Abstract. We extend the classical theory of isothermic surfaces in conformal 3-space, due to Bour, Christoffel, Darboux, Bianchi and others, to the more general context of submanifolds of symmetric R-spaces with essentially no loss of integrable structure. IntroductionBackground. A surface in R 3 is isothermic if, away from umbilics, it admits coordinates which are simultaneously conformal and curvature line or, more invariantly, if it admits a holomorphic quadratic differential q which commutes with the (trace-free) second fundamental form (then the coordinates are z = x + iy for which q = dz 2 ). Examples include surfaces of revolution, cones and cylinders; quadrics and constant mean curvature surfaces (where q is the Hopf differential).Starting with the work of Bour [9, §54], isothermic surfaces were the focus of intensive study by the geometers of the late 19th and early 20th centuries with contributions from Christoffel, Cayley, Darboux, Demoulin, Bianchi, Calapso, Tzitzéica and many others. There has been a recent revival of interest in the topic thanks to Cieśliński-Goldstein-Sym [26] who pointed out the links with soliton theory: indeed, isothermic surfaces constitute an integrable system with a particularly beautiful and intricate transformation theory, some aspects of which we list below:Conformal invariance. Since the trace-free second fundamental form is invariant (up to scale) under conformal diffeomorphisms of R 3 , such diffeomorphisms preserve the class of isothermic surfaces. Thus isothermic surfaces are more properly to be viewed as surfaces in the conformal 3-sphere.Deformations. At least locally, isothermic surfaces admit a 1-parameter deformation preserving the conformal structure and trace-free second fundamental form: this is the T -transformation of Bianchi [3] and Calapso [19]. In fact, isothermic surfaces are characterised by the existence of such a deformation [23] (see [47,17,14] for modern treatments).Darboux transformations. According to Darboux [27], given an isothermic surface, one may locally construct a 4-parameter family of new isothermic surfaces. Analytically, this is accomplished by solving an integrable system of linear differential equations; geometrically, the two surfaces envelop a conformal Ribaucour sphere congruence 1 . These transformations, the Darboux transformations, are analogous to the Bäcklund transformations of constant curvature surfaces (indeed, specialise to the latter in certain circumstances [40,43] Curved flats. Curved flats were introduced by and are an integrable system defined on submanifolds of a symmetric space G/H which, in non-degenerate cases, coincides with the G/H-system of Terng [54]. In particular, the space S 3 × S 3 \ ∆ of pairs of distinct points in S 3 is a (pseudo-Riemannian) symmetric space for the diagonal action of the conformal diffeomorphism group and curved flats in this space are the same as pairs of isothermic surfaces related by a Darboux transformation [16].Discrete theory. Bobenko-Pinkall [6] show that the combinatorics of ...

A generalization of criticality -called secondary criticality -is introduced and applied to finite-amplitude Stokes waves. The theory shows that secondary criticality signals a bifurcation to a class of steady dark solitary waves which are biasymptotic to a Stokes wave with a phase jump in between, and synchronized with the Stokes wave. We find the that the bifurcation to these new solitary waves -from Stokes gravity waves in shallow water -is pervasive, even at low amplitude. The theory proceeds by generalizing concepts from hydraulics: three additional functionals are introduced which represent non-uniformity and extend the familiar mass flux, total head and flow force, the most important of which is the wave action flux. The theory works because the hydraulic quantities can be related to the governing equations in a precise way using the multi-symplectic Hamiltonian formulation of water waves. In this setting, uniform flows and Stokes waves coupled to a uniform flow are relative equilibria which have an attendant geometric theory using symmetry and conservation laws. A flow is then 'critical' if the relative equilibrium representation is degenerate. By characterizing successively non-uniform flows and unsteady flows as relative equilibria, a generalization of criticality is immediate. Recent results on the local nonlinear behaviour near a degenerate relative equilibrium are used to predict all the qualitative properties of the bifurcating dark solitary waves, including the phase shift. The theory of secondary criticality provides new insight into unsteady waves in shallow water as well. A new interpretation of the Benjamin-Feir instability from the viewpoint of hydraulics, and the connection with the creation of unsteady dark solitary waves, is given in Part 2.

A one-parameter family of periodic orbits with frequency ! and energy E of an autonomous Hamiltonian system is degenerate when E 0 ! 0. In this paper, new features of the nonlinear bifurcation near this degeneracy are identified. A new normal form is found where the coefficient of the nonlinear term is determined by the curvature of the energy-frequency map. An important property of the bifurcating ''homoclinic torus'' is the homoclinic angle and a new asymptotic formula for it is derived. The theory is constructive, and so is useful for physical applications and in numerics.

A geometric view of criticality for two-layer flows is presented. Uniform flows are classified by diagrams in the momentum-massflux space for fixed Bernoulli energy, and cuspoidal curves on these diagrams correspond to critical uniform flows. Restriction of these surfaces to critical flow leads to new sub-surfaces in energy-massflux space. While the connection between criticality and the generation of solitary waves is well known, we find that the nonlinear properties of these bifurcating solitary waves are also determined by the properties of the criticality surfaces. To be specific, the case of two layers with a rigid lid is considered, and application of the theory to other multi-layer flows is sketched.

The time-dependent motion of water waves with a parametrically-defined free surface is mapped to a fixed time-independent rectangle by an arbitrary transformation. The emphasis is on the general properties of transformations. Special cases are algebraic transformations based on transfinite interpolation, conformal mappings, and transformations generated by nonlinear elliptic PDEs. The aim is to study the effect of transformation on variational principles for water waves such as Luke's Lagrangian formulation, Zakharov's Hamiltonian formulation, and the Benjamin-Olver Hamiltonian formulation. Several novel features are exposed using this approach: a conservation law for the Jacobian, an explicit form for surface re-parameterization, inner versus outer variations and their role in the generation of hidden conservation laws of the Laplacian, and some of the differential geometry of water waves becomes explicit. The paper is restricted to the case of planar motion, with a preliminary discussion of the extension to three-dimensional water waves.