Antoine Millecamps

Alain Batailly

Mathias Legrand

François Garcin

Snecma

Snecma's Viewpoint on the Numerical and Experimental Simulation of Blade-Tip/Casing Unilateral Contacts

Keywords: Turbomachinery, unilateral contact and friction, nonlinear dynamics, rotor-stator interactions, design process

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Competition between engine manufacturers, high gas prices and stringent environmental regulations yield tight market conditions for the development of a new generation of aircraft engines. In this context, novel engines must achieve higher performances and lower gas consumption in order to maximize their energy efficiency. Such maximization is typically twofold, it may be achieved by: (1) using lighter materials such as composite materials and (2) reducing parasitic leakage flows. The latter are closely related to the operating clearances between rotating and stationary components of an engine, they have a direct impact over the flow compression ratio from a stage of the engine to another and thus critically affects the overall engine efficiency. Theoretically, the reduction of parasitic leakage flows is achieved through the closing of operating blade-tips/casing clearances. In practice however, the minimization of operating clearances leads to more frequent structural contacts between the engine components and such events are now accepted as normal operating conditions. Because the physical phenomena at play during these contacts are still mostly unelucidated, many investigations have been recently carried out [START_REF] Padova | Experimental Results From Controlled Blade Tip/Shroud Rubs at Engine Speed[END_REF][START_REF] Schmiechen | Travelling wave speed coincidence[END_REF][START_REF] Childs | Prediction of dry-friction whirl and whip between a rotor and a stator[END_REF]. In order to mitigate the consequences of these structural contacts over the components structural integrity, a widely used solution consists of depositing an abradable coating along the casing circumference, see Fig. 1. These abradable coatings are generally laid by plasma spraying of composite powders, comprising a metal base (aluminium and silicon) and a polymer filler (polyester) generating porosities. The mechanical properties of these coatings are singular as they must satisfy the following compromise: an abradable coating should be resilient enough in order not to be damaged by the gas flow while it should act as a sacrificial material when an impact with the rotating blade occurs. The abradable microstructures are known to feature good cohesionthanks to the metal matrix-and good abrasion-thanks to the polyester and porosities [START_REF] Prchlik | Effect of the microstructure of thermally sprayed coatings on friction and wear response under lubricated and dry sliding conditions[END_REF][START_REF] Schmid | New high temperature abradables for gas turbines[END_REF].

Nevertheless, experimental observations following rotor/stator interaction simulations on full scale test benches revealed that significant damages may appear both on the blade, see cracks in Fig. 2, and the abradable coating. The critical need for ensuring structural integrity of the engine components combined with the necessity of maintaining a higher engine efficiency was the core motivation for Snecma to initiate research projects and collaborations focusing on the numerical and experimental simulation of rotor/stator interactions. First, an experimental test bench has been developped in order to simulate rotor/stator interactions on a full scale low-pressure compressor under vacuum with an abradable coating deposited along the casing circumference [START_REF] Millecamps | Influence of thermal effects during blade-casing contact experiments[END_REF]. Simultaneously, a research program focused on the development of a numerical strategy [START_REF] Legrand | Full three-dimensional investigation of structural contact interactions in turbomachines[END_REF] for the simulation of such interactions. The goal of the research program was to develop a numerical strategy accurate enough to predict experimental results and that could be integrated within Snecma's industrial environment as a first step toward the inclusion of structural contacts within the design process of compressor and turbine blades. The need for such numerical strategy called for the development of a dedicated in-house code as commercial softwares could not provide satisfying results due to several limitations such as the impossibility to combine the use of efficient component mode synthesis methods with a precise contact treatment procedure (respect of non-penetration conditions between structures) and accounting for both centrifugal and gyroscopic effects. In addition, strategies for the modeling of the abradable removal are currently non-existent in commercial softwares.

This paper aims at describing the strategy adopted by Snecma for acquiring knowledge on rotor/stator interactions initiated by blade-tip/casing or abradable coating contacts. First of all, the experimental test benches developped for the simulation of rotor/stator interactions are introduced. During the last ten years two series of tests were performed and their main results are presented herein. Secondly the original numerical approach, namely COROS, used at Snecma to study the contact between rotating part and casing is briefly detailed. Then, applications of COROS for the simulation of industrial cases are reviewed. Finally, the paper deals with on-going Research and Development projects following two guidelines: (1) the characterization of abradable materials and (2) the numerical improvement of COROS.

Experimental simulations

For the sake of confidentiality, the two experimental tests are referred to as SN 1 and SN 2 , also, all results are normalized. Test benches SN 1 and SN 2 respectively involve a full scale lowpressure compressor stage and a full scale high-pressure compressor stage. Accordingly, the blade designs are significantly different and so are the mechanical properties of the abradable coatings used for each test. Obviously, targetted angular speeds are also different in agreement with the nominal angular speed range of the stage of interest. For both test benches, one of the blade is slightly longer and is instrumented with strain gages sensitive to the blade first bending and torsional free vibration modes. Accelerometers are located on the casing. Contact with the surrounding casing is thus initiated on this longer blade as centrifugal loads are applied as the bladed disk angular speed increases. The casing is not deformed, though its shape is not perfectly circular due to manufacturing tolerances and assembly conditions yield a slight ovalization. The cold clearance between rotor and stator is zero. For each configuration the rotor speedup aims at reaching an angular speed for which the blade vibration along its first bending mode is synchronous. Also, it should be mentioned that tests are performed under vacuum, thus there are no aerodynamic loads.

Test bench SN 1

This test bench is presented in detail in [START_REF] Millecamps | Influence of thermal effects during blade-casing contact experiments[END_REF]. The geometries of parts (casing and blades) are representative of a low-pressure compressor. The targetted angular speed 1 is chosen so that it corresponds to a sixth of the first eigenfrequency of the blade as depicted on the Campbell diagram in Fig. 3. For D 1 , the first eigenfrequency of the blade intersects the sixth engine order line H6. Accordingly, when contact occurs, it is expected that the blade will mostly respond on its first free-vibration mode, the first bending mode.

At the end of the experimental simulation, cracks, heating marks and damages at the tip are witnessed on the blade. Also, a significant wear is observed on the abradable coating following a six-lobe pattern along the casing circumference as depicted in Fig. 4. Additionally, thermocouples located on the outer side of casing show significant increase of temperature, about C50 ı C. Interestingly, the accelerometers on the casing did not show any significant levels of vibration. Based on the results obtained with test bench SN 1 , it is first highlighted in [START_REF] Millecamps | Influence of thermal effects during blade-casing contact experiments[END_REF] that blade failure is observed following a blade/abradable coating interaction. The failure seems to be induced by vibratory fatigue since the blade strain gages bring to light the increase of the stress amplitude within the blade during the interaction. An extensive post-processing procedure based on a Fourier Transform of the results revealed that the first bending mode is dominant in the blade dynamics, particularly at the end of the interaction. As mentioned above, the blade dynamics along its first bending mode is synchronous with the angular speed and the number of lobes found around the casing circumference is in agreement with the targetted engine order H6. In addition, this test bench provides useful data regarding thermal effects within the abradable coating that may be related to the wear pattern.

