International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 350183, 48 pages

doi:10.1155/2012/350183

Research Article

Complex Hessian Equations on Some Compact Kähler Manifolds

Asma Jbilou

Université Internationale de Rabat (UIR), Parc Technopolis, Rocade de Rabat, Salé, 11 100 Sala Al Jadida, Morocco

Received 29 March 2012; Revised 22 July 2012; Accepted 31 July 2012

Academic Editor: Aloys Krieg

Copyright © 2012 Asma Jbilou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

On a compact connected -dimensional Kähler manifold with Kähler form , given a smooth function and an integer , we want to solve uniquely in the equation , relying on the notion of -positivity for (the extreme cases are solved: by (Yau in 1978), and trivially). We solve by the continuity method the corresponding complex elliptic th Hessian equation, more difficult to solve than the Calabi-Yau equation (), under the assumption that the holomorphic bisectional curvature of the manifold is nonnegative, required here only to derive an a priori eigenvalues pinching.

1. The Theorem

All manifolds considered in this paper are connected.

Let be a compact connected Kähler manifold of complex dimension . Fix an integer . Let be a smooth function, and let us consider the -form and the associated -tensor defined by . Consider the sesquilinear forms and on defined by and . We denote by the eigenvalues of with respect to the Hermitian form . By definition, these are the eigenvalues of the unique endomorphism of satisfying

Calculations infer that the endomorphism writes

is a self-adjoint/Hermitian endomorphism of the Hermitian space , therefore . Let us consider the following cone: , where denotes the th elementary symmetric function.

Definition 1.1. is said to be -admissible if and only if .

In this paper, we prove the following theorem.

Theorem 1.2 (the equation). Let be a compact connected Kähler manifold of complex dimension with nonnegative holomorphic bisectional curvature, and let be a function of class satisfying . There exists a unique function of class such that

Moreover the solution is -admissible.

This result was announced in a note in the Comptes Rendus de l'Acadé-mie des Sciences de Paris published online in December 2009 [1]. The curvature assumption is used, in Section 6.2 only, for an a priori estimate on as in [2, page 408], and it should be removed (as did Aubin for the case in [3], see also [4] for this case). For the analogue of on , the Dirichlet problem is solved in [5, 6], and a Bedford-Taylor type theory, for weak solutions of the corresponding degenerate equations, is addressed in [7]. Thanks to Julien Keller, we learned of an independent work [8] aiming at the same result as ours, with a different gradient estimate and a similar method to estimate , but no proofs given for the and the estimates.

Let us notice that the function appearing in the second member of satisfies necessarily the normalisation condition . Indeed, this results from the following lemma.

Lemma 1.3. Consider .

Proof. See [9, page 44].

Let us write differently.

Lemma 1.4. Consider .

Proof. Let . It suffices to prove the equality at in a -normal -adapted chart centered at . In such a chart and , so at , and . Thus

Now are distinct integers of and -forms commute therefore,

Consequently, writes:

Let us remark that corresponds to the Calabi-Yau equation , when is just a linear equation in Laplacian form. Since the endomorphism is Hermitian, the spectral theorem provides an -orthonormal basis for of eigenvectors : , . At in a chart , we have , thus . In addition, , so the equation writes locally:

Let us notice that a solution of this equation is necessarily -admissible [9, page 46]. Let us define and where is a Hermitian matrix. The function is a polynomial in the variables , specifically (sum of the principal minors of order of the matrix ). Equivalently writes:

It is a nonlinear elliptic second order PDE of complex Monge-Ampère type. We prove the existence of a -admissible solution by the continuity method.

2. Derivatives and Concavity of

2.1. Calculation of the Derivatives at a Diagonal Matrix

The first derivatives of the symmetric polynomial are given by the following: for all , where . For , let us denote and . The second derivatives of the polynomial are given by . We calculate the derivatives of the function , where denotes the set of Hermitian matrices, at diagonal matrices using the formula:

where

These derivatives are given by [9, page 48]

and all the other second derivatives of at vanish.

Consequently, the derivatives of the function at diagonal matrices with , where , are given by

and all the other second derivatives of at vanish.

2.2. The Invariance of and of Its First and Second Differentials

The function is invariant under unitary similitudes:

Differentiating the previous invariance formula (2.6), we show that the first and second differentials of are also invariant under unitary similitudes:

These invariance formulas are allowed to come down to the diagonal case, when it is useful.

