We investigate the Monge–Ampère equation on a compact Hermitian manifold. We begin by refining the challenging zeroth order estimate previously established for the equation when the right-hand side is a smooth, positive function. Subsequently, we tackle the more complex case where the right-hand side depends explicitly on the solution and its gradient, under various hypotheses. A primary contribution of this work is a comprehensive exposition of all necessary estimates, with particular emphasis on the crucial gradient and Hessian estimates. In our specific framework, the gradient estimate is a prerequisite for achieving the necessary Laplacian estimation. Furthermore, the Hessian estimate is essential either for the application of the Evans–Krylov theory to obtain Hölder estimates for the second order derivatives or to generalize Calabi estimates for third-order derivatives.
We study nonlinear Choquard equations with general radially non-increasing convolution kernels 𝒥(|x|) that interpolate between Riesz potentials I_α and I_β of order 0<α≤β
Given H∈ [0,1) and given a C^0 exterior domain Ω in a H- hypersphere of ℍ^n+1, the existence of hyperbolic Killing graphs of CMC H defined in Ω with boundary ∂Ω included in the H- hypersphere is obtained.
We prove that a normal hyperplane section of the Segre variety Σ _m, n has a non-reductive automorphism group if m n or it is not smooth. In the smooth case, this recovers Hano’s result. Our proof proceeds at the level of Lie algebras.
Abstract We give a formula for a birational map on the Schubert cell associated to each Weyl group element of $$G=\operatorname {GL}(r)$$ G = GL ( r ) . The map simplifies the UDL decomposition of matrices, providing structural insight into the Schubert cell decomposition of the flag variety G / B , where B is a Borel subgroup. An application of the formula includes a new proof of the existence of Whittaker functionals for principal series representations of $$\operatorname {GL}(r,{\mathbb {R}})$$ GL ( r , R ) via integration by parts. In this paper, we establish combinatorial properties of the birational map and prove auxiliary results.
