It is known that a surface with parallel mean curvature vector field in a Riemannian product of two surfaces of constant Gaussian curvature carries a holomorphic quadratic differential. In this paper, we consider the Riemannian product of a sphere and a hyperbolic plane of opposite Gaussian curvatures and study the parallel mean curvature surfaces for which the differential vanishes. In particular, we classify all parallel mean curvature spheres, for which the differential vanishes for topological reasons.
We study the Riesz and logarithmic energies on the Grassmannian \operatorname{Gr}_{2,4} of 2 -dimensional subspaces of \mathbb{R}^{4} . We prove that the continuous Riesz and logarithmic energies are uniquely minimized by the uniform measure, and we obtain asymptotic upper and lower bounds for the minimal discrete energies, with matching orders for the next-order terms. Additionally, we define a determinantal point process on \operatorname{Gr}_{2,4} and compute the expected energy of the points coming from this random process, thereby obtaining explicit constants in the upper bounds for the Riesz and logarithmic energies.
Let \Omega \Subset \mathbb{C}^{n} be a bounded strongly pseudoconvex domain. For any concave increasing weight \chi \colon \mathbb{R}^{-} \to \mathbb{R}^{-} such that \chi(0) = 0 , we introduce and study finite energy classes \mathcal{E}_{\chi}(\Omega) of plurisubharmonic functions, using the Orlicz space formalism. We investigate the range of the Monge–Ampère operator on these classes, and conjecture that this should lead to an integral characterization of the image of bounded plurisubharmonic functions, an open problem since the birth of pluripotential theory more than forty years ago.
In this article, we give a sufficient and necessary condition on the indices of Besov-type spaces on \mathbb{R} under which the characterization of these spaces in terms of the Haar system holds. This confirms that the sufficient condition obtained by H. Triebel is optimal. Moreover, a special case of our results also negatively answers a question, asked by H. Triebel, about the characterization of lifted local bounded mean oscillation spaces J^{s}{\mathrm{bmo}} on \mathbb{R} with s\in(-1,-1/2] in terms of the Haar system. As an application, we prove that characteristic functions of intervals are pointwise multipliers of Besov-type spaces with the aforementioned optimal indices.
We classify all positive solutions to -\Delta u=e^{-2u} in the half space \mathbb{R}^{N}_{+} , under Dirichlet boundary condition.
