科研之友
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摘要
On a compact connected -dimensional Kˋhler manifold with Kˋhler 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 Kˋhler 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 Kˋhler 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%26apos;Acad谷-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.