2.3. Concavity of

We prove in [9] the concavity of the functions and more generally when and is symmetric [9, Theorem  VII.4.2], which in particular gives the concavity of the functions [9, Corollary VII] and more generally [9, Theorem VII]. In this section, let us show by an elementary calculation the concavity of the function .

Proposition 2.1. The function is concave (this holds for all ).

Proof. The function is of class , so its concavity is equivalent to the following inequality:

Let , , and such that . We have . Let us denote :

But , and , so by concavity of at [10, page 269]. In addition, since [11], consequently , which shows that and achieves the proof.

3. The Proof of Uniqueness

Let and be two smooth -admissible solutions of such that . For all , let us consider the function with . Let , and let us denote . We have which is equivalent to . But

Therefore we obtain

We show easily that the matrix is Hermitian [9, page 53]. Besides the function is continuous on the compact manifold so it assumes its minimum at a point , so that the complex Hessian matrix of at the point , namely, , is positive-semidefinite.

Lemma 3.1. For all , ; namely, the functions are -admissible at .

Proof. Let us denote . The set is nonempty, it contains , and it is an open subset of . Let be the largest number of such that . Let us suppose that and show that we get a contradiction. Let , we have . Let us prove that for all . Fix ; the quantity is intrinsic so it suffices to prove the assertion in a particular chart at . Now at in a -unitary -adapted chart at

But since , then for all . Besides, since the matrix is positive-semidefinite. Therefore, we infer that . Consequently, we obtain that (since is -admissible). This holds for all ; we deduce then that which proves that . This is a contradiction; we infer then that .

We check easily that the Hermitian matrix is positive definite [9, page 54] and deduce then the following lemma since the map is continuous on a neighbourhood of .

Lemma 3.2. There exists an open ball centered at such that for all the Hermitian matrix is positive definite.

Consequently, the operator is elliptic on the open set . But the map is , assumes its minimum at , and satisfies ; then by the Hopf maximum principle [12], we deduce that for all . Let us denote . This set is nonempty and it is a closed set. Let us prove that is an open set: let be a point of , so , then the map assumes its minimum at the point . Therefore, by the same proof as for the point , we infer that there exists an open ball centered at such that for all so for all then , which proves that is an open set. But the manifold is connected; then , namely, for all . Besides , therefore we deduce that on namely that on , which achieves the proof of uniqueness.

4. The Continuity Method

Let us consider the one parameter family of ,

The function is a -admissible solution of :  and satisfies . For , so corresponds to .

Let us fix , and , and let us consider the nonempty set (containing ):

The aim is to prove that . For this we prove, using the connectedness of , that .

4.1. Is an Open Set of

This arises from the local inverse mapping theorem and from solving a linear problem. Let us consider the following sets:

where is a vector space and is an open set of . Using these notations, the set writes solution of }.

Lemma 4.1. The operator , , is differentiable, and its differential at a point , is equal to

Proof. See [9, page 60].

Proposition 4.2. The nonlinear operator is elliptic on .

Proof. Let us fix a function and check that the nonlinear operator is elliptic for this function . This goes back to show that the linearization at of the nonlinear operator is elliptic. By Lemma 4.1, this linearization is the following linear operator:

In order to prove that this linear operator is elliptic, it suffices to check the ellipticity in a particular chart, for example, at the center of a -normal -adapted chart. At the center of such a chart,

But for all we have on since [11], which proves that the linearization is elliptic and achieves the proof.

Let us denote the operator

As , the operator is differentiable and elliptic on of differential

Let us denote the matrix and calculate this linearization in a different way, by using the expression (2.1) of :

Thus

We infer then the following proposition.

Proposition 4.3. The linearization of the operator is of divergence type:

Proof. By (4.9) we have

Moreover

But , then

Besides, the quantity is symmetric in (indeed, and since is Kähler), and is antisymmetric in ; it follows then that , consequently .

From Proposition 4.3, we infer easily [9, page 62] the following corollary.

Corollary 4.4. The map is well defined and differentiable and its differential equals .

Now, let and let be a solution of the corresponding equation : .

Lemma 4.5. is an isomorphism.

Proof. Let with . Let us consider the equation

We have and the matrix is positive definite (since is elliptic at ); then by Theorem of [13, p. 104] on the operators of divergence type, we deduce that there exists a unique function satisfying which is solution of (4.14) and then solution of . Thus, the linear continuous map is bijective, and its inverse is continuous by the open map theorem, which achieves the proof.