Test bench SN 2

Test bench SN 2 involves a high-pressure compressor stage. The abradable coating composition contains much less polyester than for test bench SN 1 , it is representative of a coating sprayed within a high-pressure compressor stage. Similarly to test bench SN 1 , the targetted angular speed 2 is chosen so that blade vibration along its first bending free-vibration mode is synchronous. It is pictured in Fig. 5 that for D 2 , the first eigenfrequency intersects the eigth engine order line 8N. For the considered blade design, the angular speed 2 is of particular interest as it is also the location of the intersection between the first torsional mode and the twenty second engine order line 22N. The concomitance of both intersections motivates the experimental simulations carried out around this interaction point. Post-interaction observations did not reveal any blade failure: no significant damage could be seen on the longer blade eventhough stress signals time histories acquired by the different strain gages featured increasing stress levels and peaks of stresses as visible in Fig. 6. As opposed to what was observed with test bench SN 1 , the thermocouples located on the outer surface of the casing did not feature significant temperature variations (more or less 2 ı C through out the simulation). Regarding the abradable coating wear pattern, twenty-two lobes are noticeable, thus indicating that the torsional interaction may have been dominant.

A Fourier analysis of the stress signal history is conducted for three different time intervals as depicted in Fig. 6. This analysis reveals that the first bending mode (1F) is mostly dominant at the beginning of the interaction while the peak of stress located around t 2 almost solely involves the first torsional mode (1T). Following the stress peak, for t D t 3 , both the first bending mode and the first torsional mode are significantly involved in the blade dynamics. One may speculate that the concomitance of two intersections involving distinct free-vibration modes mitigates the possibility of the interaction thus explaining why, contrary to test bench SN 1 , no blade failure is observed in test bench SN 2 .

Numerical strategy and integration within snecma industrial environment

As mentioned in the introduction, the research project carried out by Snecma aims at the development of a numerical strategy, namely COROS, dedicated to the modeling of the test benches presented in the previous section. The numerical strategy must comply with the following specifications: be numerically efficient: the blade vibratory behaviour must be retrieved for many revolutions for potentially many angular speeds and contact configurations, allow for retrieving displacement and stress fields over the full finite element model at any time step of the simulation, account very accurately for non-penetration contact conditions between the structures involved in the interaction, be able to simulate a large variety of contact configurations (casing shape, clearance configuration...), account for both centrifugal and gyroscopic effects, be fully compatible with the Snecma industrial environment.

Solution method

Because the physical phenomena at play during the interaction are not fully understood, a quantitative approach based on explicit time integration is considered: the central finite differences time integration scheme is employed. Frequency based approaches, that typically require an a priori knowledge of the physical phenomena-in order to properly select useful harmonics for instance-are thus excluded. The need for a precise contact treatment procedure motivates the use of a Lagrange-multiplier based approach [START_REF] Carpenter | Lagrange constraints for transcient finite element surface contact[END_REF] instead of a penalty-based method which inconveniently allows for significant residual penetrations. In order to ensure the computational efficiency of the procedure, component mode synthesis methods are used and lead to small yet accurate models of full 3D finite element models. Fixed-interface methods are preferred [START_REF] Legrand | Full three-dimensional investigation of structural contact interactions in turbomachines[END_REF] as they allow for keeping selected degrees of freedom useful for contact management along the blade tip within the reduced space. Finally, a wear law is developped [START_REF] Legrand | Numerical investigation of abradable coating removal through plastic constitutive law in aircraft engine[END_REF] in order to model abradable coating removal as well as the contact forces applied along the blade tip during the interaction.

The reader may refer to previous publications for additional details regarding the convergence of the obtained results with respect to the time step, the modal reduction basis or the discretization of the abradable coating.

Industrial integration

In order to ensure a full integration of the numerical strategy within Snecma industrial environment, the input finite element modes are industrial 3D finite element models obtained from Samcef. The output result files are written in such way they can be read by Samcef post-processing modules: Samres and Samfac. A dedicated graphic user interface is created and advanced postprocessing tools (interaction maps, automated Fourier analyses, wear maps...) are created with Python codes.

Numerical results

Test bench SN 1

The results obtained with numerical model of test bench SN 1 have been published in [START_REF] Batailly | Numericalexperimental comparison in the simulation of rotor/stator interaction through blade-tip/abradable coating contact[END_REF]. The proposed numerical strategy was successfully used for the determination of the interaction angular speed. The abradable wear pattern along the casing circumference-featuring six distinct lobes depicted in Fig. 7-as well as the frequency content of the blade dynamics were accurately predicted. In addition, predicted maximum stress areas within the blades were in good agreement experimentally observed damaged areas.

Test bench SN 2

A numerical model was also considered for test bench SN 2 . Contrary to test bench SN 1 , the ovalization of the casing was less obvious experimentally. Yet, an ovalized casing was used in the numerical model in order to initiate contact with the rotating blade.

Based on empirical data, the wear law mechanical properties were updated in order to account for the specificity of the abradable coating of this test bench. A 22-lobe wear pattern pictured in Fig. 8 was numerically predicted in agreement with experimental observations. However, the numerical model failed to predict comparable participations of the first bending and torsional modes within the blade dynamics. Contrary to the numerical results obtained for test bench SN 1 , results for SN 2 were particularly sensitive to the casing initial profile thus hinting that the phenomenon of interest may not be as robust as for test bench SN 1 .

Optimization of a blade design

Recently, COROS was integrated to the design process of a highpressure compressor blade [START_REF] Batailly | Redesign of a high pressure compressor blade accounting for nonlinear structural interactions[END_REF]. The design improvement was driven by aerodynamic performances. The design parameter was the stacking law, see Fig. 9, as well as the cross section of the blade. The stacking law of a blade is a fictive line, from the root to the edge of a blade, giving the direction and the orientation of the stack of cross section. Following the design procedure, a few eligible blade profiles are analyzed with COROS for various contact configurations. A reduced order model is computed from the 3D finite element mesh files and contact simulations are run through out a wide angular speed range. For each angular speed, a Fourier spectrum is computed once steady state has been reached. Putting side by side these spectra leads to an interaction map such as the one pictured in Fig. 10. Peaks of amplitude define critical speeds that should be avoided.

Research and development projects and perspectives

Full-scale interaction tests

As explained in the introduction the study of Rotor/Stator contact is relatively recent at Snecma. First experimental tests were performed in 2006 and the development of the numerical approach took roughly five years. Work is in progress for Snecma and its academic partners in order to improve COROS. Beside of the numerical developments, other academic partners work with Snecma focusing on: (1) understanding and simulating wear mechanism The casing is initially scanned in order to know its shape with great precision. After the test the casing will be scanned again (ten points axially for each degree along the circumference). The only difference between the first two tests is the geometry of the blisk. The differences of the blade are a small change for the stacking law based on the numerical investigations mentioned above regarding blade design optimization. The targetted speed is slightly different because of the difference of natural frequencies. The difference between the second and third test is the abradable coating. In the second the coating is an aluminum-silicon with polyester whereas the other has only a metallic matrix composed of aluminum-silicon. This work will constitute a rich test plan to improve COROS.

