Related papers: Conformal Gauss Map Geometry and Application to Wi…
The Willmore Problem seeks closed surfaces in $\mathbb{S}^3\subset\mathbb{R}^4$ of a given topological type minimizing the squared-mean-curvature energy $W = \int |H_{\mathbb{R}^4}|^2 = area + \int |H_{\mathbb{S}^3}|^2$. The longstanding…
In the search for appropriate discretizations of surface theory it is crucial to preserve such fundamental properties of surfaces as their invariance with respect to transformation groups. We discuss discretizations based on M\"obius…
The conformal symmetry in the Liouville theory is analysed by using the Hamiltonian light--front formalism. The boundary conditions of dynamical variables are seen to involve an arbitrary function of time, so that the standard methods for…
In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds…
We study quantum cosmological models for certain classes of bang/crunch singularities, using the duality between expanding bubbles in AdS with a FRW interior cosmology and perturbed CFTs on de Sitter space-time. It is pointed out that…
The main goal of this paper is to reveal the geometric meaning of the maximal number of exceptional values of Gauss maps for several classes of immersed surfaces in space forms, for example, complete minimal surfaces in the Euclidean…
We consider quantum aspects of a class of generalized Gross-Neveu models, which in special cases reduce to sigma models. We show that, in the case of gauged models, an admissible gauge is $A_\mu=0$, which is a direct analogue of the…
We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner--Nirenberg-type problem of finding on the interior a metric…
We study the geometry of the gauged quiver quantum mechanics realizing $D(2,1;0)$ superconformal symmetry. These models arise as effective descriptions of multi-centered D-brane systems in type II Calabi-Yau compactifications, in the…
A classical theorem in the theory of minimal surfaces establishes a correspondence between minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$. A hyperbolic version of this correspondence is due to Bryant: null…
In this work a proposal for definition of twistors on generic curved spaces is exposed and investigated. We consider superpositions of nearly autoparallel and nearly geodesic maps (nearly conformal maps, nc-maps) of (pseudo-)Riemannian…
We investigate the notion of curvature in the context of Liouville quantum gravity (LQG) surfaces. We define the Gaussian curvature for LQG, which we conjecture is the scaling limit of discrete curvature on random planar maps. Motivated by…
We define discrete constant mean curvature (cmc) surfaces in the three-dimensional Euclidean and Lorentz spaces in terms of sphere packings with orthogonally intersecting circles. These discrete cmc surfaces can be constructed from…
Embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$ are reasonably well understood: From far away, they look like intersecting catenoids and planes, suitably desingularized. We consider the larger class of harmonic…
We characterize conformally removable sets in the plane with the aid of the recent developments in the theory of metric surfaces. We prove that a compact set in the plane is $S$-removable if and only if there exists a quasiconformal map…
We consider surfaces with parallel mean curvature vector field and finite total curvature in product spaces of type $\mathbb{M}^n(c)\times\mathbb{R}$, where $\mathbb{M}^n(c)$ is a space form, and characterize certain of these surfaces. When…
In this paper, we study the estimation of Gauss curvature for $K$-quasiconformal harmonic surface in ${\mathbb R}^3$ and present an accurate improvement of the previous result in [6, Theorem 5.2]. Let $X:M\rightarrow{\mathbb R}^3$ denote a…
In image processing tasks, spatial priors are essential for robust computations, regularization, algorithmic design and Bayesian inference. In this paper, we introduce weighted mean curvature (WMC) as a novel image prior and present an…
We provide a first principle definition of cosmological correlation functions for a large class of scalar toy models in arbitrary FRW cosmologies, in terms of novel geometries we name {\it weighted cosmological polytopes}. Each of these…
An interesting problem in classical differential geometry is to find methods to prove that two surfaces defined by different charts actually coincide up to position in space. In a previous paper we proposed a method in this direction for…