We deduce then by the local inverse mapping theorem that there exists an open set of containing and an open set of containing such that is a diffeomorphism. Now, let us consider a real number very close to and let us check that it belongs also to : if is sufficiently small then is small enough so that , thus there exists such that and consequently there exists of vanishing integral for which is solution of . Hence . We conclude therefore that is an open set of .

4.2. Is a Closed Set of : The Scheme of the Proof

This section is based on a priori estimates. Finding these estimates is the most difficult step of the proof. Let be a sequence of elements of that converges to , and let be the corresponding sequence of functions: is , -admissible, has a vanishing integral, and is a solution of

Let us prove that . Here is the scheme of the proof.

(1)Reduction to a estimate: if is bounded in a with , the inclusion being compact, we deduce that after extraction converges in to . We show by tending to the limit that is a solution of (it is then necessarily -admissible) and of vanishing integral for . We check finally by a nonlinear regularity theorem [14, page 467] that , which allows us to deduce that (see [9, pages 64–67] for details). (2)We show that is bounded in : first of all we prove a positivity Lemma 5.4 for , inspired by the ones of [15, page 843] (for ), but in a very different way, required since the -positivity of is weaker with (in this case, some eigenvalues can be nonpositive, which complicates the proof), using a polarization method of [7, page 1740] (cf. 5.2) and a Gårding inequality 5.3; we infer then from this lemma a fundamental inequality 5.5 as Proposition 7.18 of [13, page 262]. We conclude the proof using the Moser's iteration technique exactly as for the equation of Calabi-Yau. We deal with this estimate in Section 5. (3)We establish the key point of the proof, namely, a a priori estimate (Section 6). (4)With the uniform ellipticity at hand (consequence of the previous step), we obtain the needed estimate by the Evans-Trudinger theory (Section 7).

5. The A Priori Estimate

5.1. The Positivity Lemma

Our first three lemmas are based on the ideas of [7, Section 2].

Lemma 5.1. Let be a real -form, it then writes , with where is the symmetric tensor ; hence

Proof. The same proof as Lemma 1.4.

We consider for the map where denotes the -vector space of Hermitian square matrices of size . is a real polynomial of degree and in real variables. Moreover, it is hyperbolic (cf. [16] for the proof) and it satisfies . Let be the totally polarized form of . This polarized form is characterized by the following properties:

(i) is -linear. (ii) is symmetric. (iii)For all , .

Using these notations, we infer from Lemma 5.1 that at the center of a -unitary chart (this guarantees that the matrix is Hermitian), we have

By transition to the polarized form in this equality we obtain the following lemma.

Lemma 5.2. Let and be real -forms. These forms write , with where is the symmetric tensor . Then, at the center of a -unitary chart we have

Proof. See [9, page 71].

Theorem 5 of Gårding [16] applies to with .

Lemma 5.3 (the Gårding inequality for ). Let , for all ,

Let us recall that is the connected component of containing . The same proof as [17, pages 129, 130] implies that

Note that the Gårding inequality (Lemma 5.3) holds for .

Let us now apply the previous lemmas in order to prove the following positivity lemma inspired by the ones of [15, page 843] (for ); let us emphasize that the proof is very different since the -positivity is weaker.

Lemma 5.4 (positivity lemma). Let be a real 1-form on and , then the function defined by is nonnegative.

Proof. Let , then . Let be a real -form, it then writes . Let , hence . Similarly, we prove that , consequently . Besides, set . Now, let and be a -unitary chart centered at . Using Lemma 5.2, we infer that at in the chart :

But at , and . Indeed, since is -admissible and . Moreover, the Hermitian matrix is positive-semidefinite since for all , we have ; we then deduce that for all , we have at , . Finally, we infer by the Gårding inequality that at in the chart we have

which proves the positivity lemma.

5.2. The Fundamental Inequality

The a priori estimate is based on the following crucial proposition which is a generalization of the Proposition 7.18 of [13, page 262].

Proposition 5.5. Let be an increasing function of class defined on , and let be a   -admissible function defined on , then the following inequality is satisfied:

Proof. We have the equality . Besides, since is commutative , namely, , then . But , and so . In addition by Stokes' theorem, ; therefore,

Let us prove that (using the positivity lemma) and that . Let us apply the positivity lemma to : the function defined by is nonnegative for all . But and is an increasing function; then . Let us now calculate . Fix , and let us work in a -unitary chart centered at and satisfying at . We have then at and ; therefore,

Thus , consequently , which achieves the proof of the proposition.

