Complex Hessian Equations on Some Compact Kähler Manifolds
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 . 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 , see also  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 . Thanks to Julien Keller, we learned of an independent work  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.
Lemma 1.3. Consider .
Proof. See [9, page 44].
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  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 , 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 , 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 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 , 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 .
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.  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  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