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let @xmath0 be a closed surface and @xmath1 be a riemannian metric on @xmath0 . then the laplace - beltrami operator @xmath2 acts on the space of smooth functions on @xmath0 by the formula @xmath3 it is known that the spectrum of @xmath2 is discrete and consists only of eigenvalues . moreover , the multiplicity of any eigenvalue is finite and the sequence of eigenvalues tends to infinity . let us denote this sequence by @xmath4 where the eigenvalues are written with their multiplicities .    for a fixed @xmath0 the following quantities can be considered as functionals on the space of all riemannian metrics on @xmath0 , @xmath5 several recent papers  @xcite deal with finding supremum of these functionals in the space of all riemannian metrics on @xmath0 . an upper bound for @xmath6 in terms of genus of @xmath0 was provided in the paper  @xcite and the existence of such a bound for @xmath7 was shown in the paper  @xcite . the exact upper bounds are known for a limited number of functionals : @xmath8 ( see  @xcite ) , @xmath9 ( see  @xcite ) , @xmath10 ( see  @xcite ) , @xmath11 ( see  @xcite ) , @xmath12 ( see  @xcite ) . we refer to the introduction to the paper  @xcite for more details . the functional @xmath7 depends continously on @xmath1 but this functional is not differentiable . however , it is known that for an analytic deformation @xmath13 of the initial metric @xmath1 there exist the left and right derivatives of @xmath14 with respect to @xmath15 , see e.g. the papers  @xcite . this is a motivation for the following definition . a riemannian metric @xmath1 on a closed surface @xmath0 is called an _ extremal _ metric for a functional @xmath7 if for any analytic deformation @xmath13 such that @xmath16 the following inequality holds , @xmath17    a metric @xmath1 is called a _ metric for a functional @xmath7 if for any metric @xmath18 on @xmath0 @xmath19    a question whether there exists smooth maximal metric is not trivial . for example there is no smooth maximal metric for @xmath12 ( see  @xcite ) . the list of known extremal metrics is longer than the list of known exact upper bounds for @xmath7 , but until now their maximality has not been studied . in the present paper we investigate the maximality of the known extremal metrics . the list of currently known extremal metrics follows . * metrics on the otsuki tori @xmath20 were studied in the paper  @xcite . * metrics on the lawson tori and klein bottles @xmath21 were studied in the paper  @xcite . * metrics on the surfaces @xmath22 bipolar to lawson surfaces were studied in the paper  @xcite . * metrics on the bipolar surfaces @xmath23 to otsuki tori were studied in the paper  @xcite .    in further description a klein bottle is denoted by @xmath24 . the definitions of these surfaces are given in the following sections . the main result of the present paper is the following theorem . there are no maximal metrics among the metrics ( a)-(d ) except for @xmath25 . [ maintheorem ]    the metric on the lawson bipolar klein bottle @xmath25 is maximal for the functional @xmath26 , see  @xcite . we also prove the following proposition . the metric on the clifford torus is extremal for an infinite number of functionals @xmath7 , but it is not maximal for any of them . [ cliffordtheorem ]    the extremality of the clifford torus for an infinite number of functionals @xmath7 is known , but to the best of author s knowledge has not yet been published . in the present paper we fill this gap .    in the following description we use the notations @xmath27 , @xmath28 and @xmath29 for the elliptic integrals of the first , second and third kind respectively , see  @xcite , @xmath30    the paper is organized in the following way . in section  [ bound ] we prove lower bounds for @xmath31 and @xmath32 . these bounds are used throughout the paper in order to prove the non - maximality of metrics ( a)-(d ) . in section  [ connection ] we recall a connection between extremal metrics and minimal submanifolds of the unit sphere . section  [ otsukidef ] contains a discription of otsuki tori as an @xmath33-invariant minimal submanifolds of @xmath34 of cohomogeneity 1 . sections  [ otsukiestimate ] , [ lawsonestimate ] , [ biplawsonestimate ] , [ bipotsukiestimate ] are dedicated to estimates for extremal metrics ( a)-(d ) respectively and this finishes the proof of theorem  [ maintheorem ] . finally , section  [ cliffordestimate ] contains the proof of proposition  [ cliffordtheorem ] . the aim of this section is to prove the following proposition ( compare with corollary 4 in the paper  @xcite ) . [ bound ]    one has the following inequalities , @xmath36 @xmath37 where @xmath28 stands for the elliptic integral of the second kind . [ lowerbound ]      let @xmath0 be a compact smooth riemannian manifold of dimension @xmath38 . let us pick two distinct points @xmath39 . for @xmath40 we define    * @xmath41 union of open geodesic balls of radius @xmath42 about @xmath43 and @xmath44 , * @xmath45 , * @xmath46 .    