Wear modeling improvements

The modeling of wear in COROS is clearly a stringent limit. The material parameters: yield stress, plastic modulus and young modulus provide accurate contact forces on the blade but additional phenomena must be accounted for in order to match experimental observations. However an introduction of viscosity for the elastic behavior improves the model slightly [START_REF] Batailly | Numericalexperimental confrontation in the simulation of tool/abradable material interaction[END_REF]. A qualitative study based on a phenomenological approach is employed to take into account wear mechanisms such as: erosion, densification, plasticity, cutting, smearing or material transfer [START_REF] Borel | The wear mechanisms occurring in abradable seals of gas turbines[END_REF]. The new law is an addition of terms representing each wear mechanism. The coding in COROS is in progress. Important work will be done in order to match experimental tests performed in French universities.

A first test bench, see Fig. 11 and Fig. 12, is also performed in order to characterize the contact forces between abradable and a titanium tool [START_REF] Sutter | Dynamic analysis of the interaction between an abradable material and a titanium alloy[END_REF]. This test bench has been improved in order to measure axial, tangential and radial forces and to put a camera in-situ. These tests are very interesting to calibrate the abradable behavior in COROS since there is no dynamics movement for the titanium tool. So the numerical simulations are more or less fast and easy. A different test platform consists of a rotating cylin-

. Detailed view of the projectile and the titanium tool on the test bench [START_REF] Sutter | Dynamic analysis of the interaction between an abradable material and a titanium alloy[END_REF] der whose external surface is coated with an abradable material, see Fig. 12. The blade geometry has been simplified and the interaction generated by translating the blade toward the rotating cylinder [START_REF] Mandard | Interacting force estimation during blade/seal rubs[END_REF]. The numerical simulations of this test can settle the abradable parameters and also the contact parameters since the dynamic behavior is precisely measured through digital image correlation and strain gages.

Introduction of thermal effects

The full-scale interaction tests have shown significant increase of temperature. Then some publications highlight the influence of thermal effects through numerical approach [START_REF] Muszynska | Rotor-to-stationary part full annular contact modeling[END_REF][START_REF] Millecamps | Interaction aube/carter: contribution de l'usure de l'abradable et de la thermomécanique sur la dynamique d'aube[END_REF]. For the new test benches, tests will be performed in order to measure the temperature during a blade/abradable contact. For example Infrared thermal camera will measure temperature of the coating. Thermocouples stuck on the blade, through an inverse method, will provide temperature on the contact (on the blade tip). The associated numerical approach is still in consideration, notably on the manner to take into account the thermal expansion.

Advanced signal processing techniques

A third evolution of COROS aims at improving the Fourier analysis of the vibratory movement of the blade as detailed in [START_REF] Batailly | Conjectural Bifurcation analysis of an aircraft engine blade undergoing 3D unilateral contact constraints[END_REF]. The displacement of the blade may not always be periodic and a usual Fourier analysis may thus be inefficient. 20]. Signal is decomposed over a basis of Intrinsic Mode Functions (IMF). The IMF functions, under certain mathematical properties [START_REF] Huang | The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis[END_REF] are associated with instantaneous frequency -obtained with a Hilbert transform of the IMF. Such strategy seems promising for both live monitoring and numerical results post-processing purposes.

Full bladed disk interaction analysis

Both test benches SN 1 and SN 2 featured full bladed disks but the actual witnessed interaction phenomena only involved a single blade: vibration levels on the casing as well as on the other blades were found negligible experimentally. However, other interaction phenomena exist, such as modal interaction [START_REF] Schmiechen | Travelling wave speed coincidence[END_REF] and whirl motions [START_REF] Sinha | Rotordynamic analysis of asymmetric turbofan rotor due to fan blade-loss event with contact-impact rub loads[END_REF], that involve the full bladed disk and the surrounding casing. This section focuses on the extension of COROS to these phenomena. The modal interaction is related to the specific modal properties of cyclically symmetric structures and may occur when the mode shape geometry of the bladed disk and the casing match and their travelling wave speeds coincide. First of all, the use of cyclic symmetry conditions, assuming the bladed disks are perfectly tuned, allows for the computation of reduced-order models of full 3D finite element models of bladed disks. Contact simulations are run managing contact on the tip of each blade and displacement fields may be retreived as pictured in Fig. 14 where four sectors of a 22 sector fan stage model are pictured. For each time step of the simulation, the computation of stress fields advantageously provide information regarding critical areas within the structure when contacts occur. For example, a zoom over the stress fields within the blade roots of the aforementioned four sectors is provided in Fig. Accounting for a flexible casing is the last step before the simulation of modal interaction. Exemplarily, a cut view of the fan stage of an aircraft engine and the associated casing is pictured in Fig. 16. For such simulations, more advanced numerical developments featuring a nonlinear contact detection algorithm and the contact surface smoothing are required. The whirl/whip phenomenon [START_REF] Childs | Prediction of dry-friction whirl and whip between a rotor and a stator[END_REF] is also a concern for Snecma. It is a singular issue since under unidentified conditions the fan blades rubs slightly with the casing. Then the contact forces between blades and casing increase up to a point where the damages of the structure may be significant. As a result, the whole shaft features a dynamic movement leading potentially to important damages which could be more important than a Fan Blade Out event. To better understand these phenomena, rarely observed experimentally, the previous numerical approach (flexible casing and full bladed disk) has to be coupled with the standard studies of rotor dynamics. In rotor dynamics, studies aim at investigating the movement of flexible shaft using simplified models like cylinder/cylinder contact [START_REF] Cole | On stability of rotordynamic systems with rotor-stator contact interaction[END_REF][START_REF] Muszynska | Rotor-to-stationary element rub-related vibration phenomena in rotating machinery: Literature survey[END_REF]. In this context a recent analysis on an Industrial Whole Engine Model (WEM) has included a flexible casing [START_REF] Petrov | Multiharmonic analysis of non linear whole engine dynamics with bladed disk-casing rubbing contact[END_REF] and an other analysis introduces flexible blades through Euler-Bernoulli beams [START_REF] Parent | Whole Engine Interaction in an bladed rotor-to-stator contact[END_REF]. The objective, now is to perform analyses with COROS for flexible shaft and bearing stiffness. A first attempt was proposed [START_REF] Salvat | Two-dimensional modeling of shaft precessional motions induced by blade/casing unilateral contact in aircraft engines[END_REF] consisting of a 2D in-plane model. This model features a set of curved beams for the casing and beams for the bladed-disk (Figure 17). The flexibility of the shaft is reflected by two linear springs attached to the center node of the disk.