5.3. The Moser Iteration Technique

We conclude the proof using the Moser's iteration technique exactly as for the equation of Calabi-Yau. Let us apply the proposition to in order to obtain a crucial inequality (the inequality from which we will infer the a priori estimate of . Let be a real number. The function is admissible. Let us consider the function  : . This function is of class and , so is increasing. Therefore we infer by the previous proposition that

Besides, , so the previous inequality writes:

Let us infer from the inequality another inequality (the inequality that is required for the proof. It follows from the continuous Sobolev embedding that

where is independent of . Besides, is uniformly bounded; indeed,

Using the inequalities , , , and we obtain

where is independent of . Suppose that .

Using the Green's formula and the inequalities of Sobolev-Poincaré and of Hölder, we prove following [13] these estimates.

Lemma 5.6. There exists a constant such that for all ,

Proof. is a compact Riemannian manifold and , so by the Green's formula , where and is independent of . Here denotes the real Laplacian. Then, we infer that . But and ; then . Besides since is -admissible: indeed, at in a -normal -adapted chart, namely, a chart satisfying , and for all , , , we have so since is -admissible; consequently , but since which proves that . Therefore and . We infer then that . Now let us take in the inequality : . Besides, and has a vanishing integral; then by the Sobolev-Poincaré inequality we infer . But almost everywhere; therefore . Using the inequality with and the fact that is uniformly bounded, we obtain that . Consequently, we infer that .

Let . By the Hölder inequality we have . Therefore . But

and , thus .

Suppose without limitation of generality that . Now, we deduce from the previous lemma and the inequality , by induction, these more general estimates using the same method as [13].

Lemma 5.7. There exists a constant such that for all ,

with and where is the constant of the inequality .

Proof. We prove this lemma by induction: first we check that the inequality is satisfied for ; afterwards we show that if the inequality is true for , then it is satisfied for too. Denote . For we have , so it suffices to check that . This inequality is equivalent to ; then . But if , then (since ), and (since and ); therefore . Besides, , which proves the inequality for . Now let us fix . Suppose that and prove that . The inequality proved previously writes:

where is independent of , namely, . But since , we have by the Hölder inequality that ; therefore .

(i)If , then ; therefore . Let us check that . This inequality is equivalent to , but so it is equivalent to . Besides and , then it suffices to have , and this is satisfied since . (ii)If , we infer that , therefore by the induction hypothesis. But ; then we obtain the required inequality .

By tending to the limit in the inequality of the previous lemma, we obtain the needed a priori estimate.

Corollary 5.8. Consider

6. The A Priori Estimate

6.1. Strategy for a Estimate

First, we will look for a uniform upper bound on the eigenvalues . Secondly, we will infer from it the uniform ellipticity of the continuity equation and a uniform gradient bound. Thirdly, with the uniform ellipticity at hand, we will derive a one-sided estimate on pure second derivatives and finally get the needed bound.

6.2. Eigenvalues Upper Bound

6.2.1. The Functional

Let , and let  :  be a   -admissible solution of satisfying . Consider the following functional:

where is the unit sphere bundle associated to and is related to by: . is continuous on the compact set , so it assumes its maximum at a point . In addition, for fixed , is continuous on the compact subset (the fiber); therefore it attains its maximum at a unit vector , and by the min-max principle we can choose as the direction of the largest eigenvalue of , . Specifically, we have the following.

Lemma 6.1 (min-max principle). Consider

For fixed , we have ; therefore ; hence,

At the point , consider an -orthonormal basis of made of eigenvectors of that satisfies the following properties:

(1)-orthonormal: . (2)-diagonal: , . (3) is achieved in the direction : and .

In other words, it is a basis satisfying

(1),(2), ,(3).

Let us consider a holomorphic normal chart centered at such that and for all .

6.2.2. Auxiliary Local Functional

From now on, we work in the chart constructed at . The map is continuous on and is equal to at , so there exists an open subset such that for all . Let be the functional

We claim that assumes a local maximum at . Indeed, we have at each : (see Lemma 6.1); thus .

