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

This e-book brings jointly diversified branches of arithmetic: the speculation of Painlevé and the speculation of surfaces. Self-contained introductions to either those fields are provided. it's proven how a few classical difficulties in floor idea might be solved utilizing the fashionable thought of Painlevé equations. particularly, a necessary a part of the e-book is dedicated to Bonnet surfaces, i.e. to surfaces owning households of isometries keeping the suggest curvature functionality. an international type of Bonnet surfaces is given utilizing either components of the speculation of Painlevé equations: the idea of isomonodromic deformation and the Painlevé estate. The booklet is illustrated by means of plots of surfaces. it's 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 get to grips with one other suitable department of arithmetic.

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

**Example text**

We will treat such = Theory R3 do in Appendix not have umbilic points with M 6. 1. 1 it = 0. 2. Let P be a critical point of index J of a Bonnet surface. 2 Hazzidakis Equation j + I. 0 ) zHz (_0 ), H2 = E R. 3. In dinate w a E R,,. neighbourhood of a critical point define a conformal coor- by w The -i(p(0)10(0) 0 (z) mean = e G(z) Z, curvature where G(z) function is A a f function Of S H(w,,Cv) Proof. Observe that w is 0, w'(0) 54 0. Defining w well defined: this way dz, = G(z) yields A= f (0).

Proof. 2: Since the coefficients of the corresponding equations are analytic, smoothness of the solution yields its real anlyticity [CoL]. To prove the smoothness of H, note that due to the Painlev6 property, H may have only 0 in the case BV should be treated seperately). 2 everywhere 0 of the case BV. 1 imply the existence of a real analytic solution H(s) in a neighbourhood of s 0. 1. 2 are maximal. Proof. Let Y be a Bonnet surface with a point Po E Y. 2. 2. Since Bonnet surfaces surface. 2. 1) Y U Any point ) is also a P E Y U -P Bonnet can be 7.

48) together satisfy the Codazzi equations for w 0 0 when the mean curvature function is a function of s JW12 only. 44). 45), the fundamental forms with H(s), Q(w,Av-), e'* `) determine an immersion of a neighbourhood of w 0 into R3. 25)). 17). 44) possesses a solution 0. = holomorphic H(s). 44-3-48) are (3-51) invariant under the aEC*. 46) equation on h(w) which OER,. 1. h-40h, In addition function additional conditions s h(w). unless AH 2 0 =- 2- (3-53) * sH1 This equation does not-have solutions which a function of s JW12.