here the number @xmath42 is chosen to be less than @xmath47 of injectivity radius of @xmath0 and less than @xmath47 of a distance between @xmath43 and @xmath44 if @xmath43 and @xmath44 lie in the same connected component of @xmath0 . we say that manifold @xmath48 is obtained from @xmath0 by adding a handle across @xmath49 if    * @xmath50 is isometrically embedded in @xmath48 ; * there exists a diffeomorphism @xmath51\times \mathbb{s}^{n-1}$ ] such that @xmath52\times\mathbb{s}^{n-1}\right).\ ] ]    let us denote by @xmath53 and @xmath54 the laplace spectrum of @xmath0 and @xmath48 respectively . chavel and feldman in their paper  @xcite obtained a sufficient condition for convergence @xmath55 as @xmath42 tends to @xmath56 . in order to formulate this condition we need to give the following definition .    for any compact connected riemannian manifold @xmath57 of dimension @xmath38 , the isoperimetric constant @xmath58 is defined by @xmath59 where @xmath60 stands for @xmath61-dimensional riemannian measure , and @xmath62 ranges over all compact @xmath63-dimensional submanifolds of @xmath57 such that they divide @xmath57 into @xmath64 open submanifolds @xmath65 , @xmath66 each having boundary @xmath62 . assume that @xmath48 is connected for any @xmath42 and there exists a constant @xmath67 such that @xmath68 for all @xmath69 . then @xmath70 for all @xmath71 [ ch1 ]    the assumption in the previous theorem implies @xmath72 by picking @xmath73 .    in the same paper existence of such @xmath48 is verified for any surface @xmath0 and almost any pair of points @xmath43 , @xmath44 . let @xmath0 be a compact @xmath64-dimensional riemannian manifold , @xmath74 be its gaussian curvature and @xmath75 . then @xmath76 is open and dense in @xmath0 . suppose that @xmath77 and one of the following possibilities occur :    * @xmath0 is connected , * @xmath0 has two connected components and @xmath78 lie in different connected components .    then @xmath48 can be constructed so that assumption of theorem  [ ch1 ] holds . in particular , @xmath79 as @xmath80 . [ ch2 ]    let us remark that chavel and feldman considered only the case of a connected manifold @xmath0 . however , their arguments could be extended almost without changes to the non - connected case as stated above . consider the flat equilateral torus @xmath81 . after suitable rescaling of the metric we have @xmath82 and @xmath83 . for the euclidean sphere @xmath84 of volume @xmath85 one also has @xmath83 . let us take @xmath86 copies of @xmath84 denoted by @xmath87 , @xmath88 . thus for @xmath89 we have @xmath90 and therefore @xmath91 . consecutive application of theorem  [ ch2 ] yields the existance of the sequence @xmath48 , diffeomorphic to torus , such that @xmath92 as @xmath42 tends to @xmath56 . this observation completes the proof of the first inequality . the second inequality can be proved in the same fashion . the only difference is that instead of @xmath81 one has to use lawson bipolar klein bottle @xmath25 ( see section  [ biplawsonestimate ] for a defintion ) . it was proven in the paper  @xcite that @xmath93 . by a suitable rescaling of the metric on @xmath25 , one can assume that @xmath94 and then apply construction of the previous paragraph . let @xmath95 be a minimal immersion in the unit sphere with canonical metric @xmath96 . we denote by @xmath2 the laplace - beltrami operator on @xmath0 associated with the metric @xmath97 . let us introduce the weyl s eigenvalues counting funcion @xmath98 the following theorem provides a general approach to finding smooth extremal metrics . let @xmath99 be a minimal immersion in the unit sphere @xmath100 endowed with the canonical metric @xmath96 . then the metric @xmath97 on @xmath0 is extremal for the functional @xmath101 . [ th1 ]    therefore , one can start with minimal submanifold @xmath102 of the unit sphere then compute @xmath103 and the metric induced on @xmath102 by this immersion is extremal for the functional @xmath104 . however , for a given minimal submanifold there is no algorithm for computing the exact value of @xmath103 . nevertheless , this approach was succesfully realized by penskoi in the papers  @xcite for metrics ( a),(b ) as well as by the author in the paper  @xcite for metrics ( d ) . some ideas of this approach was partially used in the paper  @xcite for metrics ( c ) . let @xmath0 be a [ connection ] riemannian manifold equipped with a metric @xmath105 and let @xmath106 be a compact group acting on @xmath0 by isometries . for every point @xmath107 let us denote by @xmath108 the stability subgroup of @xmath109 .    