Related papers: On minimal kernels and Levi currents on weakly com…

In previous works, G. Tomassini and the authors studied and classified complex surfaces admitting a real-analytic pluri-subharmonic exhaustion function; let $X$ be such a surface and $D\subseteq X$ a domain admitting a \emph{continuous}…

Relying on the recent work of Liu-Sz\'ekelyhidi we give a weak asymptotic estimate for the Bergman kernels of polarized K\"ahler manifolds with Ricci lower bound and Sobolev constant upper bound. We will also give a simple proof for the…

A local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition,…

In this paper, a new class of Sobolev spaces with kernel function satisfying a L\'evy-integrability type condition on compact Riemannian manifolds is presented. We establish the properties of separability, reflexivity, and completeness. An…

The purpose of this note is twofold. First we show that, for weakly differentiable maps between Riemannian manifolds of any dimension, a smallness condition on a Morrey-norm of the gradient is sufficient to guarantee that the pulled-back…

If $(X,J)$ is an almost complex manifold, then a function $u$ is said to be plurisubharmonic on $X$ if it is upper semi-continuous and its restriction to every local pseudo-holomorphic curve is subharmonic. As in the complex case, it is…

Let $X$ and $Y$ be real analytic manifolds and let $\Lambda \subseteq T^*X$ and $\Sigma \subseteq T^*Y$ be closed conic subanalytic singular isotropics. Given a sheaf $K \in \mathrm{Sh}_{-\Lambda \times \Sigma}(X \times Y)$ microsupported…

We show how to build a kernel \[ K_X(x,y)=\sum_{m=0}^Xh(\lambda_m/{\lambda_X})\varphi_m(x)\overline{\varphi_m(y)} \] on a compact Riemannian manifold $M$, which is positive up to a negligible error and such that $K_X(x,x)\approx X$. Here…

We associate to any compact semi-algebraic set $X \subset \mathbb R^n$ a chain complex of currents $S_\ast (X)$ generated by integration along semi-algebraic submanifolds and we analyze the corresponding homology groups. In particular, we…

Motivated by results of Juh\'asz and van Mill in [13], we define the cardinal invariant $wt(X)$, the weak tightness of a topological space $X$, and show that $|X|\leq 2^{L(X)wt(X)\psi(X)}$ for any Hausdorff space $X$ (Theorem 2.8). As…

Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup. Then there exists a closed complex subgroup $J$ of $G$ containing $H$ such that the fibration $\pi:G/H \to G/J$ is the holomorphic reduction of $G/H$, i.e.,…

Let X be a complex manifold of dimension at least 2 which has an exhaustion function whose Levi form has at each point at least 2 positive eigenvalues. We prove that there are proper holomorphic discs in X through any given point

Let X be a hermitian manifold and let L^k be a high power of a hermitian line bundle over X. Local versions of Demailly's holomorphic Morse inequalities are presented - after integration they yield the usual inequalities. The local weak…

The Jensen envelope $J\phi$ of an upper semicontinuous function $\phi$ on a complex manifold X is defined at $x\in X$ as the infimum of $\mu(\phi)$ over all Jensen measures $\mu$ centred at x. The Poisson envelope $P\phi$ is defined by…

Semi-implicit variational inference (SIVI) extends traditional variational families with semi-implicit distributions defined in a hierarchical manner. Due to the intractable densities of semi-implicit distributions, classical SIVI often…

We prove a compactness theorem for embedded measured hyperbolic Riemann surface laminations in a compact almost complex manifold $(X, J)$. To prove compactness result, we show that there is a suitable topology on the space of measured…

In this paper, we study the existence and multiplicity of weak solutions for a general class of elliptic equations (\mathscr{P}_{\lambda}) in a smooth bounded domain, driven by a nonlocal integrodifferential operator…

We study the possible singularities of an $m$-subharmonic function $\varphi$ along a complex submanifold $V$ of a compact K\"ahler manifold, finding a maximal rate of growth for $\varphi$ which depends only on $m$ and $k$, the codimension…

We prove that the critical points of various energies such as the area, the Willmore energy, the frame energy for tori...etc among possibly branched immersions constrained to evolve within a smooth sub-manifold of the Teichm\"uller space…

For $l$-homogeneous linear differential operators $\mathcal{A}$ of constant rank, we study the implication $v_j\rightharpoonup v$ in $X$ and $\mathcal{A} v_j\rightarrow \mathcal{A} v$ in $W^{-l}Y$ implies $F(v_j)\rightsquigarrow F(v)$ in…