A. The title is designed to indicate those particular aspects of stochastic differential equations which will be considered here: these are almost equally valid when the manifold in question is ℝ n (although compactness is often a useful simplifying assumption). In fact one of the main themes here will be that stochastic differential equations, even on ℝ n , induce non-trivial differential-geometric structures and these structures are an important tool in analyzing the behaviour of the solutions of the s.d.e.

Stochastic Differential Equations on Manifolds: STOCHASTIC DIFFERENTIAL EQUATIONS ON MANIFOLDS

We consider the linearized version of the stationary Navier-Stokes equations on a subdomain Ω of a smooth, compact Riemannian manifold M. The emphasis is on regularity: the boundary of Ω is assumed to be only C1 and even Lipschitz, and the data are selected from appropriate Sobolev-Besov scales. Our approach relies on the method of boundary integral equations, suitably adapted to the variable-coefficient setting we are considering here. Applications to the stationary, nonlinear Navier-Stokes equations in this context are also discussed.

The Stationary Navier-Stokes System in Nonsmooth Manifolds: The Poisson Problem in Lipschitz and C 1 Domains

We give an account of the classical and integrable geometry of isothermic surfaces in arbitrary co-dimension. We show that the classical transformation theory of Darboux, Bianchi and Calapso goes through unchanged in arbitrary co-dimension as does the connection with the "curved flats" of Ferus and Pedit. Moreover, we identify Darboux transformations with the dressing action of "simple factors" in the sense of Terng and Uhlenbeck. In so doing, we advertise the use of Vahlen's Clifford algebra matrices as an efficient computational tool in conformal geometry.

Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems

In this article I show that every special Lagrangian cone in C 3 determines, and is determined by, a primitive harmonic surface in the 6- symmetric space SU3/SO2. For cones over tori, this allows us to use the classification theory of harmonic tori to describe the construction of all the corresponding special Lagrangian cones. A parameter count is given for the space of these, and some of the examples found recently by Joyce are put into this context.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0024610703004204

Special lagrangian cones in c 3 and primitive harmonic maps.

1 Introduction: Surfaces with prescribed mean curvature.- 2 From minimal surfaces and CMC surfaces to harmonic maps.- 2.1 Minimal surfaces.- 2.2 Constant mean curvature surfaces.- 3 Variational point of view and Noether's theorem.- 4 Working with the Hopf differential.- 4.1 Appendix.- 5 The Gauss-Codazzi condition.- 5.1 Appendix.- 6 Elementary twistor theory for harmonic maps.- 6.1 Appendix.- 7 Harmonic maps as an integrable system.- 7.1 Maps into spheres.- 7.2 Generalizations.- 7.3 A new setting: loop groups.- 7.4 Examples.- 8 Construction of finite type solutions.- 8.1 Preliminary: the Iwasawa decomposition (for)..- 8.2 Application to loop Lie algebras.- 8.3 The algorithm.- 8.4 Some further properties of finite type solutions.- 9 Constant mean curvature tori are of finite type.- 9.1 The result.- 9.2 Appendix.- 10 Wente tori.- 10.1 CMC surfaces with planar curvature lines.- 10.2 A system of commuting ordinary equations.- 10.3 Recovering a finite type solution.- 10.4 Spectral curves.- 11 Weierstrass type representations.- 11.1 Loop groups decompositions.- 11.2 Solutions in terms of holomorphic data.- 11.3 Meromorphic potentials.- 11.4 Generalizations.

/pdf/constant-mean-curvature-surfaces-harmonic-maps-and-25sedboaq0.pdf

Constant Mean Curvature Surfaces, Harmonic Maps, and Integrable Systems

The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Kodaira embedding and the Pluecker relations for linear systems are proven. Interpretations of these results in terms of the differential geometry of surfaces in 3- and 4-space are hinted at throughout the paper. Applications to estimates of the Willmore functional on constant mean curvature tori, respectively energy estimates of harmonic 2-tori, and to Dirac eigenvalue estimates on Riemannian spin bundles in dimension 2 are given.

Quaternionic holomorphic geometry: Pluecker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori

This is the first comprehensive introduction to the authors' recent attempts toward a better understanding of the global concepts behind spinor representations of surfaces in 3-space. The important new aspect is a quaternionic-valued function theory, whose "meromorphic functions" are conformal maps into quaternions, which extends the classical complex function theory on Riemann surfaces. The first results along these lines were presented at the ICM 98 in Berlin. An important new invariant of the quaternionic holomorphic theory is the Willmore energy. For quaternionic holomorphic curves in the quaternionic projective line, i.e., the conformal 4-sphere, this energy is the classical Willmore energy of conformal surfaces. Using these new techniques, the article discusses the mean curvature sphere of conformal surfaces in the 4-sphere, Willmore surfaces in the 4-sphere, Baecklund tranformations of Willmore surfaces, super-conformal surfaces and twistor projections, and a duality between Willmore surfaces in the 3-sphere and minimal surfaces in hyperbolic 3-space. Finally, a new proof of Montiel's recent classification of Willmore 2-spheres in the 4-sphere is given.

Conformal Geometry of Surfaces in the 4-Sphere and Quaternions

Following Ebin and Marsden the Navier-Stokes equation is viewed as a perturbation of a geodesic flow on the group of volume preserving diffeomorphisms on a compact Riemannian manifold. Existence and uniqueness of bounded solutions for all position time is shown by taking a higher order diffusion term.

Global Solutions of the Navier-Stokes Equation with Strong Viscosity

We construct a sequence of isometric embeddings of spherical spaceforms with cylcic fundamental groups. The standard sphere and Veronese embedding are the first two elements in this sequence. The embeddings are orbits under the unitary group and consequently the lengths of geometric quantities (like the 2nd fundamental form or the mean curvature vector) are constant.

Isometric embeddings of spherical spaceforms with cyclic fundamental groups

S 1-equivariant minimal tori inS 4 andS 1-equivariant Willmore tori inS 3

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