相关论文: A simple effective method for curvatures estimatio…

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss--Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of…

Vertex scaling of piecewise linear metrics on surfaces introduced by Luo is a straightforward discretization of smooth conformal structures on surfaces. Combinatorial $\alpha$-curvature for vertex scaling of piecewise linear metrics on…

High-fidelity 3D reconstruction of common indoor scenes is crucial for VR and AR applications. 3D Gaussian splatting, a novel differentiable rendering technique, has achieved state-of-the-art novel view synthesis results with high rendering…

In this paper we study the derived equivalences between surface algebras, introduced by David-Roesler and Schiffler. Each surface algebra arises from a cut of an ideal triangulation of an unpunctured marked Riemann surface with boundary. A…

This article investigates the application of computer vision and graph-based models in solving mesh-based partial differential equations within high-performance computing environments. Focusing on structured, graded structured, and…

This work introduces the framed curvature flow, a generalization of both the curve shortening flow and the vortex filament equation. Here, the magnitude of the velocity vector is still determined by the curvature, but its direction is given…

The analysis of curves has been routinely dealt with using tools from functional data analysis. However its extension to multi-dimensional curves poses a new challenge due to its inherent geometric features that are difficult to capture…

We present a novel method for calculating interface curvature on 3D unstructured meshes from piecewise-linear interface reconstructions typically generated in the volume of fluid method. Interface curvature is a necessary quantity to…

Halos, filaments, sheets and voids in the cosmic web can be defined in terms of the eigenvalues of the smoothed shear tensor and a threshold $\lambda_{\rm th}$. Using analytic methods, we construct mean maps centered on these types of…

We study the existence of at least one conformal metric of prescribed Gaussian curvature on a closed surface $\Sigma$ admitting conical singularities of orders $\alpha_i$'s at points $p_i$'s. In particular, we are concerned with the case…

We shall introduce the singular curvature function on cuspidal edges of surfaces, which is related to the Gauss-Bonnet formula and which characterizes the shape of cuspidal edges. Moreover, it is closely related to the behavior of the…

Accurate boundary detection in high-dimensional data remains a central challenge in unsupervised learning, particularly in the presence of non-linear structures and heterogeneous densities. In this work, we introduce Mean Curvature Boundary…

We study singularities of Gauss maps of fronts and give characterizations of types of singularities of Gauss maps by geometric properties of fronts which are related to behavior of bounded principal curvatures. Moreover, we investigate…

We define a Gauss map $\gamma:M\rightarrow\mathbb{S}^{6}$ of an oriented hypersurface $M$ of the unit sphere $\mathbb{S}^{7}$ and prove that $\gamma$ is harmonic if and only if $M$ has CMC. Results on the geometry and topology of CMC…

In this note we demonstrate how the analogy between the harmonic Gauss map of a constant mean curvature surface and the harmonic conformal Gauss map of a Willmore surface can be used to obtain results on Willmore surfaces.

We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural…

Simple properties of the Gauss map characterise important classes of surfaces in $\Rq$: $R$-surfaces, the real version of plane complex curves; Lagrangean surfaces; isoclinic surfaces.

Analysis of cosmic shear is an integral part of understanding structure growth across cosmic time, which in-turn provides us with information about the nature of dark energy. Conventional methods generate \emph{shear maps} from which we can…

Let G be a n-dimensional Lie group (n>2) with a bi-invariant Riemannian metric. We prove that if a surface of constant Gaussian curvature in G can be expressed as the product of two curves, then it must be flat. In particular, we can…

This is a short essay about some fundamental results on scalar curvature and the two key methods that are used to establish them.