In this article, we study the Hamiltonian non-displaceability of Gauss images of isopara- metric hypersurfaces in the spheres as Lagrangian submanifolds embedded in complex hyperquadrics.

No abstract uploaded!

A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying Gauss-Bonnet formula. Furthermore, the hyperbolic polyhedral metric with given curvature can be obtained using a discrete Yamabe flow with surgery. In particular, each hyperbolic polyhedral metric on a closed surface with negative Euler characteristic is discrete conformal to a unique hyperbolic metric.

A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which generalizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature metric can be found using a discrete Yamabe flow with surgery.

We study the spectral geometric properties of the scalar Laplace–Beltrami operator associated to the Weil–Petersson metric $g_{WP}$ on $\mathcal{M}_{\gamma}$ , the Riemann moduli space of surfaces of genus  $\gamma>1$. This space has a singular compactification with respect to $g_{WP}$, and this metric has crossing cusp-edge singularities along a finite collection of simple normal crossing divisors. We prove first that the scalar Laplacian is essentially self-adjoint, which then implies that its spectrum is discrete.