Related papers: Les g\'{e}om\'{e}tries de Hilbert sont \`{a} g\'{e…
In this article, we consider a bounded pseudoconvex domain in ${\bf C}^2$ satifying: (a) it admits a proper holomorphic mapping $f$ onto the unit ball $B^2$, and (b) it is simply connected and has a real analytic boundary. According to…
A (positive) locally convex curve in the 2-sphere is a curve with positive geodesic curvature (i.e., which always turns left). In the 3-sphere, it is a curve with positive torsion. In this work we discussed the topology of spaces of such…
In this paper, we show that the convex domains of the hyperbolic space which are almost extremal for the Faber-Krahn or the Payne-Polya-Weinberger inequalities are close to geodesic balls. Our proof is also valid in other space forms and…
On a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, $n>1$, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation up to the boundary is obstructed by a local CR invariant of the boundary. For a…
We show that the nearest point retraction is a uniform quasi-isometry from the Thurston metric on a hyperbolic domain in the Riemann sphere to the boundary of the convex hull of its complement. As a corollary, one obtains explicit bounds on…
We verify the three-dimensional Glassey conjecture for exterior domain (M, g), where the metric g is asymptotically Euclidean, provided that certain local energy assumption is satisfied. The radial Glassey conjecture exterior to a ball is…
We consider the Robin Laplacian in the exterior of a bounded simply-connected Lipschitz domain in the hyperbolic plane. We show that the essential spectrum of this operator is $[\frac14,\infty)$ and that, under convexity assumption on the…
We show that, given a metric space $(Y,d)$ of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $\mu$ on $Y$ giving finite mass to bounded sets, the resulting metric measure space $(Y,d,\mu)$ is…
We give, in dimensions three or greater, an example of a bounded, pseudoconvex, circular domain in complex space with smooth real analytic boundary and non-compact automorphism group which is not biholomorphically equivalent to any…
There recently has been some interest in the space of functions on an interval satisfying the heat equation for positive time in the interior of this interval. Such functions were characterised as being analytic on a square with the…
We prove that the Teichm\"{u}ller space $\mathscr{T}$ of a closed surface of genus $g \ge 2$ cannot be biholomorphic to any domain which is locally strictly convex at some boundary point.
We prove that any finitely smooth axially symmetric strictly convex domain, with everywhere positive curvature and sufficiently close to an ellipse is area spectrally rigid. This means that any area-isospectral family of domains in this…
We prove that if $(M,g)$ is a topological 3-ball with a $C^4$-smooth Riemannian metric $g$, and mean-convex boundary $\partial M$ then knowledge of least areas circumscribed by simple closed curves $\gamma \subset \partial M$ uniquely…
In a 2013 paper, Gromov proves that if smooth Riemannian metrics $g_i$ converge to a smooth Riemannian metric $g$ uniformly, and $g_i$ have scalar curvature uniformly bounded below, then $g$ shares the same scalar curvature lower bound. In…
Given a Riemannian manifold M and a hypersurface H in M, it is well known that infinitesimal convexity on a neighborhood of a point in H implies local convexity. We show in this note that the same result holds in a semi-Riemannian manifold.…
After a study of the Kobayashi metrics on certain scaled domains, we show the stabilities of the infinitesimal Kobayashi metrics and the integrated distances in different scaling processes. As an application, we prove that bounded…
We investigate the geography of Hilbert schemes parametrizing closed subschemes of projective space with specified Hilbert polynomials. We classify Hilbert schemes with unique Borel-fixed points via combinatorial expressions for their…
The extended Gaussian family is the closure of the Gaussian family obtained by completing the Gaussian family with the counterpart elements induced by degenerate covariance or degenerate precision matrices, or a mix of both degeneracies.…
It is shown that any bounded metric space can be isometrically embedded into the Gromov--Hausdorff metric class GH. This result is a consequence of local geometry description of the class GH in a sufficiently small neighborhood of a generic…
We say that a metric graph is uniformly bounded if the degrees of all vertices are uniformly bounded and the lengths of edges are pinched between two positive constants; a metric space is approximable by a uniform graph if there is one…