Torsional Rigidity on Compact Riemannian Manifolds with lower Ricci Curvature Bounds

In this article we prove a reverse H\"older inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains.


Introduction and main results
In 1972; following the spirit of the works of Faber and Krahn [1,2], Payne and Rayner [3,4] proved a reverse Hölder inequality for the norms L 1 and L 2 of the first eigenfunction of the Dirichlet problem on bounded domains D of R 2 : where 1 is the lowest eigenvalue of the fixed membrane problem, and u 1 the corresponding eigenfunction. Equality holds if and only if D is a disc. The work of Payne and Rayner was generalized by Köhler and Jobin [5] for bounded domains of R n ; n 3: In 1982, G.Chiti [6], generalized the reverse Hölder inequality for the norms L q and L p ; q p > 0 for bounded domains of R n ; n 2 0 @ Z D u q dxdy This inequality is isoperimetric in the sense that equality holds if and only if D is a ball.
The main ideas of this paper were early investigated in the PhD thesis of H.Hasnaoui [7], the first one was to establish a generalization of Chiti's reverse Hölder inequality for the norms L q and L p ; q p > 0 for compact Riemannian manifolds with Ricci curvature bounded from below and the second one is a version of the Saint Venant Theorem for such manifolds.
In fact, Modern geometric analysts, including Chavel and Gromov, have identified such manifolds as important, and have related the Ricci bound to many estimates of eigenvalues, as well as to other quantities of interest in differential geometry. Thus, it is very natural to consider similar questions about the Laplacian and the torsional rigidity in this context. Both are isoperimetric results, in the sense that the quantity of interest is dominated by the analogous expression on spheres. Much of the background and many results for the spectrum of such manifolds could be found in [8], [9], [10] and more recently in [11]. In 1856, Saint-Venant [12] observed that columns with circular cross-sections offer the greatest resistance to torsion for a given cross-sectional area. This fact was proved a century or so later by Pólya using Steiner symmetrization [13]. We also note the independent proof by Makai in 1963, one can see [14]. Torsional rigidity is a physical quantity of much interest, see for example [15,16], the recent [17,18], the more recent [19,20] and the classical papers of Payne [21,22], Payne-Weinberger [23]. As we show here, it is also a quantity of much interest for geometers as well.
Parts of our work follow an analysis of Ashbaugh and Benguria [24] for subdomains of hemispheres, one can see also [25]. In the sequel, we will introduce the main results of this paper: Let .M; g/ be a compact connected Riemannian manifold of dimension n 1 without boundary. We denote by the infimum of the Ricci curvature Ri c of .M; g/; here U T .M / is the unit tangent bundle of the manifold .M; g/. Let .S n ; g ? / be the unit sphere of the space R nC1 , endowed with the induced metric, then R.S n ; g ? / D n 1. We suppose, as done in [9], that R.M; g/ is strictly positive, and normalize the metric g so that R.M; g/ R.S n ; g ? / D n 1. In the sequel, we will denote by V .M / D R dv g the volume of .M; g/; where the element of volume is denoted dv g , ! n the volume of the unit sphere .S n ; g ? / andˇD V .M /=! n : Let D be a connected bounded domain of M with smooth boundary, and D ? the geodesic ball of S n centered at the north pole such that Vol.D/ DˇVol.D ? /: We are interested in the comparison of the fundamental solutions u and v of problems .P 1 / and .P 2 / respectively: .
B Â 1 . / is the geodesic ball of S n centered at the north pole of radius Â 1 D Â 1 . /; where is the first eigenvalue for the Dirichlet problem .P 2 /; and denotes indifferently the laplacian operator on M or S n : Let u be the decreasing rearrangement of u and u ? the corresponding radial function defined on D ? the geodesic ball of .S n ; g ? / which has the same relative volume as D; (see Section 2 for notations and details).
As a consequence of Theorem 1.1, we obtain: Corollary 1.2 (Chiti's Reverse Hölder Inequality for Compact Manifolds). Let p; q be real numbers such that q p > 0, then u and v are related by this inequality with equality if and only if the triplet .M; D; g/ is isometric to the triplet .S n ; D ? ; g ? /.
Next, we focus on the torsional rigidity T .D/ of the domain D: Recall that: where w is is the smooth solution of the boundary value problem of Dirichlet-Poisson type ( w is called the warping function of D) w D 0 on @D: We obtain the following result: the equality holds if and only if the triplet .M; D; g/ is isometric to the triplet .S n ; D ? ; g ? /.
And finally, we give a comparison formula for the warping function w which allows us to obtain directly the result of Theorem 1.3.

