Related papers: Geometric conditions which imply compactness of th…
Consider a strictly convex bounded regular domain $C$ of $\R^3$. For any arbitrary finite topological type we find a compact Riemann surface $\mathcal{M}$, an open domain $M\subset \mathcal{M}$ with the fixed topological type, and a…
Given a compact manifold $M$ with boundary $\partial M$, in this paper we introduce a global symbolic calculus of pseudo-differential operators associated to $(M,\partial M)$. The symbols of operators with boundary conditions on $\partial…
In Part I we gave a polynomial growth lower-bound for the number of nodal domains of a Hecke-Maass cuspform in a compact part of the modular surface, assuming a Lindel\"of hypothesis. That was a consequence of a topological argument and…
In this work, we investigate the geometry and topology of compact Einstein-type manifolds with nonempty boundary. First, we prove a sharp boundary estimate, as consequence we obtain under certain hypotheses that the Hawking mass is bounded…
In this article, we show that (i) any smooth function on compact Riemann surface with non-empty smooth boundary $ (M, \partial M, g) $ can be realized as a Gaussian curvature function; (ii) any smooth function on $ \partial M $ can be…
We study spectral properties of divergence form elliptic operators $-\textrm{div} [A(z) \nabla f(z)]$ with the Neumann boundary condition in planar domains (including some fractal type domains), that satisfy to the quasihyperbolic boundary…
For a bounded Lipschitz domain with Lipschitz interface we show the following compactness theorem: Any $L^2$-bounded sequence of vector fields with $L^2$-bounded rotations and $L^2$-bounded divergences as well as $L^2$-bounded tangential…
We introduce the notion of domains with uniform squeezing property, study various analytic and geometric properties of such domains and show that they cover many interesting examples, including Teichmuller spaces and Hermitian symmetric…
In this paper, the necessary and sufficient conditions in order that a smooth mapping F be a dependence of a complete solution of some second-order ordinary differential equation on Neumann conditions are deduced. These necessary and…
We obtain a conceptually new differential geometric proof of P.F. Klembeck's result that the holomorphic sectional curvature of a strictly pseudoconvex domain approaches (in the boundary limit) the constant sectional curvature of the…
We prove that any smooth Riemannian manifold of non-negative scalar curvature and with a strictly mean convex and compact boundary component can be (C^2) extended beyond the component to have non-negative scalar curvature and to enjoy…
We extend Beem's three completeness notions -- finite compactness, timelike Cauchy completeness, and Condition A -- originally defined for spacetimes, to Lorentzian length spaces and study their relationships. We prove that finite…
We investigate the geometry and topology of compact submanifolds of arbitrary codimension in space forms satisfying a certain pinching condition involving the length of the second fundamental form and the mean curvature. We prove that this…
The Diederich--Forn\ae ss index has been introduced since 1977 to classify bounded pseudoconvex domains. In this article, we derive several intrinsic, geometric conditions on boundary of domains for arbitrary indexes. Many results, in the…
In the paper we study the geometry of semitube domains in $\mathbb C^2$. In particular, we extend the result of Burgu\'es and Dwilewicz for semitube domains dropping out the smoothness assumption. We also prove various properties of…
In this paper we study optimization problems for Neumann eigenvalues $\mu_k$ among convex domains with a constraint on the diameter or the perimeter. We work mainly in the plane, though some results are stated in higher dimension. We study…
We prove that in a strongly pseudoconvex domain with smooth boundary, then the length of a geodesic for the Kobayashi-Royden infinitesimal metric between two points is bounded by a constant multiple of the Euclidean distance between the…
In the given paper we first introduce $\bar{N}_{\Delta^{-}}^{q}$ summable difference sequence spaces and prove some properties of these spaces. We then obtain the necessary and sufficient conditions for infinite matrices $A$ to map these…
In this article, we study the range of the Cauchy-Riemann operator $\bar\partial$ on domains in the complex projective space $\Bbb{CP}^n$. In particular, we show that $\bar\partial$ does not have closed range in $L^2$ for (2,1)-forms on the…
We obtain an improved version of the area theorem for not necessarily differentiable horizons which, in conjunction with a recent result on the completeness of generators, allows us to prove that under the null energy condition every…