularly interesting to know which Bäcklund-related pairs are isolated and which are not. These are parts of my on-going work Rank-2 Homogeneous. In this case, a Bäcklund manifold N is 7-dimensional; the Bäcklund ideal is algebraically generated as (3) B = {ω 0, ω 0, γ, ω 1 ω 2, ω 3 ω 4 }, where, similar to the rank-1 case, the 1-forms ω 0, ω 0, γ and the ω s comprise a local coframe on N. I have divided the problem into the following two subcases. If {dω 0, d ω 0 } has rank two modulo ω 0, ω 0, γ, I have shown that it is possible to choose coframes such that (3) still holds, and such that the following structure equations are satisfied. (4) (5) (6) dω 0 Aω 1 ω 2 + ω 3 ω 4 + (B 3 ω 3 + B 4 ω 4 ) γ mod ω 0, d ω 0 ω 1 ω 2 + ɛaω 3 ω 4 + (B 1 ω 1 + B 2 ω 2 ) γ mod ω 0, 4 4 dγ C 0 ω 0 ω 0 + C i ω i ω 0 + D i ω i ω 0 mod γ, i=1 where ɛ = ±1, A > 0 and ɛa 2 1. If {dω 0, d ω 0 } has rank one modulo ω 0, ω 0, γ, I have shown that, it is possible to choose coframes such that (3) still holds, and such that the following structure equations are satisfied. (7) (8) (9) dω 0 ω 1 ω 2 + ω 3 ω 4 + (B 3 ω 3 + B 4 ω 4 ) γ mod ω 0, d ω 0 ω 1 ω 2 + ω 3 ω 4 + (B 1 ω 1 + B 2 ω 2 ) γ mod ω 0, i=1 dγ C(ω 1 ω 2 ω 3 ω 4 ) mod ω 0, ω 0, γ. I point out that, as in the rank-1 case, the structure equations from (4) to (9) are only results of the assumption that B can be written in the form (3). In order for B to represent a Bäcklund transformation, more condition must be imposed on the structure equations

7 RESEARCH STATEMENT 7 of dω 1,..., dω 4. Once this is done, I proceed using the method of equivalence [7]. So far, I have obtained the following results. Theorem 2. Consider rank-two homogeneous Bäcklund transformations relating hyperbolic Monge-Ampère systems admitting the structure equations of the form (4) (6). i. There exist no such Bäcklund transformations with C 0, (B 1, B 2 ), (B 3, B 4 ) all being nonzero. ii. If ɛ = 1, C 0 being identically zero and (B 1, B 2 ), (B 3, B 4 ) both being nonzero, the Bäcklund transformation is either foliated by rank-one Bäcklund transformations, or is equivalent to an auto-bäcklund transformation of the linear equation z xy = z. Theorem 3. Consider rank-two homogeneous Bäcklund transformations relating hyperbolic Monge-Ampère systems admitting the structure equations (7) (9). If (B 1, B 2 ) and (B 3, B 4 ) are both nonzero and C is nonzero, then the Bäcklund transformation is equivalent to an auto-bäcklund transformation of a linear equation of the form z xy = λz, where λ is a constant. I remark that, under the assumption of Theorem 3, the case when C 0, (B 1, B 2 ), (B 3, B 4 ) are all nonzero may be regarded as generic. The theorem tells us that, surprisingly, no homogeneous examples can be found in this case. Of course, I am interested in knowing whether there exists any Bäcklund transformation in this case at all without making the homogeneity assumption. What is also interesting is that, in the cases for which I have drawn conclusions, the only new examples are Bäcklund transformations relating linear equations; and those that relate nonlinear equations all seem hidden in the statement that the Bäcklund manifold is foliated by rank-1 Bäcklund transformations. To answer whether this is true for all homogeneous rank-2 Bäcklund transformations depends on a complete classification in this case. This is also part of my on-going work. 2. Future Plan In the short-term, my main focus will remain solving problems related to Bäcklund transformations. Among these are: Considering the space of Bäcklund-related Monge-Ampère pairs, what is its shape near the pairs that are classically known? If a hyperbolic Monge-Ampère system can be put in the form z xy = F (x, y, z, z x, z y ), is it necessary that its Bäcklund pair, if any, can be put in the same form? Find an explicit Monge-Ampère system that does not have a rank-1 Bäcklund pair.

8 8 YUHAO HU Classify Bäcklund transformations relating hyperbolic Monge-Ampère systems that are of cohomogeneity-1. More ambitious problems, to name a few, are: What is the (exact) generality of rank-one Bäcklund transformations relating hyperbolic Monge-Ampère systems? Which Monge-Ampère systems are rank-2 but not rank-1 Bäcklund-related? Classify homogenerous Bäcklund transformations relating hyperbolic Monge-Ampère systems of ranks greater than 2. Study Bäcklund transformations relating systems of other types, such as those relating elliptic/parabolic Monge-Ampère systems. References [1] Robert Bryant, Phillip Griffiths, and Daniel Grossman, Exterior differential systems and Euler-Lagrange partial differential equations, University of Chicago Press, [2] Robert L Bryant, Notes on exterior differential systems, arxiv preprint arxiv: (2014). [3] Robert L Bryant, Shiing-Shen Chern, Robert B Gardner, Hubert L Goldschmidt, and Phillip A Griffiths, Exterior differential systems, Vol. 18, Springer Science & Business Media, [4] Shiing-Shen Chern and Chuu-Lian Terng, An analogue of Bäcklund s theorem in affine geometry, Rocky Mountain J. Math. 10 (1980), no. 1. [5] Jeanne N Clelland, Thomas A Ivey, et al., Bäcklund transformations and Darboux integrability for nonlinear wave equations, Asian Journal of Mathematics 13 (2009), no. 1, [6] Jeanne Nielsen Clelland, Homogeneous Bäcklund transformations of hyperbolic Monge-ampère systems, arxiv preprint math (2001). [7] Robert B Gardner, The method of equivalence and its applications, SIAM, [8] Colin Rogers and Wolfgang Karl Schief, Bäcklund and Darboux transformations: geometry and modern applications in soliton theory, Vol. 30, Cambridge University Press, 2002.