arXiv research

We prove that on compact Kähler manifolds solutions to the complex Monge-Ampère equation, with the the right hand side in $L^p, p>1,$ are Hölder continuous.

We present a somewhat new proof to the $C^{2,α}$-aprori estimate for the uniform elliptic Monge-Ampere equations, in both the real and complex settings. Our estimates do not need to differentiate the equation, and only depends on the $C^{α^{\prime}}-$norm of the right hand side of the equation, $0<α<α^{\prime}$.

We consider the complex Monge-Ampére equation on complete Kähler manifolds with cusp singularity along a divisor when the right hand side $F$ has rather weak regularity. We proved that when the right hand side $F$ is in some \emph{weighted} $W^{1,p_0}$ space for $p_0 > 2n$, the Monge-Ampére equation has a classical $W^…

Note on gradient estimates for complex Monge-Ampere equation.

Existence and uniqueness of bounded solutions to complex Monge-Ampère flows on Kähler manifolds.

Study on Monge-Ampère equations with polynomial growth rates.

Derives estimates for geometric elliptic equations on complex manifolds.

We derive a priori $C^2$ estimates for the $χ$-plurisubharmonic solutions of general complex Hessian equations with right-hand side depending on gradients.

This paper establishes the global curvature estimate for the $n-2$ curvature equation with the general right hand side which partially solves this longstanding problem.

The study finds continuous solutions to complex Hessian equations on compact Hermitian manifolds.

The main result asserts the existence of continuous solutions of the complex Monge-Ampère equation with the right hand side in $L^p, p>1$, on compact Hermitian manifolds.

Let $M$ be an $n$-dimensional Lagrangian submanifold of a complex space form. We prove a pointwise inequality $δ(n_1,\ldots,n_k) \leq a(n,k,n_1,\ldots,n_k) \|H\|^2 + b(n,k,n_1,\ldots,n_k)c,$ with on the left hand side any delta-invariant of the Riemannian manifold $M$ and on the right hand side a linear combination o…

We consider a Monge-Ampère functional and its corresponding second boundary value problem, a nonlinear fourth order PDE with two Dirichlet boundary conditions. This problem was solved by Trudinger-Wang and Le under the assumption that the right hand side of the equation is nonpositive. We remove this assumption, to set…

It is proved that solutions of the complex Monge-Ampère equation on compact Kähler manifolds with right hand side in $L^p, p>1$ are uniformly Hölder continuous under the assumption on non-negative orthogonal bisectional curvature.

Proves existence and uniqueness of viscosity solutions to complex Hessian equations on compact Hermitian manifolds.

Proves local solvability for $G_2$-structures with Poisson equations.

We introduce generalized Monge-Ampère capacities and use these to study complex Monge-Ampère equations whose right-hand side is smooth outside a divisor. We prove, in many cases, that there exists a unique normalized solution which is smooth outside the divisor.

We first obtain the interior $C^{1,1}$-regularity and solvability for the degenerate real Monge-Ampère equation in a bounded, $C^3$-smooth and strictly convex domain in $\mathbb R^d$ ($d\ge2$), assuming that the boundary data is only globally $C^{1,1}$, and the $d$-th root of the nonnegative right-hand side is globally…

Using the result by D.Gessler (Differential Geom. Appl. 7 (1997) 303-324, DIPS-9/98, http://diffiety.ac.ru/preprint/98/09_98abs.htm), we show that any invariant variational bivector (resp., variational 2-form) on an evolution equation with nondegenerate right-hand side is Hamiltonian (resp., symplectic).

In this paper, we obtain a basic Chen's inequality for a C-totally real submanifold in a generalized $(κ,μ)$-contact space forms involving intrinsic invariants, namely the scalar curvature and the sectional curvatures of the submanifold on left hand side and the main extrinsic invariant, namely the squared mean curvatu…

In this note, we derive a Liouville theorem for the complex Monge-Ampère equation. Our result states that if the global solution $u$ of the complex Monge-Ampère equation with constant right-hand side differs from a quadratic polynomial solution by $o(\abs{x}^2)$ at infinity, then $u$ is a quadratic polynomial.

Proves bounded subsolution theorem for complex Monge-Ampère equation on compact Hermitian manifolds.