Preliminary tools
Denote by .D/ the first eigenvalue of the Laplacian for the Dirichlet problem on D, and let u be the positive associated eigenfunction. Therefore, u satisfies The variational formula of .D/ is given by where jdf j is the Riemannian norm of the differential of f . We have equality if and only if f is of class C 2 and is an eigenfunction associated with the first eigenvalue .D/: The co-area formula gives Here, d is the (n-1)-dimensional Riemannian measure in .M; g/. For what follows, we will also denote the .n 1/dimensional Riemannian measure in .S n ; g ? / by d . Since D has bounded measure, the above shows that the function is integrable, and therefore the function V is absolutely continuous. Hence, V is differentiable almost everywhere and for almost all t 2 OE0; u. The function V is then a non increasing function and has an inverse which we denote by u : The function u is absolutely continuous. Now, applying the Cauchy-Schwartz inequality, we obtain Hence Next, we use the following isoperimetric inequality due to M. Gromov [26] which relates the volume of the boundaries of D and D : Lemma 2.1. Under the same hypothesis given above Vol n 1 .@D/ ˇVol n 1 .@D ? /; (14) where Vol n 1 is the .n 1/-dimensional volume relative to g and g ? . Equality holds, if and only if the triplet .M; D; g/ is isometric to the triplet .S n ; D ? ; g ? /: Let The quantity ! n 1 R Â 0 .sin / n 1 d is the n-volume of the geodesic ball of radius Â in S n . If we let, the function L.Â/ denotes the .n 1/-dimensional volume of the geodesic ball of radius Â , i.e.
Inequality (14) can then be written as where Â.s/ is the inverse function of A defined in (15). Now, applying inequality (17) to the domain D u .s/ and combining it with inequality (13), we obtain We then apply Gauss Theorem to the Dirichlet problem on D t , using the smoothness of its boundary @D t ; we get Here, we used the fact that the outward normal to D t is given by ru jruj . Remark. 8p 0; we have The change of variables Á D V . / gives Finally combining (20) for p D 1 with equalities (18) and (19), we obtain the following Lemma 2.2. Let u be a solution of problem .P 1 /. Then u , its decreasing rearrangement, satisfies the integrodifferential inequality for almost every s > 0. Let B Â 1 . / be the geodesic ball of S n centered at the north pole with radius Â 1 D Â 1 . /; such that the following problem has a solution. Let v > 0 be the first Dirichlet eigenfunction on B Â 1 . / then by lemmas 3.1 and 3.2 of [27], we conclude that v depends only on Â and is strictly decreasing on OE0; Â 1 /. Therefore, we denote by v.Â / the function v. So, in polar coordinates the problem .P 2 / can be rewritten as Integrating the equality (23), we obtain (25) Then, using (24), we can rewrite the left-hand side of (25) as for all s in OE0; A.Â 1 /. The change of variables D A.˛/ in the right-hand side of (25) gives Finally, from (26) and (27), we obtain v 0 .s/ D .ˇ! n 1 / 2 .sin Â.s// 2 2n