Conclusion

This paper aims at describing the overall methodology deployed at Snecma in order to develop new generation of aircraft engines with reduced operating clearances between rotor and stator. The methodology is twofold, it relies on: (1) an experimental approach to understand phenomena and, (2) a numerical approach for the development of predictive models. The experimental test benches highlight strong links between the wear pattern and the dynamic behavior of the blade, notably the number of lobes observed around the casing is in agreement with the targetted engine order H6 for SN 1 and H22 for SN 2 . Furthermore, significant increases of temperature are recorded involving thermal effects at the area of contact. The numerical strategy, namely COROS, has been developed for about ten years in partnership with academic partners. The goal of the numerical approach is to simulate a rotating blade in contact with a rigid casing-possibly not perfectly circularand accounts for the abradable coating. To reduce the number of degrees of freedom a first step is to use component mode synthesis methods. The second step is the simulation of blade/casing contact, taking into account the abradable wear. The contact algorithm is an explicit time scheme. This numerical approach is used to simulate the test bench presented in the first section with attractive results in terms of dynamical behavior and wear pattern matching. As a consequence COROS is a numerical tool used in design office in order to discriminate 'bad' blade geometries for which peaks of amplitudes are numerically predicted over the nominal angular speed range. Obviously, various scientific works are in progress since this research field is still fairly recent and the numerical approaches must reach higher predictivity level. Accordingly, a new experimental test plan has been defined with more accurate measures of casing shape. In order to clearly define the influence of each parameter there are few differences between tests. The wear behavior law of the abradable is improving in order to introduce more physical behavior of the abradable in COROS. Then this new law will be used and is expected to match the experimental results. Other scientific works will help Snecma to better apprehend physical phenomena as modal interaction and whirl movements. The introduction of flexible casing and a multi-bladed simulation is a way to study the first issue. For the second one, the way is to couple the COROS approach with a model shaft.

Figure 1 .

 1 Figure 1. Overview of a rotating blade and the abradable coating on the inner face of casing

Figure 2 .

 2 Figure 2. Important damages on titanium blades and wear profile on abradable coating

Figure 3 .

 3 Figure 3. First natural frequency of the blade (a stiff disk is assumed) over the angular speed range of interest and engine order lines H3 to H7. The first eigenfrequency of the blade ( ) intersects the sixth engine order line H6 ().

Figure 4 .

 4 Figure 4. Circumferential wear of the abradable coating, facing the blade trailing edge. Dashed circles represent the initial radius

Figure 5 .

 5 Figure 5. First and second natural frequencies of the blade (bending and torsional mode) depending on speed. First eigenfrequency ( ); eigth engine order line 8N ( ); first torsional mode ( ); twenty second engine order line 22N ( )

Figure 6 .

 6 Figure 6. Time evolution of strain gage signal and the FFT analysis at three different time

Figure 7 .

 7 Figure 7. Wear pattern of the abradable coating (facing the trailing edge) simulated with COROS [10] for test bench SN 1

Figure 8 .

 8 Figure 8. Wear pattern of the abradable coating (facing 8 points: from trailing to leading edge) simulated with COROS for test bench SN 2 superimposed with the initial casing profile ( )

Figure 9 .

 9 Figure 9. (a) Schematic view of the stacking law of a high-pressure compressor blade, (b) cross section of the blade

Figure 10 .

 10 Figure 10. Time-Frequency diagram highlighting critical area defined as a suddenly increasing of frequency amplitude

Figure 11 .

 11 Figure 11. Test bench developed in partnership with Snecma to measure contact under high velocity (300 m/s) [12]

Figure 13 .

 13 Figure 13. Schematic representation of the test bench developed to study the interaction forces [15]

Figure 14 .

 14 Figure 14. Finite Element Model of 4 sectors of fan stage and its associated displacement field

15

 15

Figure 15 .

 15 Figure 15. Finite Element Model of the 4 sectors focused on stress field near the dovetail

Figure 16 .

 16 Figure 16. Finite Element Model of a CFM fan stage, with the casing

Figure 17 .

 17 Figure 17. Schematic representation of the 2D in-plane model[START_REF] Salvat | Two-dimensional modeling of shaft precessional motions induced by blade/casing unilateral contact in aircraft engines[END_REF]


Vincent Alberge

A COMMENTARY ON TEICHMÜLLER'S PAPER

This is a commentary on Teichmüller's paper Ein Verschiebungssatz der quasikonformen Abbildung (A displacement theorem of quasiconformal mapping), published in 1944. We explain in detail how Teichmüller solves the problem of finding the quasiconformal mapping from the unit disc to itself, sending 0 to a strictly negative point on the real line, holding the boundary of the disc pointwise fixed and with the smallest quasiconformal dilatation. We mention also some consequences of this extremal problem and we ask a question.

Introduction

This is a commentary on Teichmüller's paper Ein Verschiebungssatz der quasikonformen Abbildung, published in 1944 1 (see [30]). We refer to the English translation which appears in this volume. The present paper is part of a series of commentaries written by various authors on papers of Teichmüller. These papers contain some ideas which are still unknown to Teichmüller theorists, see for example [START_REF] 'campo-Neuen | A commentary on Teichmüller's paper Veränderliche Riemannsche Flächen[END_REF], [START_REF] Alberge | A commentary on Teichmüller's paper Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF] and [START_REF] 'campo Neuen | A commentary on Teichmüller's paper Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen[END_REF]. The paper [30] is one of the last that Teichmüller wrote, and especially the last one about quasiconformal maps. In this paper, he solved the following geometric problem: Problem 1.1. Find and describe the quasiconformal map from the unit disc to itself such that

• its restriction to the unit circle is the identity map,

• the image of 0 is -x, where 0 < x < 1,

• its quasiconformal dilatation is as small as possible.

As Teichmüller wrote at the beginning of his paper, this extremal problem is rather different from those studied in [START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF]. It is due to the fact that mappings fix all boundary points and not only a finite number of such points. The paper [START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF] is at the foundation of the theory that we call now the classical Teichmüller theory.

In order to solve Problem 1.1, Teichmüller used an idea already contained in §23 and §24 of [START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF]. 2 Indeed, he obtained an equivalent problem (see Problem 4.1 in Subsection 4.1) by taking ramified coverings, which turns out to be simpler. Let us say a few words about that. First, he constructed, using explicit conformal maps, two 2-sheeted branched coverings of the unit disc, the first one branched at 0 and the other at -x. These two covering spaces can be conformally represented by ellipses with data depending on x. This construction shows that the main problem is equivalent to a problem of minimization of the quasiconformal dilatation for mappings between two ellipses with a particular condition on the boundary. After that, he showed, using the Cauchy-Schwarz inequality (as in the solution to the Grötzsch problem), that the extremal map between the ellipses is given by an affine transformation. Finally, he gave for the quasiconformal dilatation of the extremal map a lower bound depending on x and an asymptotic behaviour when x approaches 0.

In [30], Teichmüller did not give definitions; all the definitions he used are in [START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF]. This is why in these notes we will change the organization of the text in comparison with [30]; but we will keep all the ideas from Teichmüller.

After recalling some notation and definitions, especially about quasiconformal mappings, we will explain the Grötzsch problem and we will recall the notion of Grötzsch module. We will then give details on the proof of Teichmüller. We will conclude by some applications of this result.

Preliminaries

All along this paper, we shall be interested in planar quasiconformal mappings. Unless otherwise noted, all domains that we consider are connected subsets of the Riemann sphere C := C ∪ {∞}. There are several books which deal with quasiconformal mappings, see e.g. [START_REF] Ahlfors | Lectures on quasiconformal mappings. Second Edition with additional[END_REF], [START_REF] Lehto | Quasiconformal mappings in the plane[END_REF] or [START_REF] Fletcher | Quasiconformal maps and Teichmüller theory[END_REF].