6.2.3. Differentiating the Equation

For short, we drop henceforth the subscript of . Let us differentiate at , in a chart :

Differentiating once again, we find

Using the above chart at the point , normality yields , and . Furthermore . In this chart, we can simplify the previous expression; we get then at ,

Besides, , so still by normality, we obtain at that . Therefore we get

6.2.4. Using Concavity

Now, using the concavity of [10], we prove for Proposition 2.1 that the second sum of (6.8) is negative [9, page 84]. This is not a direct consequence of the concavity of the function since the matrix is not Hermitian.

Lemma 6.2. Consider

Hence, from (6.8) combined with Lemma 6.2 we infer

6.2.5. Differentiation of the Functional

Let us differentiate twice the functional :

Therefore at , in the above chart we find . Let us define the operator:

Thus, we have at

Combining (6.13) with (6.10), we get rid of the fourth derivatives:

Since assumes its maximum at , we have at that . So we are left with the following inequality at :

Curvature Assumption

Henceforth, we will suppose that the holomorphic bisectional curvature is nonnegative at any . Thus in a holomorphic normal chart centered at we have for all , . This holds in particular at in the previous chart . This assumption will be used only to derive an a priori eigenvalues pinching and is not required in the other sections.

Back to the inequality (6.15), we have and since , and under our curvature assumption for all . Besides, for all ; therefore . So we can get rid of the curvature terms in (6.15) and infer from it the inequality

6.2.6. A ’s Upper Bound

Here, we require elementary identities satisfied by the 's [11], namely:

Consequently, (6.16) writes:

So . But at , , then , and consequently . In other words, there exists a constant independent of such that

To proceed further, we recall the following

Lemma 6.3 (Newton inequalities). For all , :

Let us use Newton inequalities to relate to . Since for and we have and ( by convention), Newton inequalities imply then that , or else , consequently . By induction, we get for all . In particular

But ; combining this with (6.19) and (6.21) we obtain at that

Hence we may state the following.

Theorem 6.4. There exists a constant depending only on and such that for all .

Combining this result with the a priori estimate immediately yields the following.

Theorem 6.5. There exists a constant depending only on and such that for all , for all , .

6.2.7. Uniform Pinching of the Eigenvalues

We infer automatically the following pinchings of the eigenvalues.

Proposition 6.6. For all , .

Proposition 6.7. For all , for all ,.

6.3. Uniform Ellipticity of the Continuity Equation

To prove the next proposition on uniform ellipticity, we require some inequalities satisfied by the ’s.

Lemma 6.8 (Maclaurin inequalities). For all   for all .

Proposition 6.9 (uniform ellipticity). There exist constants and depending only on and such that: where .

Proof. We have where, indeed, the constant so defined depends only on , and . Let us look for a uniform lower bound on , using the identity . We distinguish two cases.

Case  1. (). When so, we have ; therefore . But and ; hence .

Case  2. (). For since and . Besides by hypothesis, therefore . From the latter, we infer by Maclaurin inequalities or else ; thus we have , consequently

Here, let us distinguish two subcases of Case 2.

(i) If , then we have the uniform lower bound that we look for. (ii) Else , thus , therefore ; then we get .

Consequently or finally , where the constant so defined depends only on and .

Similarly we prove the following.

Proposition 6.10 (uniform ellipticity). There exists constants and depending only on and such that for all , for all , .

6.4. Gradient Uniform Estimate

The manifold is Riemannian compact and , so by the Green's formula

where is the Green's function of the Laplacian . By differentiating locally under the integral sign we obtain . We infer then that at in a holomorphic normal chart, we have

Now, using the uniform pinching of the eigenvalues, we prove easily the following estimate of the Laplacian.

Lemma 6.11. There exists a constant depending on and such that .

Combining Lemma 6.11 with (6.25), we deduce that . Besides, classically [13, page 109], there exists constants and such that

We thus obtain the following result.

Proposition 6.12. There exists a constant depending on , and such that for all .

Specifically, we can choose .

6.5. Second Derivatives Estimate

Our equation is of type:

6.5.1. The Functional

Consider the following functional:

where is the real unit sphere bundle associated to . is continuous on the compact set , so it assumes its maximum at a point .

6.5.2. Reduction to Finding a One-Sided Estimate for

If we find a uniform upper bound for , since , we readily deduce that there exists a constant such that

Fix . Let be a holomorphic -normal -adapted chart centered at , namely, , and . Since , we obtain and similarly for all . Besides, we have ; therefore we obtain

Let us now bound second derivatives of mixed type . Let . Since , we have , which yields , hence as well . Similarly we prove that at , in the above chart , we have for all and for all . Consequently for all , . Therefore we deduce that

Theorem 6.13 (second derivatives uniform estimate). There exists a constant depending only on , and such that for all , .

