The Dirichlet Problem on Almost Hermitian Manifolds
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The Dirichlet Problem on Almost Hermitian Manifolds Chang Li1 · Tao Zheng2 Received: 5 May 2020 / Accepted: 9 October 2020 © Mathematica Josephina, Inc. 2020
Abstract We prove second-order a priori estimate on the boundary for the Dirichlet problem of a class of fully nonlinear equations on compact almost Hermitian manifolds with smooth boundary. As applications, we solve the Dirichlet problem of the Monge–Ampère type equation and of the degenerate Monge–Ampère equation. Keywords Dirichlet problem · Monge–Ampère type equation · Degenerate Monge–Ampère equation · Almost Hermitian manifold · A priori estimate Mathematics Subject Classification 35J25 · 35J60 · 58J05 · 58J32
1 Introduction Let (M, J , g) be an almost Hermitian manifold with dimR M = 2n and the almost complex structure J , where g is the Hermitian metric, i.e., a Riemannian metric with g(J X , J Y ) = g(X , Y ) for all vector fields X , Y ∈ X(M) (set of all the global and smooth vector fields). Then we can define a real (1, 1) form ω by ω(X , Y ) := g(J X , Y ), ∀ X , Y ∈ X(M). This form ω is determined uniquely by g and vice versa. In what follows, we will not distinguish these two terms.
Chang Li: Supported by the China post-doctoral Grant No. BX20200356. Tao Zheng: Supported by Beijing Institute of Technology Research Fund Program for Young Scholars.
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Tao Zheng [email protected] Chang Li [email protected]
1
Hua Loo-Keng Center for Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
2
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
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C. Li, T. Zheng
Fix a real (1, 1) form χ which is not necessarily positive definite. For each u ∈ C 2 (M, R), we define a new real (1, 1) form χu by χu := χ +
√
−1∂∂u + W (du),
(1.1)
where W (du) is a real (1, 1) form which depends linearly on du. Note that we do not assume that χu is positive definite, and the form χu defines an endomorphism χu of T 1,0 M which is Hermitian with respect to ω, i.e., ω X , χu (Y ) = ω(χu (X ), Y¯ ) = χu (X , Y¯ ), ∀ X , Y ∈ (T 1,0 M),
(1.2)
where (•) denotes the set of smooth sections of vector bundle •. In the following, we denote by λ(χu ) the n-tuple of eigenvalues of χu (i.e., the eigenvalues of χu with respect to the Hermitian metric ω), and use (1.2) as the definition of the operator . Given h ∈ C ∞ (M, R), we study the equation of χu given by F(χu ) = f (λ(χu )) = h, on M,
(1.3)
u ∂ M = ϕ, ϕ ∈ C ∞ (∂ M, R),
(1.4)
with the Dirichlet data
where f is a smooth symmetric function of the eigenvalues of χu . The Dirichlet problem has been extensively studied since the work of Ivochkina [29] and Caffarelli, Nirenberg & Spruck [6]; see, for example, [10,14,24–27,33,46,47,49]. We refer to [34] for recent progress and further references on this subject. We usually suppose that f is defined on an open symmetric convex cone n , with vertex at the origin 0, and that the cone satisfies that ⊃ := R n (λ1 , . . . , λn ) ∈
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