<p style='text-indent:20px;'>By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper hyperspheres. Our analysis reveals that certain singular-looking equations, such as <inline-formula><tex-mat...

Using Monge-Amp\`ere geometry, we study the singular structure of a class nonlinear equations in three dimensions, arising geophysical fluid dynamics. We extend seminal earlier work on geometry by examining role an induced metric Lagrangian submanifolds cotangent bundle. In particular, show that signature serves as classification equation, while singularities and elliptic-hyperbolic transitions...

An II x m matrix A is called bottleneck Monge matrix if max{ajj, a,,} < max{a,, ali} for all l<i<r<n, 1 < j < s < m. The matrix A is termed permuted bottleneck Monge matrix, if there exist row and column permutations such that the permuted matrix becomes a bottleneck Monge matrix. We first deal with the special case of Cl bottleneck Monge matrices. Next, we derive several fundamental properties...

Abstract. We derive a priori C estimates for a class of complex Monge-Ampère type equations on Hermitian manifolds. As an application we solve the Dirichlet problem for these equations under the assumption of existence of a subsolution; the existence result, as well as the second order boundary estimates, is new even for bounded domains in C n . Mathematical Subject Classification (2010): 58J05...

By constructing appropriate smooth supersolutions, we establish sharp lower bounds near the boundary for modulus of nontrivial solutions to singular and degenerate Monge-Ampère equations form $ \det D^2 u = |u|^q with zero condition on a bounded domain in \mathbb R^n $. These imply that currently known global Hölder regularity results these are optimal all q negative, almost 0\leq q\leq n-2 Our...

We show the optimal $C^{1,1}$ regularity of geodesics in nef and big cohomology class on K\"ahler manifolds away from non-K\"ahler locus, assuming sufficiently regular initial data. As a special case, we prove metrics compact varieties singular locus. Our main novelty is an improved boundary estimate for complex Monge-Amp\`ere equation that does not require strict positivity reference form near...

Abstract In this note, we investigate some regularity aspects for solutions of degenerate complex Monge–Ampère equations (DCMAE) on singular spaces. First, study the Dirichlet problem DCMAE Stein spaces, showing a general continuity result. A consequence our results is that Kähler–Einstein potentials are continuous at isolated singularities. Next, establish global to when reference class belong...

We study adiabatic limits of Ricci-flat Kähler metrics on a Calabi-Yau manifold which is the total space of a holomorphic fibration when the volume of the fibers goes to zero. By establishing some new a priori estimates for the relevant complex Monge-Ampère equation, we show that the Ricci-flat metrics collapse (away from the singular fibers) to a metric on the base of the fibration. This metri...

We consider degenerate Monge-Ampere equations on compact Hessian manifolds. establish compactness properties of the set normalized quasi-convex functions and show local global comparison principles for twisted operators. then use Perron method to solve whose RHS involves an arbitrary probability measure, generalizing works Cheng-Yau, Delanoe, Caffarelli-Viaclovsky Hultgren-Onnheim. The intrinsi...