This allows to deduce the needed uniform estimate:

6.5.3. Chart Choice

For fixed , is continuous on the compact subset (the fiber); therefore it assumes its maximum at a unit vector . Besides, is a symmetric bilinear form on , so by the min-max principle we have , where denotes the largest eigenvalue of with respect to ; furthermore we can choose as the direction of the largest eigenvalue . For fixed , we now have , consequently , hence .

At the point , consider a (real) basis of that satisfies the following properties:

(i),(ii),(iii).

Let be a -normal real chart at obtained from a holomorphic chart by setting where (namely, and for all ) satisfying and , so that is the direction of the largest eigenvalue .

6.5.4. Auxiliary Local Functional

From now on, we work in the real chart so constructed at .

Let be an open subset such that for all , and let be the functional

We claim that assumes its maximum at . Indeed, , so proving our claim.

Let us now differentiate twice in the real direction the equation

At the point , in a chart , we obtain

Differentiating once again

But at the point , for our function , we have since . Hence, we infer that

6.5.5. Using Concavity

The function is concave with respect to the variable . Indeed

but for a fixed point the function is affine (where denotes the set of symmetric matrices of size ); we deduce then that the composition is concave on the set , which proves our claim. Hence, since the matrix is symmetric, we obtain that

Consequently

Let us now calculate the quantity (at ). Since , we have

But at , and , then so that . Moreover , thus . Similarly, we have at  : . Consequently, we deduce that . Hence, we have at : . Besides, , which infers that at

Consequently, the inequality (6.38) becomes

6.5.6. Differentiation of the Functional

We differentiate twice the functional :

Hence, at in the chart , we obtain

Let us now calculate the different terms of this formula (at in the chart ):

Besides, we have ; therefore we deduce that ; namely, . Moreover, we have at

But at , , in addition at this point ; therefore we obtain at in the chart

Henceforth, and in order to lighten the notations, we use instead of and instead of , so we have

Let us now consider the second order linear operator:

Since the functional assumes its maximum at the point , we have at in the chart . Besides, combining the inequalities (6.41) and (6.47), we obtain

The fourth derivatives are simplified. Moreover, we have at  :  with , consequently:

Let us now express the quantities and using the components of the Riemann curvature (at the point in the normal chart ):

We then obtain

6.5.7. The Uniform Upper Bound of

By the uniform estimate of the gradient we have for all . Moreover, at in the chart : . Consequently

But for since , we obtain at in the chart that

Then

Hence

But at in the chart , ; then for all , consequently

Besides, at in the chart , we have

, so for all , therefore

Hence at in the chart , we obtain

But , and for all then

Besides

Consequently, we obtain

Let us now estimate for using . We follow the same method as for the proof of Theorem 6.13. For all , we have the inequality ; then at in a holomorphic -normal -adapted chart , namely, a chart such that , and , we deduce that for all

Since , we infer the following inequalities:

Consequently

in the chart .

Hence we infer that

But at in the chart , ; consequently we obtain

Thus

Besides

Hence

Then

But using the uniform ellipticity and the inequalities , we obtain

Then at in the chart , we have

The previous inequality means that some polynomial of second order in the variable is negative:

Set

The previous inequality writes then:

The discriminant of this polynomial of second order is equal to , which gives an upper bound for .

7. A A Priori Estimate

We infer from the estimate a estimate using a classical Evans-Trudinger theorem [18, Theorem 17.14 page 461], which achieves the proof of Theorem 1.2. Let us state this Evans-Trudinger theorem; we use Gilbarg and Trudinger's notations for classical norms and seminorms of Hölder spaces (cf. [18] and [9, page 137]).

Theorem 7.1. Let be a bounded domain (i.e., an open connected set) of , . Let one denote by the set of real symmetric matrices. is a solution of

where is elliptic with respect to and satisfies the following hypotheses.

(1) is uniformly elliptic with respect to , that is, there exist two real numbers such that

(2) is concave with respect to in the variable . Since is of class , this condition of concavity is equivalent to

Then for all , one has the following interior estimate:

where depends only on , , and and depends only on , , , , , , , et . The notation used here denotes the matrix evaluated at . It is the same for the notations , , and [18, page 457].