We give below two equivalent definitions of quasiconformal maps. Both of them are interesting and they introduce notions (module and quasiconformal dilatation) that Teichmüller used to solve Problem 1.1.

A quadrilateral Q is a Jordan domain (i.e. a simply connected domain whose boundary is a Jordan curve) with four distinct boundary points. Sometimes, we will denote by Q (a, b, c, d) such a quadrilateral, where a, b, c and d are boundary points and we shall usually assume that these four points appear on the boundary in that order. By applying successively the Riemann Mapping Theorem, the Carathéodory Theorem 3 and a suitable Schwarz-Christoffel mapping, we know that Q is conformally equivalent (i.e. there exists a holomorphic bijection) to a rectangle 3 The theorem referred to is known as the boundary correspondance theorem.

R of vertical side length 1 and horizontal length side m, for some uniquely defined m > 0. 4 We call the module of Q, denoted by mod (Q), the number m.

A doubly-connected domain C is a connected domain whose boundary is the union of two disjoint Jordan curves. As for the quadrilateral, we can associate a module to a doubly-connected domain. We know that such a domain is conformally equivalent to an annulus whose inner radius is 1 and outer radius is R, for some R > 1. We call the module of C, denoted by mod (C), the number 1 2π log (R).

Remark 2.1. Another way to introduce the module is to define it as the inverse of extremal length of a particular family of curves. This relation enables us to extend the notion of quasiconformal mapping in higher dimensions.

Definition 2.2 (Geometric definition).

Let Ω be an open set in C. Let f : Ω → f (Ω) be an orientation-preserving homeomorphism. We say that f is quasiconformal if there exists K ≥ 1 such that for any quadrilateral

Q ⊂ Ω, mod (f (Q)) ≤ K • mod (Q) .
In this case, we set

K f := sup Q mod (f (Q)) mod (Q)
and we call it the quasiconformal dilatation 5 of f . Moreover, we say that f is K f -quasiconformal.

To simplify notation, we write q.c. instead of quasiconformal.

With this definition, it is easy to see that for f 1 and f 2 respectively K 1 -q.c. and K 2 -q.c. on suitable domains,

f 1 • f 2 is K 1 K 2 -q.c.
We can show that f is conformal if and only if K = 1. Thus, if g and h are conformal, then g • f • h has the same q.c. dilatation as f .

Before giving an equivalent definition of q.c. mappings, we recall that a map f is absolutely continuous on lines (ACL) in a domain Ω if for every rectangle R := {x + i y | a < x < b, c < y < d} in Ω, f is absolutely continuous as a function of x (resp. y) on almost all segments I y := {x + i y | a < x < b} (resp. I x := {x + i y | c < y < d}). We can show that such a function f is differientiable almost everywhere (a.e.) in Ω.

The second equivalent definition of quasiconformality is the following.

Definition 2.3 (Analytic definition). Let Ω be an open set of C. Let f : Ω → f (Ω) be a homeomorphism. We say that f is K-quasiconformal if (1) f is ACL on Ω, (2) | ∂ z f | ≤ k • | ∂ z f | (a.e), where k = K -1 K + 1 .
4 To be more precise, the interior of Q is sent (conformally) onto the upper half-plane H and this map can be extended to a homeomorphism from Q to H ∪ R ∪ {∞}. Moreover, the four distinguished points of Q are sent respectively to 0, 1, λ and ∞ for some λ > 1. Finally,

z → c z dζ ζ (ζ -1) (ζ -λ)
(for a suitable c) maps the upper half-plane onto the rectangle R of vertical length side 1 and horizontal length side m. 5 This is not the term that Teichmüller used in his papers (for example [START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF], [30] and [START_REF] Teichmüller | Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen[END_REF]). He used the term "dilatation quotient." In the current literature, we can also find the terms "maximal dilatation," "dilatation," "distorsion" or "quasiconformal norm."

We recall that

         ∂ z f = 1 2 (∂ x f -i ∂ y f ) , ∂ z f = 1 2 (∂ x f + i ∂ y f ) , Jac (f ) = | ∂ z f | 2 -| ∂ z f | 2 .
For a q.c. mapping f (in the sense of Definition 2.3), we can show that

K f = ess.sup z∈Ω | ∂ z f (z) | + | ∂ z f (z) | | ∂ z f (z) | -| ∂ z f (z) | .
According to the introduction of [START_REF] Lehto | Quasiconformal mappings in the plane[END_REF], Definition 2.3 was introduced by Morrey in [START_REF] Morrey | On the solutions of quasilinear elliptic partial differential equations[END_REF]. Moreover, Definition 2.2 is due to Ahlfors (see [START_REF] Ahlfors | On quasiconformal mappings[END_REF]). Works by Bers, Mori and Yûjôbô6 show that Definition 2.2 and Definition 2.3 are equivalent. A proof can be found in [START_REF] Ahlfors | Lectures on quasiconformal mappings. Second Edition with additional[END_REF].

Grötzsch introduced q.c. mappings in [START_REF] Grötzsch | Über die Verzerrung bei schlichten nic htkonformen Abbildungen und über eine damit zusammenhaängende Erweiterung des Picardschen Satzes[END_REF] under the name "nichtkonforme Abbildungen."7 His definition is similar to Definition 2.3 but with a stronger hypothesis: he assumes the maps differentiable. Teichmüller was the first to use q.c. mappings in a substantial manner. He developed this theory to study the Teichmüller space 8 and the so-called Teichmüller metric. Note that he used in [START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF], and [30] the Grötzsch definition for q.c. mappings in a slightly different form. He considered mappings which are differentiable "up to finitely many closed analytical curve segments." Since nothing changes in Teichmüller's proof (Proposition 4.2 below) we shall use Definition 2.3 for q.c. mappings. In particular, using the point of view of Definition 2.3, we can show that for any q.c. mapping f , ∂ z f = 0 (a.e.).

Grötzsch's problem and Grötzsch's domain

3.1. Grötzsch's problem. To conclude the solution of Problem 1.1, Teichmüller used the same idea as Grötzsch used in [START_REF] Grötzsch | Über möglichst konforme Abbildungen von schlichten Bereichen[END_REF] to solve what we call now the Grötzsch problem. In fact, Teichmüller used a kind of generalization of Grötzsch's problem in his development of classical Teichmüller theory. So, let us present this problem.

Let R 1 and R 2 be two quadrilaterals. As already mentioned at the beginning of Section 2, we can suppose that R 1 and R 2 are rectangles whose modules are respectively a 1 and a 2 (see Figure 1). The Grötzsch problem is the following:

Find and describe the q.c. mapping from R 1 to R 2 which preserves sides and with the smallest q.c. dilatation.

First, we have to find a q.c. mapping from R 1 to R 2 . The simplest map is the following

z → 1 2 1 + a 2 a 1 • z + 1 2 a 2 a 1 -1 • z, a 1 a 2 R 1 R 2 1 1
Figure 1 and its q.c. dilatation is equal to max

a 1 a 2 , a 2 a 1
. By the Cauchy-Schwarz inequality, we check that if f is a q.c. mapping between these two rectangles, then

K f ≥ max a 1 a 2 , a 2 a 1
, with equality if and only if f is the affine map above. This solves this problem. Teichmüller used the same principle. He found a q.c. mapping which can be a candidate and he showed in the same manner that this map is the one with the smallest q.c. dilatation.

