By Alexander I. Bobenko, Ulrich Eitner (eds.)

ISBN-10: 3540414142

ISBN-13: 9783540414148

ISBN-10: 3540444521

ISBN-13: 9783540444527

This publication brings jointly various branches of arithmetic: the speculation of Painlevé and the idea of surfaces. Self-contained introductions to either those fields are provided. it really is proven how a few classical difficulties in floor concept may be solved utilizing the fashionable conception of Painlevé equations. particularly, an important a part of the ebook is dedicated to Bonnet surfaces, i.e. to surfaces owning households of isometries protecting the suggest curvature functionality. an international class of Bonnet surfaces is given utilizing either elements of the speculation of Painlevé equations: the speculation of isomonodromic deformation and the Painlevé estate. The booklet is illustrated by way of plots of surfaces. it truly is meant for use by way of mathematicians and graduate scholars drawn to differential geometry and Painlevé equations. Researchers operating in a single of those components can familiarize yourself with one other correct department of arithmetic.

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**Extra resources for Painlevé Equations in the Differential Geometry of Surfaces**

**Example text**

That not every surface of type B contains a critical point of curvature function. 1) in R3 can in terms of certain 3. Bonnet Surfaces 40 Painlev6 transcendents. 2. The Hazzidakis equation for Bornact surfaces of types A and B ically equivalent. Namely, HB(t) HA (7-) a a, are analytequation for a sollition of the Hazzidakis only if Bonnet surface of type B if and is is =_ - i HB (t), (3-66) t. solution of the Hazzidakis equation of type A. 1 From the Lax Moving Representation Studying Frame of Bonnet B and of Painlev6 VI BV Surfaces to the Equations Bonnet surfaces it is natural to write down their frame equations using parametrization F(w, i7v) (3-21).

Bonnet Surfaces 28 p E U and t + i7v w = C such that the mean curvature is a function of a chart w : U only. This function. H(t) sati, fles the ordinary differential equation In the coordinate w JJ ""j -H' 't) Q! H, W the Hopf differential and 1 + iT h(w) h' (W) (h(w) jh' (W) 12 2 eu(w,cv) Idw 12 are' metric (1-iTh(w)) Q (w, i7o, T) dw2 . 19) h(w)) IdW12. 19). 20). Then the fundamental forms with H (t), Q (w, Fv, T), eu(', v) determine Conversely, given a Bonnet Proof. family FT. 1). 19). 19) directly H2 (z,, ) 1 Note that the denominator in with the one f curvature function t only.

3. A branched Bonnet surface Fig. 4. Bonnet surface of immersion domain U of Bonnet surfaces of type B is mental domains U,, Indeed, 55 = JW E the fundamental forms C1 (n (see - naturally split type B into funda- 7r 1)7r2 < Im(w) Table 3. 1) are < n 2 invariant with. respect to the shift 7r i 2 are congruent in W. 3. For an appropriate choice of parameters, several copies of the fundamental domain can close up and thus comprise a closed surface with a critical point. 4 shows and thus immersed Un with different n's such with three fundamental domains.

### Painlevé Equations in the Differential Geometry of Surfaces by Alexander I. Bobenko, Ulrich Eitner (eds.)

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