Chiti's Reverse Hölder Inequality
In this section, we will prove the extension of the Chiti's Comparison Lemma, given for domains in the case of R 2 and R n in the original papers of Payne-Rayner [3,4], then, extended by Kohler-Jobin [5] and Chiti [6,28]. In [24], Proof. Using ( Now, an integration by parts in the second member of the last inequality and the fact that give Considering that is the minimum of the Rayleigh quotient on B Â 1 . / , it follows that this minimum is achieved for u ? , and so u ? is indeed an eigenfunction associated with on B Â 1 . / . Now, using the simplicity of the fundamental eigenvalue and the hypothesis of our Lemma, we get u ? D v.  Since the functions u and v are nonnegative, and A.Â 1 / Ä vol.D/ (see (30)), it is then clear that We will first prove that v .0/ u .0/. Assume that v .0/ < u .0/. In this case, 9 Ä > 1, such that Ä v .0/ D u .0/. By Lemma 3.1, we have It follows from (22) and (28) that ' satisfies From '; define a radial function in B Â 1 . / byˆ.
Then,ˆis an admissible function for the Rayleigh quotient on B Â 1 . / . Using this fact, we proceed exactly as in the proof of inequality (35), we obtain It follows that the Rayleigh quotient ofˆis equal to ; therefore,ˆis an eigenfunction for . Consequently, u D v and so u .s/ D v .s/ in OEs 1 ; s 2 , which contradicts the maximality of s 1 and hence completes the proof of the theorem. .v .Á// p dÁ: We complete the argument using the following result Lemma 3.2 ([29]). Let R; p; q be real numbers such that 0 < p Ä q, R > 0; and f; g real functions in L q .OE0; R/.
If the decreasing rearrangements of f and g satisfy the following inequality: From this, it is clear, that for all q p Finally, combining this inequality and equality (1), we obtain the desired result. Now, assume that we have equality in (4), from the normalization for the function v given in (1), we deduce that for all p > 0 R D u p dv g DˇR B Â 1 . / v p dv g ? : Hence, vol.D/ Dˇvol.B Â 1 . / /; and since D ? and B Â 1 . / are geodesic balls of S n centered both at the north pole with the same volume, it yields that D ? D B Â 1 . / : By hypothesis is the fundamental eigenvalue of B Â 1 . / ; hence of D ? ; thus we obtain that 1 .D/ D 1 .D ? / D and this is possible if and only if the triplet .M; D; g/ is isometric to the triplet .S n ; D ? ; g ? /; ( one can see Theorem 5 in [9]). The proof of Corollary 1.2 is thereby complete.

Saint-Venant Theorem
Let .M; g/ be a compact Riemannian manifold of dimension n, without boundary satisfying R.M; g/ n 1, and let D be a bounded connected domain of M with smooth boundary. We are interested in the following geometric quantity where w is the smooth solution of the boundary value problem of Dirichlet-Poisson type The geometric quantity T .D/ is called the "torsional rigidity of D", and it is customary to call the solution w of (53) the warping function. In Theorem 1.3, we will give explicit upper bounds for the torsional rigidity T .D/ which amounts to a version of the Saint-Venant Theorem for compact manifolds. Let C 1 0 .D/ denote the space of C 1 functions with compact support in D. We define the Sobolev space H 1 0 .D/ as the closure of C 1 0 .D/ in H 1 .M / the space of square integrable functions with a square integrable weak derivatives. The variational formulation for T .D/ is given by Indeed, by the scaling property of the functional,ˆ.cf / Dˆ.f / for all c > 0, one can reformulate the minimizing problem of the functionalˆas a minimizing problem of the functional R D jrf j 2 dv g subjected to the constraint R D f dv g D 1. By the above mentioned Lagrange multipliers theorem, this gives the existence of the Lagrange multiplier , such that for any h 2 Hence, f is a weak solution of the equation By standard regularity results, f is unique and smooth. Then w D 1 f is also a critical point ofˆ. Finally, the fact thatˆ.w/ D 1 T .D/ proves the equality. We will now give the proof of the Saint-Venant Theorem.
Proof of Theorem 1.3. The proof follows the steps of Talenti's method (one can see [30]) tailored to our setting. For 0 < t Ä m D supfw.x/; x 2 Dg, we define the set and introduce the following functions The smooth co-area formula gives Then differentiating (58) with respect to t; one obtain Let w be the inverse function of V; w ? the radial function defined in D ? by w ? .Â / D w .A.Â //; and Since w ? is a radial decreasing function, its level sets D ? t are geodesic balls with radius r.t / D w ? 1 .t /. Therefore and Differentiating with respect to t , we get and On one hand, multiplying (64) by (65), we obtain On the other hand, the Cauchy-Schwartz inequality gives Next, the fact thatˇZ D ?
In the sequel, we will give a comparison theorem for the warping function in the case of a smooth compact Riemannian manifold. This theorem is based on a method of Talenti ([30]). which is the solution to the problem Then, w ? the symmetrized function of w satisfies where 0 Ä t < m. We introduce the function defined by Then The function ‰ is decreasing in t then, for h > 0; we have Letting h go to zero, we obtain, for the right derivative of ‰.t / Now, for Â 2 .0; Â 0 /, integrating this inequality from Â to Â 0 ; we obtain which is the desired result. Now, assume that we have equality in (7), integrating this equality, we get Finally, by applying the Saint-Venant Theorem, we deduce that the triplet .M; D; g/ is isometric to the triplet .S n ; D ? ; g ? /; which completes the proof of the theorem.
Which is the result of Theorem 1.3.