3.2.

Grötzsch's domain and its associated module. A Grötzsch domain is an extremal domain for the following problem. Let R > 1 be a real number and let Ω ⊂ C be a doubly-connected domain separating the unit circle ∂D from {R, ∞}. Such a domain has a module and we want to know whether there exists a domain whose module is maximal. The answer is yes, and we can describe it. This domain, now called a Grötzsch domain, is C \ D ∪ [R, ∞) . We denote its module by 1 2π log (Φ (R)). Before Teichmüller, some facts about the map Φ were already known to Grötzsch (see [START_REF] Grötzsch | Über einige Extremalprobleme der konformen Abbildung[END_REF][START_REF] Grötzsch | Über einige Extremalprobleme der konformen Abbildung[END_REF]), like the fact that Φ : ]1, +∞[ → ]1, +∞[ is an increasing continuous function such that

(3.1) ∀R > 1; R < Φ (R) < 4R, and
(3.2) lim R→∞ (Φ (R) -4R) = 0.
According to [30], Teichmüller "proved in a purely geometric way" properties (3.1) and (3.2). See [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF] for these proofs.

We give below a functional relation with a sketch of proof. Let α be a positive number strictly less than 1. The doubly-connected domain D \ ([-α, α]) has a module which is (with our notation)

1 2π log Φ 1 2 α + 1 α .
Indeed, the image of D \ ([-α, α]) by the biholomorphism of D sending -α to 0 and α to 2α 1 + α 2 is D \ 0, 2α 1 + α 2 . By applying z → 1/z, we see that its module is exactly what we wrote. Moreover, the Grötzsch domain associated with 1/α 2 is equivalent to

C \ [-1, 1] ∪ 1 2 α 2 + 1 α 2 , ∞ .
To see this, we use the map z → 1 2 z + 1 z . Note that this map will be important in the solution of our problem and also that it is the bridge between the Grötzsch domain and what is now called the Teichmüller domain (see Chapter 3 of [START_REF] Ahlfors | Lectures on quasiconformal mappings. Second Edition with additional[END_REF]). Now by z → α 1 + α 2 (z + 1), we reach

C \ 0, 1 2 α + 1 α -1 ∪ 1 2 α + 1 α , ∞ .
This domain has, by reflection with respect to the unit circle, a module equal to

1 π log Φ 1 2 α + 1 α .
Thus, we obtain the following relation

(3.3) Φ 1 2 α + 1 α = Φ 1 α 2 .
The expressions "Grötzsch's domain" and "Teichmüller's domain" are used by Ahlfors in [START_REF] Ahlfors | Lectures on quasiconformal mappings. Second Edition with additional[END_REF] and also by Lehto and Virtanen in [START_REF] Lehto | Quasiconformal mappings in the plane[END_REF] whose German version was published in 1965. The author of this report does not know who was the first person to use this terminology.

4. The solution of Teichmüller's problem 4.1. A simple case. For two strictly positive real numbers α and β, we denote by E (α, β) the ellipse whose centre is the origin and the major (resp. minor) axis is equal to α (resp. β). We want to solve the following extremal problem: Problem 4.1. Is there a q.c. mapping from E (α, β) to E (β, α) with the smallest q.c. dilatation and whose restriction to the boundary coincides with the restriction of h 0 : z → 1 2

α β + β α • z + 1 2 α β - β α • z ?
It is easy to show that h 0 sends E (α, β) to E (β, α) with the good behaviour at the boundary. Moreover, its q.c. dilatation is equal to max α 2 β 2 , β 2 α 2 . We will show that this is the smallest q.c. dilatation with the given conditions. Proposition 4.2. Let f : E (α, β) → E (β, α) be a q.c. mapping such that f | ∂E(β,α) coincides with the restriction of h 0 . Then,

K f ≥ max α 2 β 2 , β 2 α 2 .
Proof. As in the solution of the Grötzsch problem, we can suppose that f is continuously differentiable in both directions. Let y ∈ ]-β, β[. We denote by l (y) the Euclidean length of the segment Im (z) = y in E (α, β). We parametrize this segment by γ y : t ∈ -l (y) 2 , l (y) 2

. Due to the hypothesis on f | ∂E(β,α) , the length of f • γ y is bigger than α β • l (y). We have the following inequality

(4.4) α β • l (y) ≤ l(y) 2 -l(y) 2 | (f • γ y ) ′ (t) |dt. But (f • γ y ) ′ (t) = ∂ z f (γ y (t)) • γ ′ y (t) + ∂ z f (γ y (t)) • γ ′ y (t), so | (f • γ y ) ′ (t) | ≤ (| ∂ z f (γ y (t)) | + | ∂ z f (γ y (t)) |) • |γ ′ y (t) | = | ∂ z f (γ y (t)) | + | ∂ z f (γ y (t)) | | ∂ z f (γ y (t)) | -| ∂ z f (γ y (t)) | • | ∂ z f (γ y (t)) | 2 -| ∂ z f (γ y (t)) | 2 1 2 ≤ (K f • Jac (f ) (γ y (t))) 1 2 . (4.5)
By applying the Cauchy-Schwarz inequality in (4.4) and using (4.5), we obtain

α 2 β 2 • l (y) ≤ K f l(y) 2 -l(y) 2 Jac (f ) (γ y (t)) dt.
Integration with respect to y gives us

α 2 β 2 ≤ K f .
If we replace the horizontal segment by the vertical segment in the ellipse, by the same method we obtain β 2 α 2 ≤ K f , and so, the proof is complete.

E (α, β) E (β, α) α α β β Figure 2.
Like in the case of rectangles, we are looking for the q.c. mapping with the smallest q.c. dilatation and a good behaviour at the boundary.

4.2.

Solution. We will explain how Teichmüller showed that Problem 1.1 is equivalent to Problem 4.1 for some (α, β) that we shall specify. We start with D \ [-x, 0]. Its preimage by the covering map, z

→ z 2 is D \ i - √ x, √
x . The interior of the latter domain can be conformally sent by ϕ onto an annulus of inner radius 1 and outer radius R. To see this, we have to map (conformally) the first quadrant of the unit disc onto the first quadrant of this annulus for some R. Such a map exists by the same arguments given in Section 2. By successive reflections with respect to the horizontal and the vertical axes, we can define ϕ. Note that according to (3.3),

(4.6) R = Φ 1 x .
We must say that Teichmüller used a nicer method to obtain (4.6). Indeed, this relation is given by the following commutative diagram

D \ i [- √ x, √ x] ϕ / / z →z 2 C (1, R) D \ [-x, x] / / C 1, Φ 1 x where C (1, R) (resp. C 1, Φ 1 x
) denotes the annulus whose inner radius is 1 and outer radius is R (resp. Φ 1 x ).

Finally,

f 1 : z → z - 1 z and f 2 : z → z + 1 z map the annulus C (1, R) onto E R - 1 R , R + 1 R \ i [-2, 2] and E R + 1 R , R - 1 R \ [-2, 2
] respectively. To simplify notation, we set

E 1 := E R - 1 R , R + 1 R and E 2 := E R + 1 R , R - 1 R .
Thus, we have two new maps,

p 1 := ϕ -1 • f -1 1 2 and p 2 := ϕ -1 • f -1 2
2 . The mapping p 1 (resp. p 2 ) can be extended to a map from E 1 (resp. E 2 ) to D such that 0 is sent to 0 (resp. -x). We denote the associated maps again by p 1 and p 2 .

