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What is the easiest (and what is the most elementary) way of proving Newlander-Nirenberg theorem for Riemannian surfaces? I was able to reduce it to existence of non-trivial harmonic functions (locally) on 2-dimensional Riemannian manifolds, but this seems to be non-trivial, too. I want to have a construction using 1-dimensional complex analysis and not much else.

A similar question was asked here.

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    $\begingroup$ the most efficient proof I know uses some fourier analysis is due to Douady with Buff taking notes . $\endgroup$ – Mohan Ramachandran Mar 7 '14 at 16:43
  • $\begingroup$ There is also Chern, Shiing-shen An elementary proof of the existence of isothermal parameters on a surface. Proc. Amer. Math. Soc. 6 (1955), 771–782. $\endgroup$ – Claudio Gorodski Mar 7 '14 at 16:54
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    $\begingroup$ I don't know about `easiest', but the proof in Spivak's Comprehensive Introduction to Differential Geometry is not hard; he is able to cover what he needs from elliptic theory fairly quickly. Also, the weaker the assumptions about the regularity of the coefficients, the harder the proof becomes, so there are really several different theorems, depending on the regularity assumed for the coefficients. When the coefficients are real-analytic, Gauss' original proof (reducing it to complex ODE) works fine. $\endgroup$ – Robert Bryant Mar 7 '14 at 19:15
  • $\begingroup$ Also, see mathoverflow.net/questions/28519/… $\endgroup$ – Deane Yang Mar 7 '14 at 19:48
  • $\begingroup$ There is a proof in the end of the first chapter of Donaldson and Kronheimer's "Geometry of 4-manifolds", if I remember well. $\endgroup$ – Albuquerque Nov 11 '14 at 17:24
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I like the discussion (only possible in this dimension) which uses (1) the fact that Calderón—Zygmund operators which are smoothing of order one transform bounded measurable functions into continuous ones with $e\log1/e$ modulus of continuity and (2) Osgood's elementary theorem that this modulus of continuity is good enough for unique solutions of ODEs.

Now in a chart with a bounded measurable change of almost complex structure thought of at each point as point in the unit disc with non-Euclidean geometry and at the origin the standard structure in the chart, draw the Poincaré geodesic between these two structures.

Applying the Cauchy transform converts the bounded measurable infinitesimal distortion of conformal structure into an Osgood vector field. We can integrate this and move along the geodesic path. This produces a composition of homeomorphisms with the required bounded conformal distortion.

To get charts with holomorphic overlap mappings modify the starting charts by these constructed homeomorphisms with bounded conformal distortion and use this fact: "homeomorphisms with bounded conformal distortion and zero distortion ae. are holomorphic". This proves Newlander—Nirenberg for bounded measurable almost complex structures (relative to a standard background).

This is not the most elementary proof of the smooth result but to me it is the conceptually easiest proof. It also gives a natural and very strong statement with lots of applications not possible using the smooth result.

Also the same proof scheme (use Calderón—Zygmund then Osgood to inch your way to a solution) also solves the Euler equation for 2D incompressible fluid motion for any fixed time which is not known in higher dimensions.

Dennis Sullivan

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There is also a proof, due to Joe Kohn, using $L^2$ methods, which is relatively easy. I think the proof, for general n, is in Hormander's book. As Mohan says, there is need to establish regularity results, but they are standard. This proof is no easier in 1d, though; it is essentially the same.

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As Mohan Ramachandran mentioned in a comment above, there is a short and clear proof in Adrien Douady and X. Buff, Le théorème d’intégrabilité des structures presque complexes, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 307–324. MR 1765096. With some local work, you show that the problem of constructing local holomorphic coordinates reduces to a Beltrami equation, close to the Cauchy--Riemann equation and equal to it outside some compact set. Then you apply elementary $L^2$ methods, using Fourier transform and convolution to construct a very explicit iteration scheme to converge to a solution.

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