Related papers: Pseudo-Einstein and Q-flat metrics with eigenvalue…

In this note we prove three rigidity results for Einstein manifolds with bounded covering geometry. (1) An almost flat manifold $(M,g)$ must be flat if it is Einstein, i.e. $\operatorname{Ric}_g=\lambda g$ for some real number $\lambda$.…

This paper provides a connection between two distinct branches of research in CR geometry -- namely, analytic and geometric conditions that suffice to establish the closed range of the Cauchy-Riemann operator and CR invariants on CR…

We show that for bounded domains in $\mathbb C^n$ with $\mathcal C^{1,1}$ smooth boundary, if there is a closed set $F$ of $2n-1$-Lebesgue measure $0$ such that $\partial \Omega \setminus F$ is $\mathcal C^{2}$-smooth and locally…

We propose the concept of Diederich--Forn{\ae}ss and Steinness indices on compact pseudoconvex CR manifolds of hypersurface type in terms of the D'Angelo 1-form. When the CR manifold bounds a domain in a complex manifold, under certain…

We prove that a relatively compact pseudoconvex domain with smooth boundary in an almost complex manifold admits a bounded strictly plurisubharmonic exhaustion function. We use this result for the study of convexity and hyperbolicity…

We show that each of the topological 4-manifolds $CP^2#k\bar{CP^2}, for $k = 6, 7$ admits a smooth structure which has an Einstein metric of scalar curvature $s > 0$, a smooth structure which has an Einstein metric with $s < 0$ and…

We study 4-dimensional second-Chern-Einstein almost-Hermitian manifolds. In the compact case, we observe that under a certain hypothesis the Riemannian dual of the Lee form is a Killing vector field. We use that observation to describe…

A Riemannian manifold is called almost positively curved if the set of points for which all $2$-planes have positive sectional curvature is open and dense. We find three new examples of almost positively curved manifolds: $Sp(3)/Sp(1)^2$,…

Let $M$ be a compact K\"ahler manifold and $N$ be a subvariety with codimension greater than or equal to 2. We show that there are no complete K\"ahler--Einstein metrics on $M-N$. As an application, let $E$ be an exceptional divisor of $M$.…

The following problem is addressed: A $3$-manifold $M$ is endowed with a triple $\Omega = \big(\Omega^1,\Omega^2,\Omega^3\big)$ of closed $2$-forms. One wants to construct a coframing $\omega = \big(\omega^1,\omega^2,\omega^3\big)$ of $M$…

We discuss a sharp lower bound for the first positive eigenvalue of the sublaplacian on a closed, strictly pseudoconvex pseudo-hermitian manifold of dimension $2m+1\geq 5$. We prove that the equality holds iff the manifold is equivalent to…

Riemannian Manifolds may be $C^{1,1}$ and the geometry of these manifolds is investigated in \cite{Groah1}. Here, a similar analysis is given for pseudohermitian, torsion-free manifolds whereby, instead of assuming that the metric is…

An extension of the estimates for the squeezing function of strictly pseudoconvex domains obtained recently by J. E. Forn\ae ss and E. Wold in \cite{FW1} is applied to derive a sharp boundary behaviour of invariant metrics and Bergman…

In this paper, we consider the CPE conjecture in the frame-work of $K$-contact and $(\kappa, \mu)$-contact manifolds. First, we prove that if a complete $K$-contact metric satisfies the CPE is Einstein and is isometric to a unit sphere…

We derive optimal estimates for the Bergman kernel and the Bergman metric for certain model domains in $\mathbb{C}^2$ near boundary points that are of infinite type. Being unbounded models, these domains obey certain geometric constraints…

For any smooth Riemannian metric on an $(n+1)$-dimensional compact manifold with boundary $(M,\partial M)$ where $3\leq (n+1)\leq 7$, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by…

Let $\Omega$ be a pseudoconvex domain with $C^2$-smooth boundary in $\mathbb CP^n$. We prove that the $\bar\partial-Neumann operator $N$ exists for $(p,q)$-forms on $\Omega$. Furthermore, there exists a $t_0>0$ such that the operators $N$,…

We prove that various spaces of constrained positive scalar curvature metrics on compact 3-manifolds with boundary, when not empty, are contractible. The constraints we mostly focus on are given in terms of local conditions on the mean…

Suppose M_t is a smooth family of compact connected two dimensional submanifolds of Euclidean space E^3 without boundary varying isometrically in their induced Riemannian metrics. Then we show that the mean curvature integrals over M_t are…

For a strongly pseudo-convex complex Finsler manifold M, a bundle U of adapted unitary frames is canonically defined. A non-linear Hermitian connection on U, invariant under local biholomorphic isometries, is given and it proved to be…