For more details, see Figure 3. Note that in Teichmüller's paper [30], there is an equivalent figure. Now, we remark that p 1 : E 1 → D (resp. p 2 : E 2 → D) is a two-sheeted ramified covering, where the branch point is 0 (resp. -x).

We now have all the elements to solve our problem. We start by recalling the problem. Let f be a q.c. mapping from D to D such that f (0) = -x and f | ∂D = id ∂D . Since f maps the branch point 0 to the branch point -x, we can lift it. We denote this lift by f (see Figure 4). It is easy to check that f : E 1 → E 2 is a q.c. mapping, with the same q.c. dilatation as f . Furthermore, f (0) = 0 and its restriction to

0 0 0 0 -x i √ x -i √ x D f -1 1 f -1 2 ϕ -1 z → z 2 E 1 E 2 1 R Figure 3. We have two different covering spaces of D \ [-x, 0]
given by E 1 and E 2 . We distinguish by crosses and points the successive inverse images of -x and 0.

the boundary coincides with the restriction to the boundary of the affine map

f0 : x + i y → R + 1 R R -1 R x + i R -1 R R + 1 R y.
Since this map is symmetric with respect to 0, we can conclude by construction of these two 2-sheeted ramified coverings that it descends to a q.c. mapping f (0,x) whith the same q.c. dilatation as f0 . Furthermore, f (0,x) satisfies the conditions of Problem 1.1 and by using Proposition 4.2 we conclude that (4.7)

K f ≥ R 2 + 1 R 2 -1 2 ,
with equality if and only if f = f (0,x) . We call f (0,x) the extremal map for Problem 1.1.

It is important to note that the existence and the uniqueness of such a mapping cannot be deduced from the so-called Teichmüller theorem, whose the first statement can be found in [START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF]. 9 Indeed, in the present case, all points (and not only a finite number) on the boundary are fixed.

However, let us observe the following interesting fact. We recall that the Beltrami differential associated with a q.c. mapping f : D → D is an element of L ∞ (D) which is defined by

µ f := ∂ z f ∂ z f . As f (0,x) • p 1 = p 2 • f0 , we conclude by using conformality of p i (i = 1, 2) that µ f (0,x) • p 1 = p ′ 1 |p ′ 1 | 2 • k (0,x) ;
where

k (0,x) = K f (0,x) +1 K f (0,x) -1
. In other words, we have

(4.8) µ f (0,x) = k (0,x) • φ |φ| ,
where φ is a meromorphic function on D with a pole of order 1 at 0. Through Equality (4.8), the knowlegeable reader will recognize the general expression of what we call the Teichmüller mapping. By the way, there are works of Strebel where Equality (4.8) is a consequence of the so-called Frame Mapping Criterion.

We refer to [START_REF] Strebel | Point shift differentials and extremal quasiconformal mappings[END_REF][START_REF] Strebel | On the existence of extremal Teichmüller mappings[END_REF]. See also [START_REF] Reich | Extremal quasiconformal mappings of the disk[END_REF] (p. 124). We have also to mention §159 of [START_REF] Teichmüller | Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen[END_REF] where Teichmüller had already guessed that the extremal map statisfies Equation (4.8). In fact, Teichmüller explained that for a given homeomorphism of the disc (i.e. a condition for all boundary points) we can always extend this map to a map with the smallest q.c. dilation and which is related to a quadratic differential by an equation of type (4.8). Note that this is at the idea of what is called the non-reduced Teichmüller theory and for which Problem 1.1 is an example.

Before getting further, let us note that to solve Problem 1.1, we are in a situation equivalent to the Grötzsch problem. Moreover, to solve the Grötzsch problem for two rectangles

Q 1 (a 1 , b 1 , c 1 , d 1 ) and Q 2 (a 2 , b 2 , c 2 , d 2 )
, we only need to consider q.c. mappings which send a 1 to a 2 , b 1 to b 2 , c 1 to c 2 and d 1 to d 2 . With this in mind, we see that if f is a q.c. mapping from D to itself sending 0 to -x, preserving the boundary and fixing 1 and -1, then its lift f : E 1 → E 2 sends the four extremal

0 0 0 -x D D f f p 1 p 2 E 1 E 2 Figure 4.
A q.c. mapping f from D to D such that f (0) = -x can be lifted to a q.c. mapping with the same q.c. dilatation.

points of E 1 to the four extremal points of E 2 . Thus, let us ask the following question:

Problem 4.3. Is it possible to find and describe the quasiconformal mapping from D to itself such that • the images of 1, -1 and 0 are respectively 1, -1 and -x where 0 < x < 1,

• its quasiconformal dilatation is smallest possible? 4.3. First consequences. We have just seen that the extremal map f (0,x) : D → D for Problem 1.1 has a q.c. dilation K f (0,x) equal to

R 2 + 1 R 2 -1 2
. Moreover, as we wrote above, R = Φ 1 x , and so (4.9)

K f (0,x) = Φ 1 x + 1 Φ 1 x -1 2 .
By using (3.2) in (4.3), we conclude that (4.10)

K f (0,x) = 0 + 1 + x + o 0 + (x) , which means that K f (0,x) -1 -x x -→ x→0 + 0.
On the other hand, the right hand side of (3.1) gives us

K f (0,x) > 1 + x 2 2 .
The last inequality leads us to Corollary 4.4. Let f : D → D be a q.c. mapping such that f | ∂D = id ∂D and f (0

) ∈ ]-1, 0]. Then |f (0) | ≤ 2 K 1 2 f -1 .
Note that there is a small mistake in Teichmüller's paper which is considered as a "misprint" by Earle and Lakic in [START_REF] Earle | Variability sets on Riemann surfaces and forgetful maps between Teichmüller spaces[END_REF]. Indeed, Teichmüller forgot to take the power 2 in the right hand side of Relation (4.9) and so he obtained a different asymptotic behaviour in (4.10) and a different upper bound in Corollary 4.4. The same mistake appears in [START_REF] Gehring | Quasiconformal mappings which hold the real axis pointwise fixed[END_REF][START_REF] Kra | On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces[END_REF]. 10 About this mistake, we refer also to the editor footnote of [START_REF] Teichmüller | Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen[END_REF].

We denote the hyperbolic distance 11 on the disc by d D (., .). According to (4.3), we can express the q.c. dilatation of f (0,x) with respect to this distance by the following formula:

(4.11) log K f (0,x) = 2 • d D 0, 1 Φ 1 x .

Some applications

In this section, we mention some applications of Teichmüller's result obtained by various authors. 5.1. Kra's distance. Before setting the Kra distance, we show an easy extension of Teichmüller's result. By extension, we mean to find for any distinct pair of points in D the q.c. mapping from D to D sending one point to the other, keeping the boundary pointwise fixed and with the smallest q.c. dilatation.