for two points @xmath110 we say that @xmath111 if @xmath112 for some @xmath113 . the orbit @xmath114 is the _ orbit of principal type _ if for any point @xmath115 one has @xmath111 . let @xmath116 stand for the union of all orbits of principal type , then @xmath116 is an open dense submanifold of @xmath0 ( see  @xcite ) . moreover , @xmath117 carries a natural riemannian metric @xmath1 defined by the formula @xmath118 , where @xmath119 are tangent vectors at @xmath120 and @xmath121 are tangent vectors at a point @xmath122 such that @xmath123 and @xmath124 are orthogonal to the orbit @xmath125 and @xmath126 . let @xmath127 be a @xmath106-invariant immersed submanifold , i.e. a manifold equipped with an action of @xmath106 by isometries such that @xmath128 for any @xmath129 . a _ cohomogeneity _ of a @xmath106-invariant immersed submanifold @xmath102 is the number @xmath130 , where @xmath131 is the dimension of the orbits of principal type . let us define for @xmath120 a volume function @xmath132 by the formula @xmath133 . also for each integer @xmath134 let us define a metric @xmath135 . let @xmath136 be a @xmath106-invariant immersed submanifold of cohomogeneity @xmath61 , and let @xmath117 be equipped with the metric @xmath137 . then @xmath136 is minimal if and only if @xmath138 is minimal . [ hsiangth ]      otsuki tori [ otsukidef ] were introduced by otsuki in the paper  @xcite . let us recall the concise description by penskoi from the paper  @xcite . for more details see section 1.2 of the paper  @xcite . consider the action of @xmath33 on the three - dimensional unit sphere @xmath139 given by the formula @xmath140 where @xmath141 is a coordinate on @xmath33 . the space of orbits @xmath142 is the closed half - sphere @xmath143 , @xmath144 where a point @xmath145 corresponds to the orbit @xmath146 . the space of principal orbits @xmath147 is the open half sphere @xmath148 let us introduce the spherical coordinates in the space of orbits , @xmath149 since we look for minimal submanifolds of cohomogeneity 1 , the hsiang - lawson s metric is given by the formula @xmath150    an immersed minimal @xmath33-invariant two - dimensional torus in @xmath34 such that its image by the projection @xmath151 is a closed geodesics in @xmath147 endowed with the metric ( [ otsukimetric ] ) is called an _ otsuki torus_. the following proposition was proved in the paper  @xcite . except one particular case given by the equation @xmath152 , otsuki tori are in[penskoiprop ] one - to - one correspondence with rational numbers @xmath153 such that @xmath154    by @xmath20 we denote the otsuki torus corresponding to @xmath153 . following the paper @xcite we reserve the term `` otsuki tori '' for the tori @xmath20 .    in order to fix notations we give a sketch of the proof of proposition  [ penskoiprop ] . let us use the standard notation for the coefficients of the metric  ( [ otsukimetric ] ) , @xmath155 as we know the velocity vector of a geodesic has a constant length . suppose this length equals 1 . then this assumption as well as the equation of geodesics for @xmath156 provides the following two equations , @xmath157 @xmath158 where @xmath159 is the minimal value of @xmath160 on the geodesic . then the geodesic is situated in the annulus @xmath161 . we choose a natural parameter @xmath15 such that @xmath162 . let us denote by @xmath163 the difference between the value of @xmath164 corresponding to @xmath165 and the closest to it value of @xmath164 corresponding to @xmath166 . it is clear that @xmath167    the geodesic is closed iff @xmath168 . the rest of the proof follows from the following properties of the function @xmath163 , see the paper  @xcite ,    * @xmath163 is continuous and monotonous on @xmath169 $ ] , * @xmath170 and @xmath171 .      according to the paper  @xcite , the metric on an otsuki torus @xmath20 is extremal [ otsukiestimate ] for the functional @xmath173 . the goal of this section is to prove the following proposition .    for @xmath174 , such that @xmath175 and @xmath176 , the following inequality holds , @xmath177 [ otsukiestimateprop ]    in order to prove proposition  [ otsukiestimateprop ] we have to prove several auxiliary propositions .    