We prove the existence of weak solutions of complex $m-$Hessian equations on compact Hermitian manifolds for the nonnegative right hand side belonging to $L^p, p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the equation has been recently solved by Szekelyhidi and Zhang. We also give a stabilit…

We prove that any two Kahler potentials on a compact Kahler manifold can be connected by a geodesic segment of C^{1,1} regularity. This follows from an a priori interior real Hessian bound for solutions of the nondegenerate complex Monge-Ampere equation, which is independent of a positive lower bound for the right hand…

Bayesian methods solve complex nonlinear PDEs efficiently.

Study solves complex equation on specific types of manifolds.

We study the Dirichlet problem for the Lagrangian phase operator, in both the real and complex setting. Our main result states that if $Ω$ is a compact domain in $\mathbb{R}^{n}$ or $\mathbb{C}^n$, then there exists a solution to the Dirichlet problem with right-hand side $h(x)$ satisfying $|h(x)| > (n-2)\fracπ{2}$ and…

In this paper, we derive estimates for scalar curvature type equations with more singular right hand side. As an application, we prove Donaldson's conjecture on the equivalence between geodesic stability and existence of cscK when $Aut_0(M,J)\neq0$. Moreover, we also show that when $Aut_0(M,J)\neq0$, the properness of …

We prove stability of solutions of the complex Monge-Ampère equation on compact Hermitian manifolds, when the right hand side varies in a bounded set in $L^p, p>1$ and it is bounded away from zero. Such solutions are shown to be Hölder continuous. As an application we extend a recent result of Székelyhidi and Tosatti o…

Proves Hölder continuity of complex Monge-Ampère solutions.

We solve the Dirichlet problem for the complex Monge-Ampère equation on a strictly pseudoconvex with the right hand side being a positive Borel measure which is dominated by the Monge-Ampère measure of a Hölder continuous plurisubharmonic function. If the boundary data is continuous, then the solution is continuous. If…

We study an equation proposed by Fu and Yau as a natural $n$-dimensional generalization of a Strominger system that they solved in dimension $2$. It is a complex Hessian equation with right hand side depending on gradients. Building on the methods of Fu and Yau, we obtain $C^0$, $C^2$ and $C^{2,α}$ a priori estimates. …

We consider the complex Monge-Ampère equation on a compact Kähler manifold $(M, g)$ when the right hand side $F$ has rather weak regularity. In particular we prove that estimate of $\tφ$ and the gradient estimate hold when $F$ is in $W^{1, p_0}$ for any $p_0>2n$. As an application, we show that there exists a classical…

Solves Dirichlet problem for specific PSH functions on Hermitian manifolds.

Solves Dirichlet problem for elliptic equations on Hermitian manifolds.

We show that the complex Monge-Ampere equation on a compact Kaehler manifold (X,ω) of dimension n admits a Holder continuous omega-psh solution if and only if its right-hand side is a positive measure with Holder continuous super-potential. This property is true in particular when the measure has locally Holder continu…

We prove that the Kontsevich tetrahedral flow $\dot{\mathcal{P}} = \mathcal{Q}_{a:b} (\mathcal{P})$, the right-hand side of which is a linear combination of two differential monomials of degree four in a bi-vector $\mathcal{P}$ on an affine real Poisson manifold $N^n$, does infinitesimally preserve the space of Poisson…

Study continuity and Hölder estimates for solutions on Stein spaces.

CODE learns ODE dynamics from sparse data, outperforming neural and kernel methods.

New method of symmetrization applied to PDEs on spheres.

Continuous solutions found for complex geometry equations.

We prove the existence and uniqueness of continuous solutions to the complex Monge-Ampère type equation with the right hand side in $L^p$, $p>1$, on compact Hermitian manifolds. Next, we generalise results of Eyssidieux, Guedj and Zeriahi \cite{EGZ09, EGZ11} to compact Hermitian manifolds which {\em a priori} are not i…

Paper proves inequalities on Hermitian manifolds with applications to bounded solutions.

The paper proves a comparison principle for complex Monge-Ampère flows and solves a uniqueness problem.

Extends boundary estimates for Monge-Ampère equations in polygonal domains.

Stability and Hölder continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.

We introduce a general constructive setting of the density ratio estimation problem as a solution of a (multidimensional) integral equation. In this equation, not only its right hand side is known approximately, but also the integral operator is defined approximately. We show that this ill-posed problem has a rigorous …

Researchers find explicit solutions to complex Monge-Ampère equation.