Let z 1 and z 2 be two distinct points in D. We denote by ϕ (z1,z2) the biholomorphism of the disc which sends z 1 to 0 and z 2 to some point -x, 0 < x < 1. It is easy to check that the extremal map 12 is

f (z1,z2) = ϕ (z1,z2) • f (0,x) • ϕ -1
(z1,z2) , where f (0,x) is the previous extremal map. The Beltrami differential of f (z1,z2) is related to a meromorphic function on the disc with a pole of order 1 at z = z 1 by a relation analogous to (4.8). Strebel calls in [START_REF] Strebel | Point shift differentials and extremal quasiconformal mappings[END_REF] such a mapping the Teichmüller shift.

Furthermore, (5.12)

d : (z 1 , z 2 ) ∈ D × D → 1 2 log K f (z 1 ,z 2 )
defines a new distance on the disc. Moreover, as Kra observed in [START_REF] Kra | On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces[END_REF], d is a complete metric.

We have all the ingredients to define the Kra distance. Let S be a hyperbolic Riemann surface of finite type (g, n), where g is the genus and n the number of punctures. We recall that a hyperbolic Riemann surface is a Riemann surface whose universal cover is the unit disc. This implies in particular that S carries a hyperbolic metric. Kra defined in [START_REF] Kra | On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces[END_REF] a new distance on S as follows. For any two points x and y in S, he sets (5.13)

d Kr (x, y) := 1 2 log inf f K f ,
where the infimum is taken over all q.c. mappings13 f isotopic to the identity mapping and sending x to y. This distance is now called the Kra distance. 14 From a compactness property of q.c. mappings we know that there always exists a q.c. mapping which attains the infimum in (5.13). Kra obtained the uniqueness of such a mapping if x and y are close enough for the hyperbolic metric on S (see Proposition 6. in [START_REF] Kra | On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces[END_REF]). Furthermore, he showed that d Kr is equivalent to the hyperbolic metric but not proportional to it unless S is the thrice-punctured sphere. In this exceptional case the two metrics coincide. It is of interest to note that the idea of Kra's distance already exists in [START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF]. Indeed, Teichmüller introduced such a distance and he showed in §27 that it coincides with the hyperbolic distance in the case of the thrice-punctured sphere. In the same paper, Teichmüller explained in §160 that up to a condition, S equipped with d Kr is a Finsler manifold. where Q D (K) denotes the set of all K-q.c. mappings from D to D which hold the boundary pointwise fixed. This problem can be addressed for any planar domain Ω with at least 3 boundary points. Indeed, for such a domain, we know that the universal cover is the unit disc, so by pushing forward the hyperbolic metric on the disc, we can define a hyperbolic metric of constant curvature -1 on Ω. We denote it by d Ω (•, •) and we may want to determine the value of (5.14) by considering d Ω instead of d D .

Krzyz gave a value for (5.14) in [START_REF] Krzyz | On an extremal problem of F. W. Gehring[END_REF]. He proved that there exists z 0 ∈ D and f K ∈ Q D (K) such that h D (K) = d D (z 0 , f K (z 0 )) .

He gave a precise value of h D (K) and he showed, by using an analogue of Corollary 4.4, that f K is the extremal map with respect to the Teichmüller problem (i.e. the extension of Problem 1.1 where the pair of points is (z 0 , f K (z 0 )).

Later, Solynin and Vuorinen showed in [START_REF] Solynin | Estimates for the hyperbolic metric of the punctured plane and applications[END_REF] that the supremum of (5.14) is attained for a unique map, the map described above.

The Gehring problem can also be addressed for domains in R n , where n > 2. See for example [START_REF] Vuorinen | A remark on the maximal dilatation of a quasiconformal mapping[END_REF] and [START_REF] Manojlović | On quasiconformal maps with identity boundary values[END_REF]. These two papers are related to a paper of Martin [START_REF] Martin | The distorsion theorem for quasiconformal mappings, Schottky's theorem and holomorphic motions[END_REF]. Furthermore, Martin worked in [START_REF] Martin | The Teichmüller problem for mean distortion[END_REF] on an extremal problem close to Teichmüller's one. To be more precise, he considered for 0 ≤ x < 1, the value He showed that if x > 0, the infimum in (5.15) cannot be attained by a q.c. mapping.

5. 2 .

 2 About a problem of Gehring and a little bit more. The Gehring problem, which could be seen as a dual of Problem 1.1, is the following. Given K > 1, we want to describe the value(5.14) h D (K) := sup {d D (z, f (z)) | z ∈ D and f ∈ Q D (K)};

  dz | f is q.c., f (0) = -x and f | ∂D = id ∂D .

Note that this paper appeared after his death which occured in 1943 on the Eastern Front.

Even if the idea was already used, the result was not known from specialists. We refer to[START_REF] Kühnau | Some historical commentaries on Teichmüller's paper "Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF] and especially to what Grötzsch told to Kühnau about this paper: "Ja...ah, das habe ich nicht gehabt [Okay, this I did not have]."

This list is not exhaustive.

The name quasiconformal is now credited to Ahlfors, but this is debatable (see the commentary related to[START_REF] Ahlfors | Zur Theorie der Überlagerungsflächen[END_REF] p. 213).

Teichmüller proved in[START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF] the uniqueness. The existence is stated there as a "conjecture." Teichmüller proved existence (for closed surfaces) later in[START_REF] Teichmüller | Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen[END_REF]. We refer to the corresponding commentaries[START_REF] Alberge | A commentary on Teichmüller's paper Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF] and[START_REF] 'campo Neuen | A commentary on Teichmüller's paper Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen[END_REF].

I. Kra informed the author of the way he discovered this mistake. Kra needed some consequences of Teichmüller results to write[START_REF] Kra | On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces[END_REF] and he used for this the Gehring paper[START_REF] Gehring | Quasiconformal mappings which hold the real axis pointwise fixed[END_REF]. Gehring found this error only after publishing[START_REF] Gehring | Quasiconformal mappings which hold the real axis pointwise fixed[END_REF] and when he knew that Kra used his paper, he informed him.

[START_REF] Grötzsch | Über einige Extremalprobleme der konformen Abbildung[END_REF] We use the metric with constant curvature

-1.[START_REF] Grötzsch | Über einige Extremalprobleme der konformen Abbildung[END_REF] We mean here the map with the smallest q.c. dilatation, sending z 1 to z 2 and keeping the boundary pointwise fixed.

A q.c. mapping on a Riemann surface is a mapping whose a lift to the universal cover is a q.c. mapping and the q.c. dilatation is the q.c. dilatation of this lift.

It seems that this name appears for the first time in[START_REF] Shen | Some notes on Teichmüller shift mappings and the Teichmüller density[END_REF].

Acknowledgements. The author would like to express his sincere gratitude to Professor Athanase Papadopoulos for giving an opportunity for writing this commentary and especially for his patience and kindness. The author wants to thanks Professor Ken'ichi Ohshika for his helpful comments on a previous version. The author also thanks Professor Irwin Kra for his interest and for a correspondence on these results. The author would like to thank Professor Hideki Miyachi for his suggestions.

This work was partially supported by the French ANR grant FINSLER. The author acknowledges support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 "RNMS: GEometric structures And Representation varieties" (the GEAR Network).