Related papers: On the Levi-flat Plateau problem

Let $\Gamma$ be a compact codimension-two submanifold of $\mathbb{R}^n$, and let $L$ be a nontrivial real line bundle over $X = \mathbb{R}^n \setminus \Gamma$. We study the Allen--Cahn functional, \[E_\varepsilon(u) = \int_X \varepsilon…

Let $M$ be a real analytic hypersurface in $\bC^N$ which is finitely nondegenerate, a notion that can be viewed as a generalization of Levi nondegenerate, at $p_0\in M$. We show that if $M'$ is another such hypersurface and $p'_0\in M'$,…

The paper gives a complete description of local automorphism groups for Levi degenerate hypersurfaces of finite type in $\mathbb{C}^2$. We also prove that, with the exception of hypersurfaces of the form $v = |z|^k$, local automorphisms are…

In a recent paper, the author and I. Zelenko introduce the concept of modified CR symbols for organizing local invariants of $2$-nondegenerate CR structures. In this paper, we consider homogeneous hypersurfaces in $\mathbb{C}^4$, a natural…

Following on from ``Hyperbolic Plateau problems'' (by the same author), we provide a complete geometric description of solutions to the Plateau problem $(S,\phi)$ when $S$ is a compact Riemann surface with a finite number of points removed.

We investigate 3-nondegenerate CR structures in the lowest possible dimension 7 and show that 8 is the maximal dimension for the Lie algebra of symmetries of such structures. The next possible symmetry dimension is 6, and for the…

We study implications and consequences of well-posed solutions of Cauchy problems of a Novikov equation describing pseudospherical surfaces. We show that if the co-frame of dual one-forms satisfies certain conditions for a given periodic…

Motivated by recent works in Levi degenerate CR geometry, this article endeavours to study the wider and more flexible para-CR structures for which the constraint of invariancy under complex conjugation is relaxed. We consider…

That is, given a compact set $B \subset \mathbb{R}^n$ (the boundary) and a subgroup $L$ of the \v{C}ech homology group $\check{H}_{d-1}(B;G)$ of dimension $d$ over some commutative group $G$, we find a compact set $E \supset B$ such that…

Assume that $M$ is a CR compact manifold without boundary and CR Yamabe invariant $\mathcal{Y}(M)$ is positive. Here, we devote to study a class of sharp Hardy-Littlewood-Sobolev inequality as follows \begin{equation*} \Bigl| \int_M\int_M…

Using a geometric construction, we solve Plateau's Problem in the Heisenberg group $\mathbb{H}^{1}$ for intrinsic graphs defined on a convex domain $D$, under a smallness condition either on the boundary $\partial D$ or on the Lipschitz…

We prove the existence of a normal form for a real-analytic Levi-flat hypersurface defined by the vanishing of the real part of a holomorphic function with a Morse-Bott singularity. As a consequence, we recover the Burns-Gong normal form…

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a compact manifold with boundary $(M^{n+1},\partial M)$, $3\leq (n + 1)\leq 7$, we prove that, for any open subset $V$ of $\partial M$, there exists a compact, properly…

We consider a complex Plateau problem for strongly pseudoconvex contours in non K\"ahler manifolds. A positive solution in the case of manifolds carrying a pluriclosed Hermitian metric forms is given. For the general case we propose a…

In this paper, we show how to construct a special class of ruled hypersurfaces in the nonflat complex space forms $\mathbb{CP}^n$ and $\mathbb{C}H^n$. This is done by taking an arbitrary smooth curve in a totally geodesic (complex)…

On a real ($\mathbb F=\mathbb R$) or complex ($\mathbb F=\mathbb C$) analytic connected 2-manifold $M$ with empty boundary consider two vector fields $X,Y$. We say that $Y$ {\it tracks} $X$ if $[Y,X]=fX$ for some continuous function…

For the completion B of a local geometric normal domain, V. Srinivas asked which subgroups of Cl B arise as the image of the map from Cl A to Cl B on class groups as A varies among normal geometric domains with B isomorphic to the…

We study/construct (proper and non-proper) Morse functions on complete Riemannian manifolds, the level hypersurfaces of which have positive mean curvatures at all non-critical points. We show, for instance, that if a complete Rieannin…

Consider a compact Lie group $G$ and a closed subgroup $H<G$. Suppose $\mathcal M$ is the set of $G$-invariant Riemannian metrics on the homogeneous space $M=G/H$. We obtain a sufficient condition for the existence of $g\in\mathcal M$ and…

In this paper, we provide {\em effective} results on the non-embeddability of real-analytic hypersurfaces into a hyperquadric. We show that, for any $N >n \geq 1$, the defining functions $\varphi(z,\bar z,u)$ of all real-analytic…