7.1. The Evans-Trudinger Method

Let us suppose that there exists a constant such that for all , we have . In the following, we remove the index from to lighten the notations. In order to construct a estimate with , we prepare the framework of application of Theorem 7.1.

Let be a finite covering of the compact manifold by charts, and let be a partition of unity of class subordinate to this covering. The family of continuity equations writes in the chart where is a fixed integer as follows:

Besides, we have where the s denotes real derivatives; thus our equation writes:

This map is concave in the variable as the map appearing in the estimate (cf. (6.36)), (namely, for all fixed of , is concave on ). For all , let us consider a bounded domain of strictly included in :

The notation means that is strictly included in , namely, that . We will explain later how these domains are chosen. The map is of class and the solution since with . The equation on is now written in the form corresponding to the Theorem 7.1; it remains to check the hypotheses of this theorem on , namely, that

(1) is uniformly elliptic with respect to ; that is, there exist two real numbers such that

Moreover, we will impose ourselves to find real numbers independent of . (2) is concave with respect to in the variable . Since is of class , this concavity condition is equivalent to

This has been checked before. (3)The derivatives , , , and are controlled (these quantities are evaluated at ).

Once these three points checked, and since we have a estimate of by , Theorem 7.1 allows us to deduce that for all open set there exist two real numbers and depending only on , , , , on the uniform estimate of , and on the uniform estimates of the quantities , , , and , so in particular and are independent of , such that

The Choice of and

Let us denote by the support of the function :

The set is compact, and it is included in the open set of , and is separated locally compact; then by the theorem of intercalation of relatively compact open sets, applied twice, we deduce the existence of two relatively compact open sets and such that

The set is required to be connected: for this, it suffices that be connected even after restriction to a connected component in of a point of ; indeed, this connected component is an open set of since is locally connected (as an open set of ); moreover it is bounded since is bounded.

Application of the Theorem

Let ; the norm is submultiplicative; then

But, by (7.8) we have where depends only on , , , , (the constant of the estimate) and the uniform estimates of the quantities , , , and . We obtain consequently the needed estimate:

Let us now check the hypotheses and above.

7.2. Uniform Ellipticity of on

Let and :

Let us recall that (cf. (6.36)); we consequently obtain

In the following, we denote . Thus

Besides, let us denote and instead of in order to lighten the formulas. We obtain by the invariance formula (2.7) that

But by Proposition 6.10 and the inequalities , we have for (6.72)

Combining (7.16) and (7.17), we obtain

But

And ; then

Consequently, and since , the checking of the hypothesis of uniform ellipticity of the Theorem 7.1 is reduced to find two real numbers such that

By the min-max principle applied on to the Hermitian form relatively to the canonical one, we have

But the functions and are continuous on which is compact since it is a closed set of the compact manifold (cf. (7.5) for the choice of the domains ), so these functions are bounded and reach their bounds; thus

By the inequalities (7.18) and (7.23), we deduce that

The real numbers and depend on , , , , , , , and and are independent of , and , which achieves the proof of the global uniform ellipticity.

7.3. Uniform Estimate of , , , and

In this subsection, we estimate uniformly the quantities , ,, and (recall that these quantities are evaluated at ) by using the same technique as in the previous subsection for the proof of uniform ellipticity (7.24).

We have

For (7.14), we obtain

and for (7.16), we infer then by the invariance formula (2.7) that

where such that and . We can then write:

But ; then for all , , consequently

which gives the needed uniform estimate for :

Similarly

And we have

where is the matrix whose all coefficients are equal to zero except the coefficient which is equal to , and the matrix is obtained from by the formula , thus

where    and    are as before for  .

Since for all , we obtain for that

where , which gives the needed uniform estimate for :

Concerning , we have

A calculation shows that

All the terms are uniformly bounded; it remains to justify that the term in second derivative is also uniformly bounded:

But we know the second derivatives of at a diagonal matrix by (2.5). Besides, we have by (7.17), and since , it remains only to control the quantities with to prove that is uniformly bounded. But since , we have [11]. Moreover, by the pinching of the eigenvalues, we deduce automatically that

which achieves the checking of the fact that is uniformly bounded.

Similarly, we establish a uniform estimate of using this calculation:

which achieves the proof of the estimate.

Acknowledgment

This work was accomplished when the author was a Ph.D. student at the Nice Sophia-Antipolis University in France.

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