Related papers: Envelopes with Prescribed Singularities

In this paper, we prove the $C^{1, 1}$-regularity of the plurisubharmonic envelope of a $C^{1,1}$ function on a compact Hermitian manifold. We also present examples to show this regularity is sharp.

The goal of this work is to prove the regularity of certain quasi-plurisubharmonic upper envelopes. Such envelopes appear in a natural way in the construction of hermitian metrics with minimal singularities on a big line bundle over a…

We establish the C^{1,1} regularity of quasi-psh envelopes in a Kahler class, confirming a conjecture of Berman.

We investigate envelopes of positive metrics with a prescribed singularity type. First we generalise work of Berman to this setting, proving C^{1,1} regularity of such envelopes, showing their Monge-Ampere measure is supported on a certain…

We prove a $C^{1,1}$ estimate for solutions of complex Monge-Amp\`ere equations on compact K\"ahler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong-Sturm. As…

We develop a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Amp\`ere equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. Our method allows…

Let $(X,\omega)$ be a compact K\"ahler manifold. We prove the existence and uniqueness of solutions to complex Monge-Amp\`ere equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is…

Let $X$ be a compact complex manifold of complex dimension $n$ and $\alpha$ be a smooth closed real form on $X$ such that its cohomology class $\{ \alpha \}\in H^{1,1}(X, \mathbb{R})$ is big. In this paper we prove that, given a bounded…

We show the optimal $C^{1,1}$ regularity of geodesics in nef and big cohomology class on K\"ahler manifolds away from the non-K\"ahler locus, assuming sufficiently regular initial data. As a special case, we prove the $C^{1,1}$ regularity…

We show that in $\mathbb{C}^2$ if the set of strongly regular points are closed in the boundary of a smooth bounded pseudoconvex domain, then the domain is c-regular, that is, the plurisubharmonic upper envelopes of functions continuous up…

We obtain higher order estimates for a parabolic flow on a compact Hermitian manifold. As an application, we prove that a bounded $\hat{\omega}$-plurisubharmonic solution of an elliptic complex Monge-Amp\`{e}re equation is smooth under an…

We study the plurisubharmonic envelopes of functions in the setting of domains in $\mathbb C^n$. In particular we prove a complex analogue of a result of De Philippis and Figalli concerning the optimal regularity of such envelopes in smooth…

We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the…

We establish plurisubharmonicity of the envelope of Poisson and Lelong functionals on almost complex manifolds. That is, we generalize the corresponding results for complex manifolds and almost complex manifolds of complex dimension two. We…

For a compact spinc manifold $X$ with boundary $b_1(\partial X)=0$, we consider moduli spaces of solutions to the Seiberg-Witten equations in a generalized double Coulomb slice in $L^2_1$ (i.e., $W^{1,2}$) Sobolev regularity. We prove they…

Let (X,L) be a polarized compact manifold, i.e. L is an ample line bundle over X and denote by H the infinite dimensional space of all positively curved Hermitian metrics on L equipped with the Mabuchi metric. In this short note we show,…

Let $X$ be a compact K\"ahler manifold. We study plurisupported currents on $X$, i.e. closed, positive $(1,1)$-currents which are supported on a pluripolar set. In particular, we are able present a technical generalization of…

We prove $\varepsilon$-regularity theorems for varifolds with capillary boundary condition in a Riemannian manifold. These varifolds were first introduced by Kagaya-Tonegawa \cite{KaTo}. We establish a uniform first variation control for…

We prove that the regularity of the extremal function of a compact subset of a compact K\"ahler manifold is a local property, and that the continuity and H\"older continuity are equivalent to classical notions of the local $L$-regularity…

In this note we investigate the regularity of geodesics in the space of convex and plurisubharmonic functions. In the real setting we prove (optimal) local C^{1,1} regularity. We construct examples which prove that the global C^{1,1}…