for @xmath178 such that @xmath168 one has @xmath179 [ valueotsuki ]    let us use the notations of proposition  [ penskoiprop ] . as we know , @xmath180 where @xmath181 . therefore , the length of the segment on the geodesic @xmath182 between the closest points with @xmath165 and @xmath183 is equal to @xmath184 , where @xmath185    let us express @xmath186 in terms of elliptic integrals , @xmath187    here the following changes of variables were used , @xmath188 since the maps @xmath189 and @xmath190 are isometries , the length of the geodesic @xmath182 is equal to @xmath191 . by proposition 13 from the paper  @xcite , @xmath172 is equal to the doubled length of the geodesic @xmath182 .    for @xmath192 $ ] one has the following inequality , @xmath193 [ mainlemma ]    let us expand the left hand side using the definitions of @xmath194 and @xmath74 , @xmath195    since the integrand is negative on @xmath196 and positive on @xmath197 , one has @xmath198    let us introduce the notation @xmath199    the function @xmath200 is non - decreasing and @xmath201 for any @xmath202 . in particular , @xmath203 . [ phiprop ] one has @xmath204 [ ineqotsuki ]    * remark . * let us remark that during the preparation of the manuscript inequality  ( [ inotsuki ] ) appeared in the paper  @xcite . let us recall the following formulae for the derivatives of elliptic integrals , @xmath205 @xmath206    let us introduce a notation @xmath207 . one obtains @xmath208 now the monotonicity of the function @xmath200 follows from proposition  [ mainlemma ] .    for the proof of the second part , let us go back to formula  ( [ dphi ] ) . one has @xmath209    this finishes the proof of proposition  [ phiprop ] . the function @xmath210 is increasing on the interval @xmath211 .    in the paper  @xcite the following formula was proved , @xmath212 using formulae  ( [ dpi ] ) one obtains the following formula , @xmath213    let us recall the notation @xmath214 . then one has @xmath215 @xmath216    moreover , by formula  ( [ dphi ] ) one has @xmath217 the inequality @xmath218 and proposition  [ mainlemma ] imply the inequality @xmath219    for @xmath220 $ ] one has @xmath221 [ cor1 ]    using the tables of elliptic integrals , e.g. the book  @xcite , one obtains the inequality @xmath222 the rest of the proof follows from the monotonicity of the function on the left hand side .    for @xmath223 $ ] one has @xmath224 [ prop1 ]    by formula  ( [ domega ] ) for @xmath223 $ ] one has @xmath225 in the last inequality we used the facts that @xmath27 is increasing function and @xmath28 as well as @xmath226 are decreasing functions . the table of the elliptic integrals in the book  @xcite provides the inequality @xmath227 which completes the proof . we want to prove that @xmath228 where @xmath168 . this inequality is equivalent to the following one @xmath229 since @xmath168 , it is sufficient to prove that @xmath230 since @xmath231 , the application of corollary  [ cor1 ] provides inequality  ( [ mainineq ] ) for @xmath220 $ ] . in order to prove inequality for @xmath232 $ ] let us note that by proposition  [ phiprop ] @xmath233 for some @xmath234 . moreover , @xmath235 or @xmath236 therefore , inequality  ( [ mainineq ] ) follows from the inequality @xmath237 or the inequality @xmath238 the last inequality easily follows from proposition  [ prop1 ] . a lawson tau - surface @xmath21 is an immersed surface in the sphere @xmath34 [ lawsonestimate ] defined by the double - periodic immersion of @xmath239 given by the formula @xmath240    it was introduced by lawson in the paper  @xcite . he also proved that for each pair @xmath241 , such that @xmath242 and @xmath243 , the surface @xmath21 is a distinct compact minimal surface in @xmath34 . let us assume that @xmath243 then if both @xmath244 and @xmath61 are odd then @xmath21 is a torus , we call it a lawson torus . otherwise @xmath21 is a klein bottle , we call it a lawson klein bottle . let @xmath21 be a lawson surface . then the induced metric on @xmath21 is an extremal metric for the functional @xmath245 , where @xmath246 + m + k - 1 , \label{j}\ ] ] @xmath247 if both @xmath248 are odd and @xmath249 otherwise . the corresponding value of the functional is @xmath250    let @xmath251 be defined by formula  ( [ j ] ) . if @xmath21 is a lawson torus , then @xmath252 if @xmath21 is a klein bottle , then @xmath253 [ lawsonprop ]    it is sufficient to obtain the inequality @xmath254    let us remark that the function @xmath255 is positive on the interval @xmath256 $ ] . indeed , @xmath257    let us divide both sides of inequality ( [ lawsonineq ] ) by @xmath244 and denote by @xmath109 the ratio @xmath258 $ ] . since @xmath259\geqslant \left[\frac{m+k}{2}\right]\geqslant\left[\frac{m+1}{2}\right ] > \frac{m}{2},\ ] ] one has that inequality ( [ lawsonineq ] ) follows from the positivity of @xmath260 let @xmath261 be a minimal immersion . [ biplawsonestimate ] a gauss map @xmath262 is defined pointwise as the image of the unit normal in @xmath34 translated to the origin in @xmath263 . then the exterior product @xmath264 is an immersion of @xmath102 in @xmath265 . lawson proved in the paper  @xcite that this immersion is minimal . the image @xmath266 is called a bipolar surface to @xmath102 . let us denote by @xmath267 the bipolar surface to the surface @xmath21 . lapointe proved in the paper  @xcite that    * if @xmath268 then @xmath267 is a torus carrying the extremal metric for the functional @xmath269 and @xmath270 * if @xmath271 then @xmath267 is a torus carrying the extremal metric for the functional @xmath272 and @xmath273 * if @xmath274 then @xmath267 is a klein bottle carrying the extremal metric for the functional @xmath275 and @xmath276    if @xmath277 then the following inequality holds @xmath278 if @xmath279 then the following inequality holds @xmath280 if @xmath274 and @xmath281 then the following inequality holds @xmath282 [ biplawsonprop ]    in order to prove the first inequality it is sufficient to prove that @xmath283 it is well - known that @xmath284 for @xmath285 $ ] . this implies that it is sufficient to prove that @xmath286 this inequality holds for @xmath287 . the statement for @xmath288 follows from the fact that @xmath289 is a clifford torus and @xmath290 .    in the same way , in order to prove the second inequality in proposition  [ biplawsonprop ] it is sufficient to prove that @xmath291 this inequality holds for @xmath292 . the third inequality is equivalent the following one @xmath293 since @xmath294 it is sufficient to prove that @xmath295 this inequality holds for @xmath296 . for the exceptional case @xmath297 one verifies the third inequality explicitly using the tables of elliptic integrals in the book  @xcite . in the paper  @xcite the following proposition was proved . [ bipotsukiestimate ]    the bipolar surface @xmath23 to an otsuki torus @xmath20 is a torus . if @xmath298 is odd then the metric on bipolar otsuki torus @xmath23 is extremal for the functional @xmath299 and @xmath300 . if @xmath298 is even then the metric on bipolar otsuki torus @xmath23 is extremal for the functional @xmath301 and @xmath302.[bipotsuki ]    if @xmath298 is even then the following inequality holds , @xmath303 if @xmath298 is odd then one has the following inequality , @xmath304 [ bipotsukiprop ]    if @xmath298 is even , then we have @xmath305 we used the inequalities @xmath306 and @xmath307 in order to prove the last inequality . in the same way , if @xmath298 is odd , then we have @xmath308    now it is easy to see that propositions  [ otsukiestimateprop ] ,  [ lawsonprop ] ,  [ biplawsonprop ] ,  [ bipotsukiprop ] together with proposition  [ lowerbound ] imply theorem  [ maintheorem ] . let us represent the clifford torus as a flat torus with the square lattice with edges equal to @xmath309 . in this case [ cliffordestimate ] the laplace - beltrami coinsides up to a sign with the classical two - dimensional laplace operator . therefore , using the separation of variables one obtains that the eigenfunctions are @xmath310 where @xmath311 and @xmath312 . then , the eigenvalues are equal to @xmath313 , @xmath314 , @xmath315 , @xmath316 respectively . it is easy to check that the set of functions @xmath324 form an isometrical immersion of clifford torus in the unit sphere . the same is true for the set @xmath325 and the set @xmath326 where @xmath327 . therefore , according to theorem  [ th1 ] , the metric on the clifford torus is extremal for the functionals @xmath328 , where @xmath329 with @xmath330 , and @xmath331 . so , it is sufficient to prove that @xmath335 and this inequality holds for @xmath336 . and for @xmath337 holds the inequality @xmath338 . this inequality can be obtained by the direct enumeration of all possible values of @xmath339 . this completes the proof of proposition  [ cliffordtheorem ] . 99 a. el soufi , h. giacomini , m. jazar , a unique extremal metric for the least eigenvalue of the laplacian on the klein bottle . _ duke math . j. _ * 135*:1 ( 2006 ) , 181 - 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el soufi - ilias theorem establishes a connection between minimal submanifolds of spheres and extremal metrics for eigenvalues of the laplace - beltrami operator . 
 recently , this connection was used to provide several explicit examples of extremal metrics . 
 we investigate the maximality of these metrics and prove that all of them are not maximal . 
 + _ 2010 mathematics subject classification . 
 _ 58e11 , 58j50 , 35p15 . 
 + _ key words and phrases . _ 
 extremal metric , otsuki tori , lawson tau - surfaces , bipolar surface .